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101 |
Magnetism
|
Consider an uncharged ellipsoid subjected to a uniform external electric field. When the external electric field is oriented in any direction relative to the ellipsoid's $x, y, z$ axes, find the charge distribution $\sigma$ on its surface.
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[] |
Expression
|
{"$\\sigma$": "surface charge distribution", "$x$": "x-axis of ellipsoid", "$y$": "y-axis of ellipsoid", "$z$": "z-axis of ellipsoid", "$\\varphi$": "electric potential", "$\\xi$": "ellipsoidal coordinate", "$\\eta$": "ellipsoidal coordinate", "$\\zeta$": "ellipsoidal coordinate", "$h_1$": "metric coefficient for $\\xi$", "$h_2$": "metric coefficient for $\\eta$", "$h_3$": "metric coefficient for $\\zeta$", "$R_\\xi$": "radius for ellipsoidal coordinate $\\xi$", "$R_\\eta$": "radius for ellipsoidal coordinate $\\eta$", "$R_\\zeta$": "radius for ellipsoidal coordinate $\\zeta$", "$s$": "integration variable", "$a$": "semi-axis length of ellipsoid", "$\\nu_x$": "direction cosine for x-axis", "$\\mathscr{E}$": "external electric field", "$n^{(x)}$": "normalization factor for x-component", "$n_{ik}$": "matrix of normalizing factors", "$\\nu_i$": "direction cosine in arbitrary direction", "$\\mathfrak{C}_{k}$": "component of external electric field in direction $k$", "$\\mathfrak{C}_{x}$": "x-component of external electric field", "$\\mathfrak{C}_{y}$": "y-component of external electric field", "$\\mathfrak{C}_{z}$": "z-component of external electric field"}
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Electrostatics of Conductors
|
conducting ellipsoid
|
|
102 |
Magnetism
|
For a prolate rotational ellipsoid conductor, find the external potential $\varphi$ when its symmetry axis is perpendicular to the external field (specifically referring to the scenario described in the solution where the field is in the $z$ axis direction).
|
[] |
Expression
|
{"$\\varphi$": "external potential", "$\\varphi_0$": "initial potential", "$z$": "coordinate axis direction (z-axis)", "$a$": "semi-major axis of the ellipsoid", "$b$": "semi-minor axis of the ellipsoid", "$\\xi$": "local coordinate parameter", "$R_s$": "radius related function", "$\\mathfrak{C}$": "proportionality constant"}
|
Electrostatics of Conductors
|
conducting ellipsoid
|
|
103 |
Magnetism
|
For an oblate rotating ellipsoidal conductor, when its symmetry axis is perpendicular to the external field (specifically referring to the case described in the solution where the field is in the $x$ axis direction), find the potential $\varphi$ outside the conductor.
|
[] |
Expression
|
{"$\\varphi$": "potential outside the conductor", "$\\varphi_0$": "reference potential", "$a$": "semi-major axis of the oblate ellipsoid", "$c$": "semi-minor axis of the oblate ellipsoid", "$x$": "x axis direction", "$\\xi$": "integration variable", "$\\mathfrak{c}$": "constant related to the system"}
|
Electrostatics of Conductors
|
conducting ellipsoid
|
|
104 |
Magnetism
|
Consider a uniform electric field of magnitude \( \mathfrak{C} \) existing along the positive z-axis in the half-space \( z<0 \) (i.e., at \( z \to -\infty \), the electric field is \( \vec{E} = \mathfrak{C}\hat{k} \), corresponding to the potential \( \varphi = -\mathfrak{C}z \)). This electric field is constrained by a grounded conductive plane with a circular hole of radius \( a \) centered at the origin, \( z=0 \). Determine the expression for the potential \( \varphi \) throughout the entire space (which can be expressed in either oblate spheroidal coordinates \( \xi, \eta \) or Cartesian coordinates \( z \)).
|
[] |
Expression
|
{"$\\mathfrak{C}$": "magnitude of the electric field", "$z$": "position coordinate along the z-axis", "$\\varphi$": "electric potential", "$a$": "radius of the circular hole", "$\\pi$": "mathematical constant pi", "$\\xi$": "oblate spheroidal coordinate", "$\\eta$": "oblate spheroidal coordinate"}
|
Electrostatics of Conductors
|
conducting ellipsoid
|
|
105 |
Magnetism
|
In the same physical scenario as the previous sub-question (that is, in the half-space $z<0$, there exists a uniform electric field $\mathfrak{C}$ along the positive $z$-axis, constrained by a grounded conductive plane at $z=0$ with a circular hole of radius $a$), the expression for the electric potential $\varphi$ is known to be $\varphi=-\mathfrak{C} \frac{z}{\pi}[\arctan \frac{a}{\sqrt{\xi}}-\frac{a}{\sqrt{\xi}}]$. Try to find the expression for the surface charge density $\sigma$ on the lower side of the conductive plane ($z=0^-, \rho > a$) in terms of the cylindrical radial distance $\rho$ and hole radius $a$.
|
[] |
Expression
|
{"$z$": "coordinate along the z-axis", "$\\mathfrak{C}$": "uniform electric field along the positive z-axis", "$a$": "radius of the circular hole", "$\\varphi$": "electric potential", "$\\sigma$": "surface charge density", "$\\rho$": "cylindrical radial distance", "$\\xi$": "variable related to the radial distance"}
|
Electrostatics of Conductors
|
conducting ellipsoid
|
|
106 |
Magnetism
|
Assume a charged spherical conductor is cut in half, try to determine the mutual repulsive force between the two hemispheres.
|
[] |
Expression
|
{"$F$": "force", "$E$": "electric field", "$\\theta$": "angle", "$e$": "total charge", "$a$": "radius of the sphere"}
|
Electrostatics of Conductors
|
Force on a Conductor
|
|
107 |
Magnetism
|
A spherical conductor is cut into two halves, determine the mutual repulsive force between the two hemispheres. The conductor sphere is uncharged and is in a uniform external electric field $\mathfrak{C}$ perpendicular to the interface.
|
[] |
Expression
|
{"$\\mathfrak{C}$": "external electric field", "$E$": "electric field on the sphere surface", "$\\theta$": "angle with respect to the perpendicular interface", "$F$": "mutual repulsive force", "$a$": "radius of the sphere"}
|
Electrostatics of Conductors
|
Force on a Conductor
|
|
108 |
Magnetism
|
For waves propagating on the charged surface of a liquid conductor in a gravitational field, determine the stability conditions of this surface.
|
[] |
Expression
|
{"$\\omega$": "frequency", "$k$": "wave number", "$\\rho$": "density of the liquid", "$g$": "gravitational acceleration", "$\\sigma_{0}$": "surface charge density", "$\\alpha$": "surface tension"}
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Electrostatics of Conductors
|
Force on a Conductor
|
|
109 |
Magnetism
|
Find the stability condition for a charged spherical droplet with respect to small deformations.
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[] |
Expression
|
{"$x$": "horizontal position along the x-axis", "$z$": "vertical position along the z-axis", "$\\zeta$": "vertical displacement of the liquid surface", "$a$": "amplitude of the vertical displacement", "$k$": "wave number", "$\\omega$": "angular frequency of the wave", "$t$": "time", "$E_z$": "electric field in the z-direction", "$E$": "electric field magnitude", "$\\sigma_0$": "surface charge density", "$\\varphi$": "electric potential", "$\\varphi_1$": "small correction to the electric potential", "$F_s$": "additional negative pressure acting on the surface", "$\\rho_0$": "initial or reference liquid density", "$\\rho$": "liquid density", "$g$": "acceleration due to gravity", "$\\alpha$": "surface tension coefficient", "$\\Phi$": "velocity potential of the liquid", "$A$": "amplitude related to the velocity potential"}
|
Electrostatics of Conductors
|
Force on a Conductor
|
|
110 |
Magnetism
|
Find the stability condition (Rayleigh, 1882) of a charged spherical droplet relative to splitting into two identical smaller droplets (large deformation). Assume the original droplet has a charge of $e$ and a radius of $a$, while each smaller droplet after splitting has a charge of $e/2$ and a radius of $a/2^{1/3}$.
|
[] |
Expression
|
{"$e$": "charge of the original droplet", "$a$": "radius of the original droplet", "$\\sigma_0$": "surface charge density", "$k$": "wave number", "$\\omega$": "angular frequency of the wave", "$\\rho$": "liquid density", "$\\alpha$": "surface tension coefficient", "$g$": "acceleration due to gravity", "$E_z$": "electric field above the surface", "$E$": "electric field", "$\\varphi$": "electric potential", "$\\zeta$": "vertical displacement of the liquid surface", "$\\Phi$": "velocity potential of the liquid", "$A$": "amplitude of the velocity potential"}
|
Electrostatics of Conductors
|
Force on a Conductor
|
|
111 |
Magnetism
|
An infinitely long straight charged wire (with charge per unit length of $e$) is parallel to the interface between two media with different dielectric constants ($\varepsilon_1$ and $\varepsilon_2$ respectively), and the distance from the interface is $h$. Determine the potential $\varphi_1$ in medium 1 ($\varepsilon_1$).
|
[] |
Expression
|
{"$e$": "charge per unit length of the wire", "$\\varepsilon_1$": "dielectric constant of medium 1", "$\\varepsilon_2$": "dielectric constant of medium 2", "$h$": "distance from the interface", "$\\varphi_1$": "electric potential in medium 1", "$e'$": "image charge per unit length", "$r$": "distance from observation point to the original wire", "$r'$": "distance from observation point to the image wire"}
|
Electrostatics of Dielectrics
|
Dielectric constant
|
|
112 |
Magnetism
|
An infinitely long straight conductor (with a linear charge of $e$) is parallel to the interface between two media with different dielectric constants ($\varepsilon_1$ and $\varepsilon_2$ respectively) and at a distance $h$ from the interface. Determine the potential $\varphi_2$ within medium 2 ($\varepsilon_2$).
|
[] |
Expression
|
{"$e$": "linear charge density of the conductor", "$\\varepsilon_1$": "dielectric constant of medium 1", "$\\varepsilon_2$": "dielectric constant of medium 2", "$h$": "distance from the conductor to the interface", "$\\varphi_2$": "potential within medium 2", "$r$": "distance from the observation point to the original wire"}
|
Electrostatics of Dielectrics
|
Dielectric constant
|
|
113 |
Magnetism
|
Find the torque $K$ acting on a rotational ellipsoid in a uniform electric field $\mathfrak{C}$, with a dielectric constant of $\varepsilon$. The volume of the ellipsoid is $V$, $\alpha$ is the angle between the direction of $\mathfrak{C}$ and the symmetry axis of the ellipsoid, and $n$ is the depolarization coefficient along this axis.
|
[] |
Expression
|
{"$K$": "torque", "$\\mathfrak{C}$": "uniform electric field", "$\\varepsilon$": "dielectric constant", "$V$": "volume of the ellipsoid", "$\\alpha$": "angle between the electric field and symmetry axis", "$n$": "depolarization coefficient along the symmetry axis", "$\\mathscr{P}$": "dipole moment of the ellipsoid"}
|
Electrostatics of Dielectrics
|
Dielectric Ellipsoid
|
|
114 |
Magnetism
|
For a conductive rotating ellipsoid ($\varepsilon \rightarrow \infty$), in a uniform electric field $\mathfrak{C}$, find the torque $K$ acting on it. The volume of the ellipsoid is $V$, $\alpha$ is the angle between the direction of $\mathfrak{C}$ and the symmetry axis of the ellipsoid, and $n$ is the depolarization factor along this axis.
|
[] |
Expression
|
{"$\\varepsilon$": "permittivity", "$\\mathfrak{C}$": "uniform electric field", "$K$": "torque", "$V$": "volume of the ellipsoid", "$\\alpha$": "angle between the electric field direction and the ellipsoid's symmetry axis", "$n$": "depolarization factor along the symmetry axis", "$\\mathscr{P}$": "dipole moment of the ellipsoid"}
|
Electrostatics of Dielectrics
|
Dielectric Ellipsoid
|
|
115 |
Magnetism
|
A hollow dielectric sphere (dielectric constant $\varepsilon$, with inner and outer radii $b$ and $a$, respectively) is placed in a uniform external electric field $\mathfrak{E}$. Determine the field inside the cavity of the sphere.
|
[] |
Expression
|
{"$\\varepsilon$": "dielectric constant", "$b$": "inner radius of the sphere", "$a$": "outer radius of the sphere", "$\\mathfrak{E}$": "external electric field", "$A$": "constant related to potential in region 1", "$B$": "constant related to potential in region 3", "$C$": "constant related to potential in region 2", "$D$": "constant related to potential in region 2", "$\\boldsymbol{E}_{3}$": "electric field inside the cavity"}
|
Electrostatics of Dielectrics
|
Dielectric Ellipsoid
|
|
116 |
Magnetism
|
Determine the height $h$ by which the liquid surface inside a vertical parallel-plate capacitor rises.
|
[] |
Expression
|
{"$h$": "height of the liquid surface", "$\\widetilde{F}$": "symbol representing a force or energy term", "$\\rho$": "density of the liquid", "$g$": "acceleration due to gravity", "$\\varepsilon$": "relative permittivity of the liquid", "$E$": "electric field strength"}
|
Electrostatics of Dielectrics
|
Thermodynamic Relations for Dielectrics in an Electric Field
|
|
117 |
Magnetism
|
If the object is not in a vacuum, but in a medium with a dielectric constant of $\varepsilon^{(e)}$, find the formula of $\mathscr{F}-\mathscr{F}_{0}$.
If the answer exists in an integral, then find the integrand
|
[] |
Expression
|
{"$\\varepsilon^{(e)}$": "dielectric constant of the medium", "$\\mathscr{F}$": "current force in medium", "$\\mathscr{F}_{0}$": "force in vacuum", "$\\mathfrak{C}$": "electric displacement field source term", "$\\boldsymbol{D}$": "electric displacement vector", "$\\boldsymbol{E}$": "electric field vector"}
|
Electrostatics of Dielectrics
|
Total free energy of dielectric
|
|
118 |
Magnetism
|
Consider a capacitor composed of two conducting surfaces separated by a distance $h$, with $h$ being smaller than the dimensions of the capacitor plates. The space between the capacitor plates is filled with a material of dielectric constant $\varepsilon_{1}$. A small sphere with radius $a \ll h$ and dielectric constant $\varepsilon_{2}$ is placed inside the capacitor. Determine the change in the capacitance of the capacitor.
|
[] |
Expression
|
{"$h$": "distance between the capacitor plates", "$\\varepsilon_{1}$": "dielectric constant of the material between the capacitor plates", "$a$": "radius of the small sphere", "$\\varepsilon_{2}$": "dielectric constant of the small sphere", "$\\varphi$": "potential difference between the capacitor plates", "$\\widetilde{\\mathscr{F}}$": "free energy when the potential is constant", "$C_{0}$": "original capacitance of the capacitor", "$\\mathfrak{C}$": "electric field strength", "$\\mathscr{F}$": "free energy", "$\\mathscr{F}_{0}$": "initial free energy without the sphere", "$\\boldsymbol{D}$": "electric displacement vector", "$\\varepsilon^{(e)}$": "effective dielectric constant", "$\\boldsymbol{E}$": "electric field vector", "$\\mathbf{E}^{(i)}$": "electric field inside the sphere", "$C$": "capacitance with the sphere"}
|
Electrostatics of Dielectrics
|
Electrostriction of Isotropic Dielectrics
|
|
119 |
Magnetism
|
Try to determine the potential $\varphi$ produced by a point charge $e$ inside an anisotropic homogeneous medium (with the point charge located at the origin and the principal axes of the dielectric tensor $\varepsilon_{ik}$ aligned along the $x, y, z$ axes), and express it using the principal dielectric constants $\varepsilon^{(x)}, \varepsilon^{(y)}, \varepsilon^{(z)}$.
|
[] |
Expression
|
{"$\\varphi$": "potential", "$e$": "point charge", "$\\varepsilon_{ik}$": "dielectric tensor component", "$\\varepsilon^{(x)}$": "principal dielectric constant in the x direction", "$\\varepsilon^{(y)}$": "principal dielectric constant in the y direction", "$\\varepsilon^{(z)}$": "principal dielectric constant in the z direction", "$x$": "Cartesian coordinate along the x-axis", "$y$": "Cartesian coordinate along the y-axis", "$z$": "Cartesian coordinate along the z-axis"}
|
Electrostatics of Dielectrics
|
Dielectric properties of crystals
|
|
120 |
Magnetism
|
Determine the potential $\varphi$ generated by a point charge $e$ in an anisotropic homogeneous medium, using tensor notation that does not depend on the choice of coordinate system.
|
[] |
Expression
|
{"$\\varphi$": "potential", "$e$": "point charge", "$\\varepsilon^{(x)}$": "permittivity along the x-axis", "$\\varepsilon^{(y)}$": "permittivity along the y-axis", "$\\varepsilon^{(z)}$": "permittivity along the z-axis", "$|\\varepsilon|$": "determinant of permittivity tensor", "$\\varepsilon_{i k}$": "permittivity tensor", "$\\varepsilon_{i k}^{-1}$": "inverse permittivity tensor", "$x$": "x-coordinate", "$y$": "y-coordinate", "$z$": "z-coordinate", "$x_{i}$": "coordinate component i", "$x_{k}$": "coordinate component k"}
|
Electrostatics of Dielectrics
|
Dielectric properties of crystals
|
|
121 |
Magnetism
|
An anisotropic dielectric sphere with a radius of $a$ (principal values of the dielectric tensor are $\varepsilon^{(x)}, \varepsilon^{(y)}, \varepsilon^{(z)}$, with principal axes along the $x, y, z$ axes, respectively) is in a uniform external electric field $\boldsymbol{\mathfrak{C}} = (\mathfrak{C}_x, \mathfrak{C}_y, \mathfrak{C}_z)$ (in vacuum). Determine the $x$ component of the torque $K_x$ acting on the sphere.
|
[] |
Expression
|
{"$a$": "radius of the sphere", "$\\varepsilon^{(x)}$": "dielectric tensor principal value along the x-axis", "$\\varepsilon^{(y)}$": "dielectric tensor principal value along the y-axis", "$\\varepsilon^{(z)}$": "dielectric tensor principal value along the z-axis", "$\\mathfrak{C}_x$": "x component of the external electric field", "$\\mathfrak{C}_y$": "y component of the external electric field", "$\\mathfrak{C}_z$": "z component of the external electric field", "$K_x$": "x component of the torque acting on the sphere", "$\\mathcal{P}_y$": "y component of the total dipole moment", "$\\mathcal{P}_z$": "z component of the total dipole moment"}
|
Electrostatics of Dielectrics
|
Dielectric properties of crystals
|
|
122 |
Magnetism
|
An anisotropic dielectric sphere with a radius of $a$ (the principal values of the dielectric tensor are $\varepsilon^{(x)}, \varepsilon^{(y)}, \varepsilon^{(z)}$, with principal axes along the $x, y, z$ directions, respectively) is placed in a uniform external electric field $\boldsymbol{\mathfrak{C}} = (\mathfrak{C}_x, \mathfrak{C}_y, \mathfrak{C}_z)$ (in vacuum). Determine the $y$ component of the torque $K_y$ acting on the sphere.
|
[] |
Expression
|
{"$a$": "radius of the sphere", "$\\varepsilon^{(x)}$": "dielectric tensor principal value along x-axis", "$\\varepsilon^{(y)}$": "dielectric tensor principal value along y-axis", "$\\varepsilon^{(z)}$": "dielectric tensor principal value along z-axis", "$\\mathfrak{C}_x$": "x-component of the external electric field", "$\\mathfrak{C}_y$": "y-component of the external electric field", "$\\mathfrak{C}_z$": "z-component of the external electric field", "$E_i$": "electric field intensity component inside the sphere", "$P_i$": "polarization intensity component of the sphere", "$\\mathcal{P}_i$": "total dipole moment component of the sphere", "$K_y$": "y-component of the torque acting on the sphere"}
|
Electrostatics of Dielectrics
|
Dielectric properties of crystals
|
|
123 |
Magnetism
|
An anisotropic dielectric sphere with a radius of $a$ (principal values of the dielectric tensor are $\varepsilon^{(x)}, \varepsilon^{(y)}, \varepsilon^{(z)}$, with principal axes along the $x, y, z$ axes respectively) is placed in a uniform external electric field $\boldsymbol{\mathfrak{C}} = (\mathfrak{C}_x, \mathfrak{C}_y, \mathfrak{C}_z)$ in vacuum. Determine the $z$ component $K_z$ of the torque acting on the sphere.
|
[] |
Expression
|
{"$a$": "radius of the sphere", "$\\varepsilon^{(x)}$": "principal value of the dielectric tensor along the x-axis", "$\\varepsilon^{(y)}$": "principal value of the dielectric tensor along the y-axis", "$\\varepsilon^{(z)}$": "principal value of the dielectric tensor along the z-axis", "$x$": "x-axis", "$y$": "y-axis", "$z$": "z-axis", "$\\mathfrak{C}$": "external electric field vector", "$\\mathfrak{C}_x$": "x component of the external electric field", "$\\mathfrak{C}_y$": "y component of the external electric field", "$\\mathfrak{C}_z$": "z component of the external electric field", "$E_i$": "component of the electric field intensity inside the sphere", "$i$": "index representing x, y, or z axis", "$P_i$": "component of the polarization intensity", "$\\mathcal{P}_i$": "component of the total dipole moment", "$\\boldsymbol{K}$": "torque vector acting on the sphere", "$K_z$": "z component of the torque on the sphere"}
|
Electrostatics of Dielectrics
|
Dielectric properties of crystals
|
|
124 |
Magnetism
|
Consider a dielectric sphere (with radius $a$) placed in a uniform external electric field $\mathfrak{C}$, sliced into two halves by a plane perpendicular to the field direction. Find the attraction force between the two hemispheres.
|
[] |
Expression
|
{"$a$": "radius of the dielectric sphere", "$\\mathfrak{C}$": "external electric field", "$\\varepsilon$": "dielectric constant of the sphere", "$\\mathbf{E}$": "electric field strength near the sphere surface", "$\\boldsymbol{E}^{(i)}$": "electric field strength inside the sphere", "$\\boldsymbol{D}^{(i)}$": "electric displacement within the slit", "$E_{r}$": "radial component of the electric field on the outer surface", "$E_{\\theta}$": "tangential component of the electric field on the outer surface", "$\\theta$": "angle between the radial vector and the electric field direction", "$F$": "attraction force between the two hemispheres"}
|
Electrostatics of Dielectrics
|
Electricity in solids
|
|
125 |
Magnetism
|
Try to determine the shape change of a dielectric sphere in a uniform external electric field.
|
[] |
Expression
|
{"$\\mathscr{P}$": "polarization vector", "$\\mathfrak{C}$": "external electric field strength", "$V$": "volume of the sphere", "$\\varepsilon^{(x)}$": "dielectric constant along the x axis", "$n$": "effective medium parameter", "$\\varepsilon_{ik}$": "permittivity tensor", "$\\varepsilon_{0}$": "permittivity of free space", "$\\delta_{ik}$": "Kronecker delta", "$a_{1}$": "elastic constant related to strain", "$u_{ik}$": "strain tensor component", "$a_{2}$": "elastic constant related to strain", "$u_{ll}$": "trace of strain tensor", "$u_{x x}$": "strain component in the x direction", "$u_{y y}$": "strain component in the y direction", "$a$": "semi-major axis of the ellipsoid", "$b$": "semi-minor axis of the ellipsoid", "$R$": "radius of the original sphere", "$\\mu$": "shear modulus"}
|
Electrostatics of Dielectrics
|
Electricity in solids
|
|
126 |
Magnetism
|
Determine the Young's modulus of a non-pyroelectric piezoelectric material parallel plate thin slab under the following conditions (the ratio of tensile stress to relative tensile strain): The slab is under tensile stress between the plates of a short-circuited capacitor.
|
[] |
Expression
|
{"$\\boldsymbol{E}$": "electric field", "$\\sigma_{ik}$": "stress tensor component", "$\\sigma_{zz}$": "tensile stress (z-axis)", "$u_{zz}$": "relative tensile strain (z-axis)", "$\\mu_{zzzz}$": "material coefficient related to Young's modulus (zzzz component)", "$E$": "Young's modulus", "$E_{x}$": "electric field component along x-axis", "$E_{y}$": "electric field component along y-axis", "$E_{z}$": "electric field component along z-axis", "$D_{z}$": "electric displacement field component along z-axis", "$\\varepsilon_{zz}$": "permittivity component (zz)", "$\\gamma_{z,zz}$": "piezoelectric coupling coefficient (z, zz component)", "$\\sigma_{xx}$": "tensile stress (x-axis)", "$u_{xx}$": "relative tensile strain (x-axis)", "$\\mu_{xxxx}$": "material coefficient related to Young's modulus (xxxx component)", "$\\gamma_{z,xx}$": "piezoelectric coupling coefficient (z, xx component)"}
|
Electrostatics of Conductors
|
Piezoelectric
|
|
127 |
Magnetism
|
Try to derive the expression or equation for the velocity of sound within a piezoelectric medium.
|
[] |
Expression
|
{"$u_{i k}$": "displacement component", "$\\sigma_{i k}$": "stress tensor component", "$\\widetilde{F}$": "modified free energy", "$F_{0}$": "initial free energy", "$\\lambda_{i k l m}$": "elasticity tensor component", "$\\varepsilon_{i k}$": "dielectric tensor component", "$E_{i}$": "electric field component", "$D_{i 0}$": "electric displacement component", "$\\beta_{i, k l}$": "piezoelectric tensor component", "$\\rho$": "density of the medium", "$\\boldsymbol{u}$": "displacement vector", "$\\nu_{i k}$": "strain component", "$\\varphi$": "electric potential", "$\\omega$": "angular frequency", "$k_{k}$": "wave vector component", "$\\boldsymbol{k}$": "wave vector", "$\\delta_{i k}$": "Kronecker delta", "$l$": "subscript index indicating a specific component", "$m$": "another subscript index indicating a specific component", "$p$": "another subscript index indicating a specific component", "$r$": "another subscript index indicating a specific component", "$s$": "another subscript index indicating a specific component", "$q$": "another subscript index indicating a specific component"}
|
Electrostatics of Dielectrics
|
Piezoelectric
|
|
128 |
Magnetism
|
The piezoelectric crystals belonging to the $C_{6 v}$ crystal class are constrained by the surface plane ( $xz$ plane) of the symmetry axis ($z$ axis). Try to determine the velocity of surface waves propagating perpendicular to the symmetry axis (along the $x$ axis) which undergo displacement $u_{z}$ and potential $\varphi$ oscillations
|
[] |
Expression
|
{"$C_{6 v}$": "crystal class", "$x$": "coordinate axis perpendicular to the symmetry axis", "$z$": "symmetry axis", "$u_{z}$": "displacement oscillation", "$\\varphi$": "potential oscillation", "$\\rho$": "density", "$\\sigma_{i k}$": "stress tensor component", "$\\lambda_{i k l m}$": "elastic stiffness tensor component", "$\\beta_{l, i k}$": "piezoelectric tensor component", "$E_{l}$": "electric field component", "$\\varepsilon_{i k}$": "permittivity tensor component", "$\\sigma_{zx}$": "stress component in the zx plane", "$\\sigma_{zy}$": "stress component in the zy plane", "$D_{x}$": "electric displacement in the x direction", "$D_{y}$": "electric displacement in the y direction", "$E_{x}$": "electric field component in the x direction", "$E_{y}$": "electric field component in the y direction", "$u_{zx}$": "strain component in the zx plane", "$u_{zy}$": "strain component in the zy plane", "$\\beta$": "piezoelectric coupling constant", "$\\lambda$": "effective stiffness constant", "$\\varepsilon$": "permittivity", "$D_{z}$": "pyroelectric displacement", "$D_{0}$": "constant pyroelectric displacement", "$t$": "time", "$\\Delta$": "Laplacian operator", "$\\bar{\\lambda}$": "modified stiffness constant", "$\\psi$": "auxiliary function", "$\\varphi^{(i)}$": "internal potential", "$\\varphi^{(e)}$": "external potential", "$\\varkappa$": "decay constant", "$k$": "wave number", "$\\omega$": "angular frequency", "$\\Lambda$": "dimensionless constant"}
|
Electrostatics of Dielectrics
|
Piezoelectric
|
|
129 |
Magnetism
|
Given the second-order tensor $\sigma_{ik}$, with its symmetric part $s_{ik}$ and its antisymmetric part formed by the axial vector $\boldsymbol{a}$ (specific definitions are found in the symbols table), express the determinant $|\sigma|$ of the tensor $\sigma_{ik}$ in terms of the components of $s_{ik}$ and $\boldsymbol{a}$.
|
[] |
Expression
|
{"$\\sigma_{ik}$": "second-order tensor", "$s_{ik}$": "symmetric part of the tensor", "$\\boldsymbol{a}$": "axial vector", "$s_{xx}$": "component of the symmetric tensor along x", "$s_{yy}$": "component of the symmetric tensor along y", "$s_{zz}$": "component of the symmetric tensor along z", "$a_x$": "component of the axial vector along x", "$a_y$": "component of the axial vector along y", "$a_z$": "component of the axial vector along z", "$s_{i k}$": "general component of the symmetric part of the tensor", "$a_{i}$": "component of the axial vector", "$a_{k}$": "component of the axial vector"}
|
Constant current
|
Hall Effect
|
|
130 |
Magnetism
|
Given a second-order tensor $\sigma_{ik}$, its symmetric part is $s_{ik}$, and the antisymmetric part is formed by the axial vector $\boldsymbol{a}$. Given its determinant as $|\sigma|=|s|+s_{i k} a_{i} a_{k}$. Try to express the axial vector $b_i$ of the antisymmetric part of its inverse tensor $\sigma_{ik}^{-1}$ using the components of $s_{i k}$ and $\boldsymbol{a}$ (i.e., $\sigma_{ik}^{-1} = \rho_{ik} + \epsilon_{ikl}b_l$, where $\epsilon_{ikl}b_l$ is the antisymmetric part).
|
[] |
Expression
|
{"$\\sigma_{ik}$": "second-order tensor", "$s_{ik}$": "symmetric part of the tensor", "$\\boldsymbol{a}$": "axial vector forming the antisymmetric part", "$|\\sigma|$": "determinant of the tensor", "$b_i$": "axial vector of the antisymmetric part of the inverse tensor", "$\\rho_{ik}$": "symmetric part of inverse tensor", "$\\epsilon_{ikl}$": "Levi-Civita symbol", "$a_i$": "component of the axial vector", "$a_k$": "component of the axial vector", "$b_l$": "component of the axial vector forming the antisymmetric part", "$\\sigma_{ik}^{-1}$": "inverse tensor"}
|
Constant current
|
Hall Effect
|
|
131 |
Magnetism
|
Assuming two parallel planar plates (made of the same metal $A$) are immersed in an electrolyte solution $AX$. In the case of a very small current $j$, derive the expression for the effective resistivity of the solution $\frac{\mathscr{E}}{l j}$.
|
[] |
Expression
|
{"$A$": "metal of the plates", "$j$": "current density", "$l$": "distance between the plates", "$c_1$": "surface concentration at one plate", "$c_2$": "surface concentration at the other plate", "$\\rho$": "resistivity", "$D$": "diffusion coefficient", "$\\beta$": "constant related to the system", "$m$": "mass of the ion", "$e$": "elementary charge", "$\\mathscr{E}$": "potential difference between the two plates", "$\\sigma$": "conductivity", "$\\zeta$": "electric potential related to concentration"}
|
Constant current
|
Diffusive electricity phenomenon
|
|
132 |
Magnetism
|
Consider a circular line current with a radius $a$. Try to find the radial component $B_r$ of the magnetic field in cylindrical coordinates.
|
[] |
Expression
|
{"$a$": "radius of the circular line current", "$B_r$": "radial component of the magnetic field", "$r$": "radial distance in cylindrical coordinates", "$\\varphi$": "angular coordinate in cylindrical coordinates", "$z$": "vertical coordinate in cylindrical coordinates", "$A_{\\varphi}$": "azimuthal component of the vector potential", "$J$": "current density", "$c$": "speed of light", "$R$": "distance from line element to observation point", "$\\theta$": "angle variable in elliptic integral transformation", "$k$": "parameter in elliptic integrals", "$K$": "complete elliptic integral of the first kind", "$E$": "complete elliptic integral of the second kind"}
|
Static Magnetic Field
|
Magnetic field of constant current
|
|
133 |
Magnetism
|
Consider a circular line current with a radius of $a$. Try to find the axial component $B_z$ of the magnetic field it produces in cylindrical coordinates.
|
[] |
Expression
|
{"$a$": "radius of the circular line current", "$B_z$": "axial component of the magnetic induction", "$r$": "radial distance in the cylindrical coordinate system", "$\\varphi$": "angular coordinate in the cylindrical coordinate system", "$z$": "axial coordinate in the cylindrical coordinate system", "$A_{\\varphi}$": "azimuthal component of the vector potential", "$J$": "current density", "$c$": "speed of light in vacuum", "$R$": "distance from element of current loop to point of interest", "$\\theta$": "variable introduced to simplify integral", "$k$": "parameter for elliptic integrals", "$K$": "complete elliptic integral of the first kind", "$E$": "complete elliptic integral of the second kind"}
|
Static Magnetic Field
|
Magnetic field of constant current
|
|
134 |
Magnetism
|
Try to find the 'internal' part of the self-inductance $L_i$ of a closed thin wire with a circular cross-section.
\footnotetext{
(1) The assertion in the main text that the self-inductance does not depend on the current distribution actually applies not only to the approximation (34.1), but also to subsequent approximations that do not contain large logarithmic terms (this corresponds to considering the coefficient in front of $l / a$ in the argument of the logarithm); see the exercises in this section.
(2) In exercises $1-6$, it is assumed that the susceptibility of the medium is $\mu_{e}=1$.
}
|
[] |
Expression
|
{"$L_i$": "internal part of the self-inductance", "$r$": "distance to the wire axis", "$a$": "radius of the wire", "$H$": "magnetic field", "$J$": "current density", "$c$": "speed of light in vacuum (used in calculations with units involving electromagnetism)", "$l$": "length of the closed wire", "$\\mu_i$": "permeability inside the wire", "$\\mu_{e}$": "susceptibility of the medium"}
|
Static Magnetic Field
|
Self-inductance of a wire conductor
|
|
135 |
Magnetism
|
Try to determine the self-inductance of a thin circular ring (radius $b$) made from a wire with a circular cross-section (radius $a$).
|
[] |
Expression
|
{"$b$": "radius of the thin circular ring", "$a$": "radius of the wire's circular cross-section", "$L_e$": "self-inductance associated with external magnetic field", "$L_{i}$": "self-inductance associated with internal magnetic field", "$L$": "total self-inductance", "$\\varphi$": "central angle subtended by the chord on the ring", "$\\varphi_{0}$": "initial central angle for integration", "$\\mu_{i}$": "relative permeability of the material"}
|
Static Magnetic Field
|
Self-inductance of a wire conductor
|
|
136 |
Magnetism
|
A current flows through a wire loop $(\mu_{i}=1)$; try to determine the elongation of the loop under the magnetic field generated by this current.
|
[] |
Expression
|
{"$\\mu_{i}$": "permeability (set to 1 in this context)", "$J$": "current density", "$L$": "inductance", "$q$": "generalized coordinate", "$a$": "radius of the wire", "$b$": "radius parameter for the loop", "$\\sigma_{\\|}$": "stress parallel to the wire axis", "$\\sigma_{\\perp}$": "stress perpendicular to the wire axis", "$c$": "speed of light", "$E$": "Young's modulus", "$\\sigma$": "Poisson's ratio"}
|
Static Magnetic Field
|
Self-inductance of a wire conductor
|
|
137 |
Magnetism
|
Seek the first-order correction value of the cylindrical helical tube's self-inductance, due to the field distortion near both ends of the cylindrical helical tube when $l / h$ (with $\mu_{e}=1$).
|
[] |
Expression
|
{"$l$": "characteristic length related to the cylindrical helical tube", "$h$": "length of the cylindrical helical tube", "$\\mu_{e}$": "magnetic permeability, given as 1 in this problem", "$L$": "self-inductance of the cylindrical helical tube", "$J$": "surface current density", "$\\boldsymbol{g}_{1}$": "tangential vector on the helical surface", "$\\boldsymbol{g}_{2}$": "tangential vector on the helical surface", "$R$": "distance function in the integral", "$g$": "surface current density", "$n$": "number of turns per unit length of the helix", "$b$": "radius of the cylindrical helical tube", "$\\varphi$": "angle between diametrical planes", "$z_{1}$": "position variable along the helical tube", "$z_{2}$": "position variable along the helical tube", "$\\zeta$": "difference $z_{2} - z_{1}$"}
|
Static Magnetic Field
|
Self-inductance of a wire conductor
|
|
138 |
Magnetism
|
If a planar circuit is placed on the surface of a semi-infinite medium with a permeability of $\mu_{e}$, find the factor by which the self-inductance of the planar circuit changes. We neglect the internal part of the conductor's self-inductance.
|
[] |
Expression
|
{"$\\mu_{e}$": "permeability of the medium", "$\\boldsymbol{H}_{0}$": "magnetic field due to the current in absence of medium", "$\\boldsymbol{H}$": "magnetic field in the vacuum half-space", "$\\boldsymbol{B}$": "magnetic field in the medium", "$B_{n}$": "normal component of the magnetic field", "$\\boldsymbol{H}_{t}$": "tangential component of the magnetic field"}
|
Static Magnetic Field
|
Self-inductance of a wire conductor
|
|
139 |
Magnetism
|
Given a straight wire carrying a current $J$ parallel to an infinitely long cylindrical conductor with a radius $a$ (permeability $\mu$) at a distance $l$ from the axis of the cylinder, determine the force on the straight wire.
|
[] |
Expression
|
{"$J$": "current", "$a$": "radius of the cylindrical conductor", "$\\mu$": "permeability of the cylindrical conductor", "$l$": "distance from the axis of the cylinder", "$J^{\\prime}$": "symbolic current related to $J$ and $\\mu$", "$J^{\\prime \\prime}$": "another symbolic current related to $J$ and $\\mu$", "$b$": "another distance parameter relevant to the configuration"}
|
Static Magnetic Field
|
Forces in a magnetic field
|
|
140 |
Magnetism
|
Try to determine the average magnetization intensity of a polycrystal in a strong magnetic field $(H \gg 4 \pi M)$, where the microcrystals have uniaxial symmetry.
|
[] |
Expression
|
{"$H$": "magnetic field intensity", "$M$": "magnetization intensity", "$\\theta$": "angle between easy magnetization direction and magnetization vector", "$\\psi$": "angle between easy magnetization direction and magnetic field vector", "$\\vartheta$": "small angle close to difference between $\\theta$ and $\\psi$", "$\\beta$": "parameter related to the material's magnetic properties", "$\\overline{M}$": "average magnetization intensity"}
|
Ferromagnetism and Antiferromagnetism
|
Ferromagnetism and Antiferromagnetism
|
|
141 |
Magnetism
|
For cubic symmetric microcrystals, try to determine the average magnetization intensity of a polycrystal in a strong magnetic field $(H \gg 4 \pi M)$.
|
[] |
Expression
|
{"$M_x$": "x-component of magnetization", "$M_y$": "y-component of magnetization", "$M_z$": "z-component of magnetization", "$H_x$": "x-component of magnetic field", "$H_y$": "y-component of magnetic field", "$H_z$": "z-component of magnetic field", "$\\lambda$": "Lagrange multiplier", "$M$": "magnitude of magnetization", "$H$": "magnitude of magnetic field", "$\\beta$": "coefficients related to magnetization properties", "$\\vartheta$": "angle between magnetization and magnetic field", "$\\bar{M}$": "average magnetization"}
|
Ferromagnetism and Antiferromagnetism
|
Ferromagnetism and Antiferromagnetism
|
|
142 |
Magnetism
|
Try to find out the relative elongation of a ferromagnetic cubic crystal depending on the magnetization direction $\boldsymbol{m}$ and the measurement direction $\boldsymbol{n}$.
|
[] |
Expression
|
{"$\\boldsymbol{m}$": "magnetization direction", "$\\boldsymbol{n}$": "measurement direction", "$u_{i k}$": "strain tensor component", "$a_{1}$": "coefficient of magnetization direction (x-component)", "$a_{2}$": "coefficient of magnetization direction (mixed components)", "$m_x$": "x-component of magnetization direction", "$m_y$": "y-component of magnetization direction", "$m_z$": "z-component of magnetization direction", "$n_x$": "x-component of measurement direction", "$n_y$": "y-component of measurement direction", "$n_z$": "z-component of measurement direction", "$\\nu_{xx}$": "normal stress component (x-axis)", "$\\nu_{xy}$": "shear stress component in xy-plane"}
|
Ferromagnetism and Antiferromagnetism
|
Magnetostriction of ferromagnets
|
|
143 |
Magnetism
|
The easy magnetization axes of a cubic ferromagnet align along the three edges of the cube (specifically the $x, y, z$ axes). The magnetic domains are magnetized parallel or antiparallel to the $z$ axis, and the domain walls are distributed parallel to the (100) plane. Determine the surface tension of the domain wall in this case.
|
[] |
Expression
|
{"$x$": "x-axis coordinate", "$y$": "y-axis coordinate", "$z$": "z-axis coordinate", "$M$": "magnetization vector", "$\\alpha$": "parameter related to inhomogeneous anisotropy", "$K$": "anisotropy constant", "$\\beta$": "parameter related to anisotropy energy", "$m_{x}$": "component of magnetization in the x-direction", "$m_{y}$": "component of magnetization in the y-direction", "$m_{z}$": "component of magnetization in the z-direction", "$\\theta$": "angle between magnetization vector and the z-axis", "$\\Delta_{(100)}$": "surface tension of the domain wall parallel to the (100) plane"}
|
Ferromagnetism and Antiferromagnetism
|
Surface tension of domain walls
|
|
144 |
Magnetism
|
The easy magnetization axes of a cubic ferromagnet are along the three edges of the cube (namely the $x, y, z$ axes). The magnetic domains are magnetized parallel or antiparallel to the $z$-axis, and the domain walls are distributed parallel to the (110) plane. Determine the surface tension of the domain walls in this scenario.
|
[] |
Expression
|
{"$x$": "x-axis of the cubic ferromagnet", "$y$": "y-axis of the cubic ferromagnet", "$z$": "z-axis of the cubic ferromagnet", "$\\boldsymbol{M}$": "magnetization vector", "$M$": "magnitude of the magnetization vector", "$\\theta$": "angle between magnetization vector and z-axis", "$\\nu_{\\mathrm{non}-\\mathrm{u}}$": "non-uniform anisotropy energy", "$A$": "constant involving magnetic parameters", "$B$": "constant related to magnetic anisotropy", "$\\beta$": "anisotropy constant", "$\\alpha$": "constant related to surface tension", "$\\xi$": "coordinate perpendicular to the domain wall plane", "$\\Delta$": "surface tension of the domain wall", "$\\Delta_{(110)}$": "surface tension of the domain wall in the (110) plane"}
|
Ferromagnetism and Antiferromagnetism
|
Surface tension of domain walls
|
|
145 |
Magnetism
|
If the transition between magnetic domains is not achieved through the rotation of $M$ but by changing the magnitude of $M$ (i.e., when $M$ changes sign after passing through zero), determine the surface tension of the domain wall in a uniaxial crystal. The free energy's dependence on $M$ (at $\boldsymbol{H}=0$) takes the expanded form corresponding to the situation near the Curie point as given by Equation \begin{align*}
\tilde{\Phi} = \Phi_0 + AM^2 + BM^4 - MH - \frac{H^2}{8\pi},
\end{align*}.
|
[] |
Expression
|
{"$M$": "magnetization", "$H$": "magnetic field", "$\\tilde{\\Phi}$": "free energy density", "$\\Phi_0$": "initial free energy density", "$A$": "coefficient of quadratic term in free energy expansion", "$B$": "coefficient of quartic term in free energy expansion", "$\\alpha_{1}$": "coefficient related to inhomogeneity in magnetization", "$M_{z}$": "component of magnetization along the z-axis", "$M_{0}$": "equilibrium value of magnetization within the domain", "$m$": "normalized magnetization vector", "$T_c$": "Curie temperature", "$x$": "axis perpendicular to the domain wall plane", "$\\Delta$": "surface tension of the domain wall", "$T$": "temperature", "$\\beta$": "dimensionless parameter related to the energy"}
|
Ferromagnetism and Antiferromagnetism
|
Surface tension of domain walls
|
|
146 |
Magnetism
|
The parallel plane magnetic domains extend perpendicularly to the surface of the ferromagnetic material without changing the direction of magnetization . Try to derive and present the exact mathematical expression for the magnetic field energy per unit surface area near the surface of the ferromagnet (expressed in terms of $\zeta(3)$).
|
[] |
Expression
|
{"$\\zeta(3)$": "Apéry's constant, which is the value of the Riemann zeta function at 3", "$\\sigma$": "surface charge density", "$M$": "magnetization or magnetic polarization", "$z$": "coordinate perpendicular to the surface plane", "$x$": "coordinate parallel to the surface plane", "$a$": "width of the magnetic domain", "$c_{n}$": "Fourier series coefficient", "$b_{n}$": "coefficient for series solution of potential field"}
|
Ferromagnetism and Antiferromagnetism
|
Magnetic domain structure of ferromagnets
|
|
147 |
Superconductivity
|
Find the magnetic moment ${ }^{(1)}$ of a superconducting disk perpendicular to the external magnetic field.
|
[] |
Expression
|
{"$c$": "radius of the disk approaching zero", "$a$": "radius of the spheroid", "$\\varepsilon^{(i)}$": "permittivity of the dielectric (internal)", "$n^{(x)}$": "demagnetizing factor along the x-axis", "$\\mathscr{M}$": "magnetic moment", "$\\mathfrak{H}$": "external magnetic field"}
|
Superconductivity
|
Superconducting current
|
|
148 |
Superconductivity
|
Seek the heat capacity of a superconducting ellipsoid in the intermediate state.
|
[] |
Expression
|
{"$\\tilde{\\phi}_t$": "thermodynamic potential", "$V$": "volume", "$H_{\\text{cr}}$": "critical magnetic field", "$n$": "demagnetization factor", "$\\mathfrak{H}$": "external magnetic field", "$T$": "temperature", "$\\mathscr{C}_{t}$": "total heat capacity", "$\\mathscr{C}_{s}$": "heat capacity in superconducting state", "$H_{\\mathrm{cr}}^{\\prime}$": "derivative of critical magnetic field with respect to temperature", "$H_{\\mathrm{cr}}^{\\prime \\prime}$": "second derivative of critical magnetic field with respect to temperature"}
|
Superconductivity
|
Intermediate structure
|
|
149 |
Magnetism
|
An isotropic conducting sphere with radius $a$ is in a uniform periodic external magnetic field. Determine the expression for its magnetic polarizability $\alpha$.
|
[] |
Expression
|
{"$a$": "radius of the sphere", "$\\alpha$": "magnetic polarizability", "$k$": "wave number", "$\\mathfrak{H}$": "external magnetic field strength", "$\\beta$": "constant related to axial vector", "$V$": "volume of the sphere, expressed as $\\frac{4 \\pi a^3}{3}$", "$r$": "radial distance from the center of the sphere", "$\\boldsymbol{H}^{(i)}$": "magnetic field inside the sphere", "$\\boldsymbol{H}^{(e)}$": "external magnetic field", "$\\boldsymbol{n}$": "unit vector in the radial direction", "$\\boldsymbol{A}$": "vector potential", "$f$": "spherically symmetric solution function", "$\\varphi$": "scalar potential function"}
|
Quasi-static electromagnetic field
|
Depth of magnetic field penetration into a conductor
|
|
150 |
Magnetism
|
An isotropic conducting sphere with a radius of $a$ is in a uniform periodic external magnetic field, with its magnetic susceptibility given by $\alpha = \alpha^{\prime} + \mathrm{i} \alpha^{\prime \prime}$. Determine the expression for the real part of its magnetic susceptibility $\alpha^{\prime}$.
|
[] |
Expression
|
{"$a$": "radius of the sphere", "$\\alpha$": "magnetic susceptibility", "$\\alpha^{\\prime}$": "real part of the magnetic susceptibility", "$\\alpha^{\\prime \\prime}$": "imaginary part of the magnetic susceptibility", "$k$": "wave number related to the sphere", "$\\delta$": "skin depth"}
|
Quasi-static electromagnetic field
|
Depth of magnetic field penetration into a conductor
|
|
151 |
Magnetism
|
An isotropic conductive sphere with a radius of $a$ is in a uniform periodic external magnetic field. Its magnetic susceptibility is $\alpha = \alpha^{\prime} + \mathrm{i} \alpha^{\prime \prime}$. Determine the expression for the imaginary part of its magnetic susceptibility $\alpha^{\prime \prime}$.
|
[] |
Expression
|
{"$a$": "radius of the sphere", "$\\alpha$": "magnetic susceptibility", "$\\alpha^{\\prime}$": "real part of the magnetic susceptibility", "$\\alpha^{\\prime \\prime}$": "imaginary part of the magnetic susceptibility", "$k$": "wave number related to skin depth", "$\\delta$": "skin depth"}
|
Quasi-static electromagnetic field
|
Depth of magnetic field penetration into a conductor
|
|
152 |
Magnetism
|
Find the smallest value of the attenuation coefficient for the magnetic field inside a conducting sphere.
|
[] |
Expression
|
{"$H$": "magnetic field magnitude", "$H_{r}$": "radial component of the magnetic field", "$\\gamma$": "attenuation coefficient", "$k$": "real scalar constant relating to wave number", "$r$": "radial distance from center of the sphere", "$a$": "radius of the sphere", "$\\sigma$": "conductivity of the sphere", "$c$": "speed of light"}
|
Quasi-static electromagnetic field
|
Depth of magnetic field penetration into a conductor
|
|
153 |
Magnetism
|
Two inductively coupled circuits respectively contain self-inductances $L_{1}$ and $L_{2}$ and capacitances $C_{1}$ and $C_{2}$. Determine the intrinsic frequencies of electric oscillations within these coupled circuits (we neglect resistances $R_{1}$ and $R_{2}$).
|
[] |
Expression
|
{"$L_{1}$": "self-inductance of the first circuit", "$L_{2}$": "self-inductance of the second circuit", "$L_{12}$": "mutual inductance between the two circuits", "$C_{1}$": "capacitance of the first circuit", "$C_{2}$": "capacitance of the second circuit", "$R_{1}$": "resistance of the first circuit", "$R_{2}$": "resistance of the second circuit", "$\\omega_{1}$": "intrinsic frequency of the first circuit", "$\\omega_{2}$": "intrinsic frequency of the second circuit", "$c$": "speed of light (used in calculation context)"}
|
Quasi-static electromagnetic field
|
Capacitors in a quasi-constant current loop
|
|
154 |
Magnetism
|
A uniformly magnetized sphere rotates uniformly around an axis parallel to the magnetization direction. Determine the unipolar induced electromotive force between one pole of the uniformly magnetized sphere and the equator.
|
[] |
Expression
|
{"$\\mathbf{B}$": "magnetic field vector", "$\\mathbf{v}$": "velocity vector", "$\\mathbf{H}$": "magnetic field strength vector", "$\\sigma$": "electrical conductivity", "$c$": "speed of light", "$\\mathcal{E}$": "electromotive force (EMF) along a path", "$\\mathbf{r}$": "position vector", "$\\boldsymbol{\\Omega}$": "angular velocity vector", "$d\\mathbf{l}$": "infinitesimal line element", "$B_{0}$": "magnetic induction inside the sphere", "$a$": "radius of the sphere", "$H$": "magnetic field strength", "$M$": "magnetization", "$\\mathscr{M}$": "total magnetic moment of the sphere", "$\\mathscr{E}$": "induced electromotive force"}
|
Quasi-static electromagnetic field
|
Movement of a conductor in a magnetic field
|
|
155 |
Superconductivity
|
Determine the current generated inside the superconducting ring when its uniform rotation stops.
|
[] |
Expression
|
{"$J$": "current density", "$m$": "mass", "$c$": "speed of light", "$e$": "elementary charge", "$L$": "self-inductance", "$\\Phi_e$": "external magnetic flux", "$\\Phi_0$": "constant reference magnetic flux", "$\\Omega$": "angular velocity", "$b$": "radius of the superconducting ring", "$a$": "radius of the wire cross-section"}
|
Quasi-static electromagnetic field
|
The excitation of current by acceleration
|
|
156 |
Magnetism
|
Try to determine the absorption coefficient of Alfven waves in an incompressible fluid (assuming this coefficient is very small).
|
[] |
Expression
|
{"$\\gamma$": "absorption coefficient", "$\\bar{Q}$": "average energy dissipated per unit volume per unit time", "$\\bar{q}$": "average energy flux density of the wave", "$\\rho$": "density", "$T$": "temperature", "$s$": "entropy per unit mass", "$\\mathbf{v}$": "fluid velocity vector", "$\\sigma'$": "stress tensor", "$\\kappa$": "thermal conductivity", "$c$": "speed of light", "$\\sigma$": "conductivity", "$\\mathbf{H}$": "magnetic field vector", "$Q$": "dissipation", "$\\eta$": "dynamic viscosity", "$\\boldsymbol{v}$": "wave-induced velocity", "$\\boldsymbol{h}$": "wave-induced magnetic field", "$q_{x}$": "energy flux density in x-direction", "$H_{x}$": "magnetic field component in x-direction", "$\\nu$": "wave frequency", "$u_{A}$": "Alfven speed", "$\\omega$": "angular frequency of the wave"}
|
Magnetohydrodynamics
|
Magnetohydrodynamics waves
|
|
157 |
Magnetism
|
Try to find the law of rotational discontinuity expanding with time.
|
[] |
Expression
|
{"$x$": "spatial coordinate", "$v_{x}$": "velocity component in the x-direction", "$H_{x}$": "magnetic field component in the x-direction", "$\\mathbf{v}_1$": "velocity vector on one side of the discontinuity", "$\\mathbf{H}_1$": "magnetic field vector on one side of the discontinuity", "$\\mathbf{v}_2$": "velocity vector on the other side of the discontinuity", "$\\mathbf{H}_2$": "magnetic field vector on the other side of the discontinuity", "$u_{x}$": "velocity component u in the x-direction", "$\\boldsymbol{u}$": "some velocity field", "$\\mathbf{u}'$": "perturbation of the velocity field u", "$\\mathbf{v}'$": "perturbation of the velocity field v", "$\\rho$": "density", "$P$": "pressure", "$P'$": "perturbation of the pressure", "$\\mathbf{H}$": "magnetic field vector", "$c$": "speed of light", "$\\sigma$": "conductivity", "$\\nu$": "kinematic viscosity", "$\\boldsymbol{u}_{t}$": "transverse component of the velocity field u", "$\\boldsymbol{v}_{t}$": "transverse component of the velocity field v", "$t$": "time", "$\\delta$": "width of the discontinuity"}
|
Magnetohydrodynamics
|
Tangential and rotational discontinuities
|
|
158 |
Magnetism
|
A dielectric sphere in vacuum rotates in a constant magnetic field $\mathfrak{H}$, determine the electric field produced around the sphere.
|
[] |
Expression
|
{"$\\mathfrak{H}$": "constant magnetic field", "$\\boldsymbol{H}^{(i)}$": "magnetic field inside the sphere", "$\\varepsilon$": "permittivity of the sphere", "$\\mu$": "permeability of the sphere", "$c$": "speed of light in vacuum", "$\\boldsymbol{\\Omega}$": "angular velocity of rotation", "$a$": "radius of the sphere", "$\\boldsymbol{n}$": "unit vector in the direction of position vector", "$D_{i k}$": "electric quadrupole moment tensor of the sphere", "$\\mathfrak{H}_{i}$": "component of constant magnetic field", "$\\Omega_{k}$": "component of angular velocity"}
|
Electromagnetic wave equation
|
Electrodynamics of Moving Dielectrics
|
|
159 |
Magnetism
|
A magnetized dielectric sphere (with dielectric constant $\varepsilon$) rotates uniformly in a vacuum around its own axis parallel to the magnetization direction (the $z$-axis) with angular velocity $\Omega$. This rotation generates an electric field around the sphere. To describe this electric field, it is necessary to calculate the electric quadrupole moment. Determine the $D_{zz}$ component of the electric quadrupole tensor generated by this rotating sphere. The sphere has a radius $a$, total magnetic moment $\mathscr{M}$, and the speed of light in vacuum $c$.
|
[] |
Expression
|
{"$\\varepsilon$": "dielectric constant", "$z$": "axis parallel to the magnetization direction", "$\\Omega$": "angular velocity", "$a$": "radius of the sphere", "$\\mathscr{M}$": "total magnetic moment of the sphere", "$c$": "speed of light in vacuum", "$B^{(i)}$": "magnetic field inside the sphere", "$H^{(i)}$": "magnetic field intensity inside the sphere", "$M$": "constant magnetization", "$\\boldsymbol{D}$": "electric displacement field", "$\\boldsymbol{E}$": "electric field", "$\\boldsymbol{v}$": "velocity vector", "$\\boldsymbol{B}$": "magnetic field vector", "$\\boldsymbol{H}$": "magnetic field intensity", "$\\varphi^{(e)}$": "electric potential outside the sphere", "$\\varphi^{(i)}$": "electric potential inside the sphere", "$D_{zz}$": "electric quadrupole moment component", "$\\theta$": "angle between the normal and the z-axis"}
|
Electromagnetic wave equation
|
Electrodynamics of Moving Dielectrics
|
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160 |
Magnetism
|
A magnetized metallic sphere (considered as the case of dielectric constant $\varepsilon \rightarrow \infty$) rotates uniformly in vacuum around its own axis parallel to the magnetization direction (the $z$-axis) with an angular velocity $\Omega$. This rotation will generate an electric field around the sphere. Determine the $D_{zz}$ component of the electric quadrupole moment tensor produced by the metallic sphere to describe its external electric field. The sphere radius is $a$, the total magnetic moment is $\mathscr{M}$, and the speed of light in vacuum is $c$.
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[] |
Expression
|
{"$\\varepsilon$": "dielectric constant", "$z$": "axis parallel to magnetization direction", "$\\Omega$": "angular velocity", "$a$": "sphere radius", "$\\mathscr{M}$": "total magnetic moment", "$c$": "speed of light in vacuum", "$\\mathbf{M}$": "magnetization", "$B^{(i)}$": "magnetic field inside the sphere", "$H^{(i)}$": "magnetic intensity inside the sphere", "$\\mathbf{D}$": "electric displacement vector", "$\\mathbf{E}$": "electric field", "$\\mathbf{v}$": "velocity in the rotating frame", "$D_{zz}$": "component of the electric quadrupole moment tensor", "$\\varphi^{(e)}$": "electric field potential outside the sphere", "$\\varphi^{(i)}$": "electric field potential inside the sphere", "$\\theta$": "angle between normal and direction of $z$-axis", "$D_{xx}$": "component of the electric quadrupole moment tensor", "$D_{yy}$": "component of the electric quadrupole moment tensor"}
|
Electromagnetic wave equation
|
Electrodynamics of Moving Dielectrics
|
|
161 |
Magnetism
|
Try to find the dispersion relation of magnetostatic oscillations in an unbounded medium.
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[] |
Expression
|
{"$\\mu_{ik}$": "permeability tensor component", "$\\omega$": "angular frequency", "$\\psi$": "scalar field function related to magnetostatic oscillations", "$x_i$": "Cartesian coordinate (i-component)", "$x_k$": "Cartesian coordinate (k-component)", "$\\mu_{xx}$": "permeability tensor component (xx)", "$\\mu_{yy}$": "permeability tensor component (yy)", "$\\mu_{zz}$": "permeability tensor component (zz)", "$\\mu_{xy}$": "permeability tensor component (xy)", "$\\mu_{yx}$": "permeability tensor component (yx)", "$\\omega_M$": "characteristic angular frequency related to magnetization", "$\\omega_H$": "angular frequency related to external magnetic field", "$\\beta$": "parameter related to propagation in the medium", "$\\mu(\\omega)$": "frequency-dependent permeability", "$\\theta$": "angle between wave vector and easy magnetization axis", "$\\boldsymbol{k}$": "wave vector", "$\\boldsymbol{r}$": "position vector", "$\\gamma$": "gyromagnetic ratio", "$M$": "magnetization"}
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Electromagnetic wave equation
|
Dispersion of magnetic permeability
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|
162 |
Magnetism
|
The surface of an infinite parallel plate is perpendicular to the easy magnetization axis, and an external magnetic field $\mathfrak{H}$ is applied along this axis direction. Determine the non-uniform resonance frequency within this plate.
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[] |
Expression
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{"$\\mathfrak{H}$": "external magnetic field", "$\\psi^{(i)}$": "internal potential field", "$\\psi^{(e)}$": "external potential field", "$\\varphi^{(i)}$": "internal potential", "$\\varphi^{(e)}$": "external potential", "$z$": "coordinate axis perpendicular to the plate", "$L$": "half the thickness of the plate", "$A$": "amplitude constant for the internal potential", "$B$": "amplitude constant for the external potential", "$k_{z}$": "wave vector component in the z direction", "$k_{x}$": "wave vector component in the x direction", "$\\mu$": "relative permeability", "$n^{(z)}$": "demagnetizing coefficient in the z direction", "$M$": "magnetization", "$\\beta$": "scaling factor for the magnetization", "$\\omega_M$": "magnetization-related angular frequency", "$\\omega_H$": "field-related angular frequency", "$\\mu_{xx}$": "permeability component xx", "$\\mu_{yy}$": "permeability component yy", "$\\mu_{zz}$": "permeability component zz", "$\\mu_{xy}$": "permeability component xy", "$\\mu_{yx}$": "permeability component yx", "$\\omega$": "vibration frequency", "$\\gamma$": "gyromagnetic ratio", "$\\theta$": "angle between wave vector and z axis", "$\\boldsymbol{k}$": "wave vector"}
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Electromagnetic wave equation
|
Dispersion of magnetic permeability
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163 |
Magnetism
|
Calculate the reflection coefficient when light is almost grazing the surface of a material with $\varepsilon$ close to 1 from a vacuum.
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[] |
Expression
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{"$R_{\\perp}$": "perpendicular reflection coefficient", "$R_{\\parallel}$": "parallel reflection coefficient", "$\\theta_0$": "angle of incidence", "$\\theta_2$": "angle of refraction", "$\\varphi_{0}$": "incidence angle offset from grazing", "$\\varepsilon$": "relative permittivity"}
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Propagation of electromagnetic waves
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Reflection and refraction of waves
|
|
164 |
Magnetism
|
Find the reflection coefficient $R_{\perp}$ when a wave is incident from vacuum onto the surface of a medium where both $\varepsilon$ and $\mu$ are different from 1, with the electric field vector perpendicular to the plane of incidence.
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[] |
Expression
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{"$R_{\\perp}$": "reflection coefficient for electric field perpendicular to the plane of incidence", "$\\varepsilon$": "permittivity of the medium", "$\\mu$": "permeability of the medium", "$\\theta_{0}$": "angle of incidence"}
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Propagation of electromagnetic waves
|
Reflection and refraction of waves
|
|
165 |
Magnetism
|
Find the reflection coefficient $R_{\|}$ when a wave is incident on the surface of a medium with both $\varepsilon$ and $\mu$ different from 1, and when the electric field vector is parallel to the plane of incidence.
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[] |
Expression
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{"$R_{\\|}$": "reflection coefficient for parallel polarization", "$\\varepsilon$": "permittivity", "$\\mu$": "permeability", "$\\theta_{0}$": "angle of incidence"}
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Propagation of electromagnetic waves
|
Reflection and refraction of waves
|
|
166 |
Magnetism
|
For metals with impedance determined by formula \begin{align*}
\zeta = (1 - i)\sqrt{\frac{\omega\mu}{8\pi\sigma}}
\end{align*} (a special case with a flat surface having low impedance), and assuming its permeability $\mu=1$, try to determine the ratio of its thermal radiation intensity to the absolute blackbody surface radiation intensity ($I/I_0$). The ratio should be expressed in terms of the angular frequency of radiation $\omega$ and the conductivity of the metal $\sigma$.
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[] |
Expression
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{"$I$": "thermal radiation intensity from the flat surface", "$I_{0}$": "absolute blackbody surface radiation intensity", "$\\zeta$": "impedance", "$\\mu$": "permeability", "$\\omega$": "angular frequency of radiation", "$\\sigma$": "conductivity of the metal"}
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Propagation of electromagnetic waves
|
Surface impedance of metal
|
|
167 |
Magnetism
|
Determine the dependence of the radiation intensity of a dipole emitter immersed in a homogeneous isotropic medium on the medium's permittivity $\varepsilon$ and magnetic permeability $\mu$.
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[] |
Expression
|
{"$\\varepsilon$": "permittivity", "$\\mu$": "magnetic permeability", "$I$": "radiation intensity", "$I_{0}$": "radiation intensity in vacuum", "$\\boldsymbol{E}$": "electric field", "$\\boldsymbol{H}$": "magnetic field", "$\\omega$": "angular frequency", "$c$": "speed of light in vacuum", "$\\mathbf{B}$": "magnetic flux density", "$\\mathbf{D}$": "electric displacement field", "$\\mathbf{j}_{\\text{ex}}$": "external current density", "$\\boldsymbol{A}^{\\prime}$": "vector potential in medium", "$R_{0}$": "distance from the source", "$\\boldsymbol{k}^{\\prime}$": "wave vector in medium", "$\\boldsymbol{H}_{0}$": "magnetic field in vacuum", "$\\boldsymbol{E}_{0}$": "electric field in vacuum"}
|
Propagation of electromagnetic waves
|
Reciprocity principle
|
|
168 |
Magnetism
|
For the E wave in a circular waveguide with a radius of $a$, provide the expression for its attenuation coefficient $\alpha$.
|
[] |
Expression
|
{"$a$": "radius of the circular waveguide", "$\\alpha$": "attenuation coefficient", "$\\omega$": "angular frequency", "$\\zeta^{\\prime}$": "characteristic impedance of the medium", "$c$": "speed of light in vacuum", "$k_z$": "propagation constant along the z-axis", "$\\kappa$": "wavenumber in the medium", "$E_z$": "electric field component in the z direction", "$H_z$": "magnetic field component in the z direction"}
|
Propagation of electromagnetic waves
|
Propagation of electromagnetic waves in waveguides
|
|
169 |
Magnetism
|
Provide the expression for the attenuation coefficient $\alpha$ of H modes in a circular (radius $a$) cross-section waveguide.
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[] |
Expression
|
{"$\\alpha$": "attenuation coefficient", "$a$": "radius of the circular cross-section waveguide", "$\\omega$": "angular frequency", "$\\zeta'$": "some characteristic impedance factor", "$\\kappa$": "propagation constant transversal to the direction of energy flow", "$k_z$": "propagation constant in the z-direction", "$c$": "speed of light in vacuum", "$E_z$": "z-component of the electric field", "$H_z$": "z-component of the magnetic field", "$n$": "mode number", "$\\varkappa$": "another propagation constant or related factor"}
|
Propagation of electromagnetic waves
|
Propagation of electromagnetic waves in waveguides
|
|
170 |
Magnetism
|
Linearly polarized light is scattered by small particles with random orientations, where the particles' polarizability tensor has three distinct principal values. It is known that the scalar constants describing the average of the electric dipole moment tensor are $A = \frac{1}{15}(2 \alpha_{i i} \alpha_{k k}^{*}-\alpha_{i k} \alpha_{i k}^{*})$ and $B = \frac{1}{30}(3 \alpha_{i k} \alpha_{i k}^{*}-\alpha_{i i} \alpha_{k k}^{*})$. Determine the expression for the depolarization ratio $\frac{I_y}{I_x}$ of the scattered light, where $\theta$ is the angle between the incident light electric field $\boldsymbol{E}$ and the scattering direction $\boldsymbol{n}$.
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[] |
Expression
|
{"$A$": "scalar constant related to the electric dipole moment tensor", "$B$": "scalar constant related to the electric dipole moment tensor", "$\\alpha_{i i}$": "principal value of polarizability tensor", "$\\alpha_{k k}^{*}$": "complex conjugate of principal value of polarizability tensor", "$\\alpha_{i k}$": "element of polarizability tensor", "$\\alpha_{i k}^{*}$": "complex conjugate of element of polarizability tensor", "$I_y$": "intensity of scattered light in the y-direction", "$I_x$": "intensity of scattered light in the x-direction", "$\\theta$": "angle between the incident light electric field and the scattering direction", "$\\boldsymbol{E}$": "incident light electric field", "$\\boldsymbol{n}$": "scattering direction", "$\\omega$": "angular frequency of light", "$c$": "speed of light", "$R$": "distance from scattering particle", "$\\mathscr{P}$": "polarization of the scattered light", "$\\delta_{i l}$": "Kronecker delta function"}
|
Propagation of electromagnetic waves
|
Scattering of electromagnetic waves by small particles
|
|
171 |
Magnetism
|
Try to express the components of the ray vector $s$ in terms of the components of $\boldsymbol{n}$ within the principal dielectric axes.
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[] |
Expression
|
{"$s$": "components of the ray vector", "$\\boldsymbol{n}$": "unit vector in principal dielectric axes", "$n_x$": "x-component of unit vector", "$n_y$": "y-component of unit vector", "$n_z$": "z-component of unit vector", "$s_x$": "x-component of the ray vector", "$s_y$": "y-component of the ray vector", "$s_z$": "z-component of the ray vector", "$\\varepsilon^{(x)}$": "dielectric permittivity along the x-axis", "$\\varepsilon^{(y)}$": "dielectric permittivity along the y-axis", "$\\varepsilon^{(z)}$": "dielectric permittivity along the z-axis"}
|
Electromagnetic waves in anisotropic media
|
Optical properties of uniaxial crystals
|
|
172 |
Magnetism
|
Find the polarization of the reflected light when linearly polarized light is perpendicularly incident from vacuum onto the surface of an anisotropic object induced by a magnetic field.
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[] |
Expression
|
{"$\\boldsymbol{H}$": "magnetic field vector", "$E$": "electric field", "$E_{z}$": "z-component of the electric field", "$H_{x}$": "x-component of the magnetic field", "$E_{y}$": "y-component of the electric field", "$H_{y}$": "y-component of the magnetic field", "$E_{x}$": "x-component of the electric field", "$n$": "refractive index", "$\\boldsymbol{E}_{1}$": "electric field vector of the reflected wave", "$\\boldsymbol{E}_{0}$": "electric field vector of the incident wave", "$\\boldsymbol{E}_{0}^{+}$": "component of the incident electric field vector with positive circular polarization", "$\\boldsymbol{E}_{0}^{-}$": "component of the incident electric field vector with negative circular polarization", "$E_{0 x}^{+}$": "x-component of the positively polarized incident electric field", "$E_{0 y}^{+}$": "y-component of the positively polarized incident electric field", "$E_{0 x}^{-}$": "x-component of the negatively polarized incident electric field", "$E_{0 y}^{-}$": "y-component of the negatively polarized incident electric field", "$n_{+}$": "refractive index for positive polarization", "$n_{-}$": "refractive index for negative polarization", "$n_{0}$": "initial refractive index", "$G_z$": "anisotropic parameter affecting the refractive indices", "$g_z$": "component of the anisotropic parameter", "$E_{1 x}$": "x-component of the reflected electric field", "$E_{1 y}$": "y-component of the reflected electric field", "$g$": "anisotropic parameter vector", "$\\theta$": "angle between the incident direction and the vector $\\boldsymbol{g}$"}
|
Electromagnetic waves in anisotropic media
|
Magneto-optical effect
|
|
173 |
Superconductivity
|
Attempt to find the limiting law of the dependence of the surface tension coefficient $\alpha$ of liquid nitrogen near absolute zero on temperature
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[] |
Expression
|
{"$\\alpha$": "surface tension coefficient", "$\\alpha_{0}$": "surface tension coefficient at zero temperature", "$T$": "temperature", "$\\omega_{\\alpha}$": "frequency related to surface vibrations", "$\\rho$": "liquid density", "$k$": "wave vector of vibrations", "$\\hbar$": "reduced Planck's constant", "$\\omega$": "vibration frequency"}
|
Superfluidity
|
Superfluidity
|
|
174 |
Superconductivity
|
Try to find the dispersion relation of impurity particles in a moving superfluid $\varepsilon_{\text {imp}}(p)$, given that in a stationary fluid the dispersion relation is $\varepsilon_{\text {imp}}^{(0)}(p)$.
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[] |
Expression
|
{"$\\varepsilon_{\\text{imp}}(p)$": "dispersion relation of impurity particles in a moving superfluid", "$\\varepsilon_{\\text{imp}}^{(0)}(p)$": "dispersion relation of impurity particles in a stationary fluid", "$p$": "momentum", "$p_{0}$": "initial momentum of the impurity atom", "$m$": "mass of the impurity atom", "$v$": "velocity of the moving superfluid", "$M$": "mass associated with the fluid's motion", "$m_{\\text{eff}}^{*}$": "effective mass of impurity particles in the moving fluid"}
|
Superfluidity
|
Superfluidity
|
|
175 |
Superconductivity
|
Try to find the dispersion relation of small oscillations of a rectilinear vortex line (W. Thomson, 1880).
|
[] |
Expression
|
{"$z$": "coordinate axis along vortex line", "$r$": "displacement vector of points on vortex line", "$t$": "time", "$k$": "wave number", "$\\omega$": "angular frequency", "$\\kappa$": "circulation strength", "$R_{0}$": "radius of curvature", "$\\lambda$": "wavelength", "$a$": "core radius of vortex line", "$b$": "binormal vector magnitude", "$n_{z}$": "unit vector along z axis", "$n_{\\varepsilon}$": "perturbation direction unit vector"}
|
Superfluidity
|
Quantum vortex
|
|
176 |
Superconductivity
|
A neutron with an initial velocity $v$ scatters within a liquid. Determine the conditions under which an excitation with momentum $p$ and energy $\varepsilon(p)$ can be produced.
|
[] |
Expression
|
{"$v$": "initial velocity of the neutron", "$p$": "momentum of the excitation", "$\\varepsilon(p)$": "energy of the excitation with momentum p", "$m$": "mass of the neutron", "$P$": "initial momentum of the neutron", "$V$": "velocity component relevant to the condition being determined", "$\\theta$": "angle between the initial momentum P and the excitation momentum p", "$p_{\\mathrm{c}}$": "critical momentum value", "$q$": "momentum within the interval related to phonon radiation"}
|
Superfluidity
|
Fission of quasiparticles
|
|
177 |
Superconductivity
|
Try to find the magnetic moment of a superconducting sphere of radius $R \ll \delta$ in a magnetic field under the London situation.
|
[] |
Expression
|
{"$R$": "radius of the superconducting sphere", "$\\delta$": "London penetration depth", "$\\mathfrak{G}$": "external magnetic field", "$n_{s}$": "density of superconducting electrons", "$e$": "elementary charge", "$m$": "electron mass", "$c$": "speed of light", "$M$": "magnetic moment"}
|
Superconductivity
|
Superconductivity current
|
|
178 |
Superconductivity
|
For superconductors with parameter $\kappa \ll 1$, find the first-order correction to the penetration depth in a weak magnetic field.
|
[] |
Expression
|
{"$\\kappa$": "parameter for superconductors", "$\\psi$": "superconducting wavefunction", "$A$": "magnetic vector potential", "$B$": "magnetic field", "$\\mathfrak{S}$": "boundary value of magnetic field derivative", "$H_{\\mathrm{e}}$": "external magnetic field", "$\\delta_{\\text{eff}}$": "effective penetration depth", "$\\delta$": "penetration depth at zero correction", "$\\psi_{1}$": "first-order correction to superconducting wavefunction", "$A_{1}$": "first-order correction to magnetic vector potential"}
|
Superconductivity
|
Surface tension at the boundary between the superconducting phase and the normal phase
|
|
179 |
Superconductivity
|
Attempt to find the critical field of a superconducting small sphere with radius $R \ll \delta$.
|
[] |
Expression
|
{"$R$": "radius of the small sphere", "$\\delta$": "characteristic penetration depth", "$n$": "normal phase", "$s$": "superconducting phase", "$e$": "elementary charge", "$A$": "vector potential", "$c$": "speed of light", "$m$": "mass of the electron", "$\\hbar$": "reduced Planck's constant", "$\\boldsymbol{A}$": "vector potential in bold", "$\\mathfrak{S}$": "magnitude of vector potential related to the field", "$E_{0}$": "average energy of perturbation", "$\\mathfrak{G}$": "magnitude related to the uniform field", "$a$": "parameter related to critical field condition", "$H_{\\mathrm{c}}^{\\text {(sphere) }}$": "critical field for a superconducting sphere", "$H_{\\mathrm{o}}$": "reference field strength"}
|
Superconductivity
|
Two types of superconductors
|
|
180 |
Superconductivity
|
Calculate the interaction energy of two vortices separated by $d \gg \xi$.
|
[] |
Expression
|
{"$d$": "distance between two vortices", "$\\xi$": "coherence length", "$F_{\\text{vertex}}$": "free energy of the vortex system", "$\\mathbf{B}$": "magnetic field", "$\\delta$": "penetration depth", "$\\phi_0$": "magnetic flux quantum", "$\\varepsilon_{12}$": "interaction energy between two vortices", "$B_{1}$": "magnetic field of first vortex", "$B_{2}$": "magnetic field of second vortex", "$r_{0}$": "radius of cylindrical surface around the vortex", "$f_{1}$": "surface surrounding the first vortex", "$f_{2}$": "surface surrounding the second vortex"}
|
Superconductivity
|
Mixed structure
|
|
181 |
Superconductivity
|
A thin film (thickness $d \ll \xi(T)$) is placed in a weak magnetic field perpendicular to its plane. Find the magnetic moment of the film when the temperature $T$ $>T_{\mathrm{c}}, T-T_{\mathrm{c}} \ll T_{\mathrm{c}}$.
|
[] |
Expression
|
{"$d$": "thickness of the film", "$\\xi(T)$": "temperature-dependent coherence length", "$T$": "temperature", "$T_{\\mathrm{c}}$": "critical temperature", "$p_{z}$": "quantized momentum component perpendicular to the film", "$E$": "energy", "$e$": "elementary charge", "$\\hbar$": "reduced Planck's constant", "$\\mathfrak{H}$": "magnetic field strength", "$m$": "mass of charge carriers", "$c$": "speed of light", "$a$": "energy related constant", "$5 S$": "multiplicative factor involving the area", "$S$": "area of the film", "$\\Delta F$": "change in free energy", "$M$": "magnetic moment of the film", "$\\mathscr{Q}$": "generalized coordinate for the free energy", "$\\mathscr{G}$": "given function related to the magnetic moment expression", "$\\alpha$": "positive constant related to the temperature dependence"}
|
Superconductivity
|
Diamagnetic susceptibility above the phase transition point
|
|
182 |
Superconductivity
|
Under the conditions of the previous question, determine the magnetic moment of a small sphere with radius $R \ll \xi(T)$
|
[] |
Expression
|
{"$R$": "radius of the small sphere", "$\\xi(T)$": "temperature-dependent parameter", "$m$": "mass", "$e$": "elementary charge", "$\\hbar$": "reduced Planck's constant", "$c$": "speed of light in vacuum", "$\\mathbf{A}$": "vector potential", "$E$": "energy eigenvalue", "$E_{0}$": "minimum energy eigenvalue", "$T$": "temperature", "$T_{\\mathrm{c}}$": "critical temperature", "$a$": "constant parameter", "$\\mathfrak{S}$": "function related to magnetic moment calculations", "$\\mathscr{S}_{g}$": "scalar function specific to the material or geometry", "$\\alpha$": "coefficient function of temperature"}
|
Superconductivity
|
Diamagnetic susceptibility above the phase transition point
|
|
183 |
Magnetism
|
Find the energy spectrum of spin wave quanta in an uniaxial ferromagnet of the 'easy magnetization plane' type $(K<0)$.
|
[] |
Expression
|
{"$M_{0}$": "equilibrium magnetization", "$M$": "magnetization", "$\\varepsilon$": "energy of spin wave quanta", "$\\beta$": "constant (beta)", "$\\alpha$": "constant (alpha)", "$k$": "wave number", "$K$": "anisotropy constant", "$n_{x}$": "unit vector along x-axis", "$n_{y}$": "unit vector along y-axis", "$m$": "vector in yz plane", "$\\theta$": "polar angle", "$\\varphi$": "azimuthal angle"}
|
magnetic
|
Spin wave quantum spectroscopy in ferromagnets
|
|
184 |
Magnetism
|
Calculate the spin wave quantum part of thermodynamic quantities (energy) at temperature $T \ll \varepsilon(0)$.
|
[] |
Expression
|
{"$T$": "temperature", "$\\varepsilon(0)$": "energy at zero wave vector", "$k$": "quasi-momentum", "$\\theta$": "angle related to propagation direction", "$\\varepsilon(k)$": "energy at wave vector k", "$\\beta$": "constant related to the material", "$K$": "anisotropy constant", "$M$": "magnetization", "$A$": "coefficient related to crystal type", "$\\alpha$": "parameter related to cubic crystals", "$\\alpha_{2}$": "parameter related to uniaxial crystals", "$V$": "volume", "$E_{\\operatorname{mag}}$": "magnetic energy", "$M_{\\operatorname{mag}}$": "magnetic magnetization", "$C_{\\operatorname{mag}}$": "magnetic specific heat"}
|
magnetic
|
Spin wave quantum thermodynamic quantities in ferromagnets
|
|
185 |
Magnetism
|
In the exchange approximation, determine the spatial correlation function of magnetization fluctuations at distances $r \gg a$.
|
[] |
Expression
|
{"$\\hat{m}_{x}$": "magnetization operator in the x-direction", "$\\hat{m}_{y}$": "magnetization operator in the y-direction", "$\\beta$": "inverse temperature or thermodynamic beta", "$M$": "magnetization", "$\\mathbf{r}$": "position vector", "$k$": "index representing x or y direction", "$n_{k}$": "occupation number of spin-wave quantum states", "$\\varphi_{i k}$": "spatial correlation function of magnetization fluctuations", "$T$": "temperature", "$\\alpha$": "constant related to cubic ferromagnet", "$i$": "index representing x or y direction"}
|
magnetic
|
Spin wave quantum thermodynamic quantities in ferromagnets
|
|
186 |
Magnetism
|
Given $S \gg 1$, try to find the correction terms related to the interaction of heat capacity of a cubic lattice. In this lattice, only the exchange integral between a pair of neighboring atoms (along the cubic axes) is non-zero.
|
[] |
Expression
|
{"$S$": "total spin of the system", "$J_{mn}$": "exchange integral between atoms at positions m and n", "$k$": "wave vector in reciprocal space", "$\\mathbf{r}_m$": "position vector of atom m", "$\\mathbf{r}_n$": "position vector of atom n", "$J_{0}$": "exchange integral between a pair of nearest neighbor atoms", "$a$": "lattice constant", "$V$": "volume of the system", "$k_{1x}$": "x-component of wave vector k1", "$k_{2x}$": "x-component of wave vector k2", "$k_{1y}$": "y-component of wave vector k1", "$k_{2y}$": "y-component of wave vector k2", "$k_{1z}$": "z-component of wave vector k1", "$k_{2z}$": "z-component of wave vector k2", "$U(k_{1}, k_{2} ; k_{1}, k_{2})$": "energy correction term involving wave vectors k1 and k2", "$\\varepsilon(k)$": "energy of magnon quanta with wave vector k", "$\\beta$": "inverse temperature (1/kT)", "$\\mathfrak{H}$": "external magnetic field", "$M_{\\text {int }}$": "interaction part of magnetization", "$M$": "magnetization", "$C_{\\text{int}}$": "interaction part of heat capacity", "$N$": "number of lattice sites or atoms", "$T$": "temperature"}
|
magnetic
|
Interaction of spin wave quanta
|
|
187 |
Magnetism
|
Try to determine the interaction law between an atom and a metal wall at 'large' distances.
|
[] |
Expression
|
{"$\\varepsilon_{10}$": "permittivity of medium 1", "$\\varepsilon_{20}$": "permittivity of medium 2", "$\\hbar$": "reduced Planck's constant", "$c$": "speed of light", "$l$": "separation distance between atom and wall", "$L$": "distance between the atom and the wall", "$a$": "interaction constant", "$n$": "number density of atoms", "$\\alpha_{2}$": "polarizability of medium 2"}
|
Electromagnetic Fluctuations
|
Molecular forces and limiting cases of solid-solid interactions
|
|
188 |
Strongly Correlated Systems
|
Determine the correlation function $\nu(r)$ of a Bose liquid at temperature $T \ll T_{\lambda}$, at distances $r \gtrsim \hbar u / T$.
|
[] |
Expression
|
{"$\\nu(r)$": "correlation function of the Bose liquid at distance r", "$r$": "distance", "$T$": "temperature", "$T_{\\lambda}$": "lambda transition temperature for Bose liquid", "$\\hbar$": "reduced Planck's constant", "$u$": "speed of sound in the medium", "$k$": "wave number", "$m$": "mass of particles in the liquid", "$a$": "interatomic distance", "$\\omega$": "angular frequency"}
|
Fluid Dynamics Fluctuations
|
Summation Rule for Shape Factors
|
|
189 |
Strongly Correlated Systems
|
In a Bose superfluid, the condensate wave function exhibits fluctuations. Try to find the asymptotic form of this fluctuation correlation function at large distances.
|
[] |
Expression
|
{"$\\Phi$": "condensate wave function", "$\\Omega$": "total thermodynamic potential", "$V$": "volume", "$T$": "temperature", "$\\mu$": "chemical potential", "$\\rho_{s}$": "superfluid density", "$v_{s}$": "superfluid velocity", "$m$": "mass", "$\\hbar$": "reduced Planck's constant", "$k$": "wave vector", "$\\delta \\Phi_{k}$": "fluctuation amplitude in Fourier space", "$G(r)$": "fluctuation correlation function", "$n_{0}$": "condensate density", "$r$": "distance", "$r_{\\mathrm{c}}$": "correlation radius", "$\\sigma$": "critical exponent", "$\\zeta$": "appropriate critical exponent", "$\\beta$": "critical exponent related to order parameter", "$\\nu$": "critical exponent related to correlation length", "$\\xi$": "exponent in scaling relation", "$T_{\\lambda}$": "lambda point temperature"}
|
Fluid Dynamics Fluctuations
|
Summation Rule for Shape Factors
|
|
190 |
Semiconductors
|
If only the interaction between nearest neighbors is considered, the energy band of $s$-state electrons in a body-centered cubic lattice derived by the tight-binding approximation method is
\begin{align*}
E(\mathbf{k}) = E_0 - A - \delta J(\cos\pi a k_x \cos\pi a k_y \cos\pi a k_z)，
\end{align*}
where $J$ is the overlap integral. Calculate the bandwidth of the body-centered cubic lattice;
|
[] |
Expression
|
{"$k_{x}$": "wave vector component in x-direction", "$k_{y}$": "wave vector component in y-direction", "$k_{z}$": "wave vector component in z-direction", "$E_{\\min }$": "minimum energy in the band", "$E_{0}$": "reference energy level", "$A$": "energy offset", "$J$": "exchange interaction energy", "$E_{\\max }$": "maximum energy in the band", "$\\Delta E$": "bandwidth"}
|
Electronic states in semiconductors
|
Electronic states in semiconductors
|
|
191 |
Semiconductors
|
If only the interaction between nearest neighbors is considered, the energy band of $s$-state electrons in a body-centered cubic lattice derived by the tight-binding approximation method is
\begin{align*}
E(\mathbf{k}) = E_0 - A - \delta J(\cos\pi a k_x \cos\pi a k_y \cos\pi a k_z)，
\end{align*}
where $J$ is the overlap integral. Calculate the effective mass of electrons at the bottom of the band;
|
[] |
Expression
|
{"$k$": "wave vector magnitude", "$E(\\boldsymbol{k})$": "energy as a function of wave vector", "$E_{0}$": "baseline energy level", "$A$": "some constant", "$J$": "exchange energy", "$a$": "lattice constant", "$k_{x}$": "x-component of wave vector", "$k_{y}$": "y-component of wave vector", "$k_{z}$": "z-component of wave vector", "$E_{\\min}$": "minimum energy", "$h$": "Planck's constant", "$m_{\\mathrm{b}}^{*}$": "effective mass of electrons at the bottom of the band", "$E_{\\max}$": "maximum energy", "$\\delta k_{x}$": "deviation in x-component of wave vector", "$\\delta k_{y}$": "deviation in y-component of wave vector", "$\\delta k_{z}$": "deviation in z-component of wave vector", "$m_{\\mathrm{t}}^{*}$": "effective mass of electrons at the top of the band"}
|
Electronic states in semiconductors
|
Electronic states in semiconductors
|
|
192 |
Semiconductors
|
If only the interaction between nearest neighbors is considered, the energy band of $s$-state electrons in a body-centered cubic lattice derived by the tight-binding approximation method is
\begin{align*}
E(\mathbf{k}) = E_0 - A - \delta J(\cos\pi a k_x \cos\pi a k_y \cos\pi a k_z)，
\end{align*}
where $J$ is the overlap integral. Calculate the effective mass of electrons at the top of the band;
|
[] |
Expression
|
{"$k$": "wave vector", "$E$": "energy", "$E_{0}$": "initial energy offset", "$A$": "constant energy term", "$J$": "exchange interaction term", "$a$": "lattice constant", "$k_{x}$": "x-component of wave vector", "$k_{y}$": "y-component of wave vector", "$k_{z}$": "z-component of wave vector", "$E_{\\min}$": "minimum energy level", "$h$": "Planck's constant", "$m_{\\mathrm{b}}^{*}$": "effective mass at the bottom of the band", "$E_{\\max}$": "maximum energy level", "$m_{\\mathrm{t}}^{*}$": "effective mass at the top of the band", "$\\delta k_{x}$": "small deviation in x-component of wave vector", "$\\delta k_{y}$": "small deviation in y-component of wave vector", "$\\delta k_{z}$": "small deviation in z-component of wave vector"}
|
Electronic states in semiconductors
|
Electronic states in semiconductors
|
|
193 |
Semiconductors
|
By considering only the nearest-neighbor interactions and using the tight-binding method, the energy band of s-state electrons in a simple cubic crystal is derived as
$$
E(\boldsymbol{k})=E_{0}-A-2 J(\cos 2 \pi a k_{x}+\cos 2 \pi a k_{y}+\cos 2 \pi a k_{z})
$$ Find the bandwidth $(\Delta E)$ ;
|
[] |
Expression
|
{"$\\Delta E$": "bandwidth", "$E$": "energy", "$\\boldsymbol{k}$": "wave vector", "$E_{0}$": "energy level constant", "$A$": "atomic energy correction", "$\\boldsymbol{R}_{\\mathrm{s}}$": "position vector of the reference atom", "$\\boldsymbol{R}_{\\mathrm{n}}$": "position vector of the nearest neighbors", "$J_{\\mathrm{sn}}$": "overlap integral between s-state electrons", "$a$": "lattice constant", "$E_{\\min}$": "minimum energy", "$E_{\\max}$": "maximum energy", "$J$": "interaction energy constant", "$k_{x}$": "x-component of wave vector", "$k_{y}$": "y-component of wave vector", "$k_{z}$": "z-component of wave vector"}
|
Electronic states in semiconductors
|
Electronic states in semiconductors
|
|
194 |
Semiconductors
|
The energy $E$ near the top of the valence band of a semiconductor crystal can be expressed as: $E(k)=E_{\text {max }}-10^{26} k^{2}(\mathrm{erg})$. Now, remove an electron with the wave vector $k=10^{7} \mathrm{i} / \mathrm{cm}$, and find the speed of the hole left by this electron.
|
[] |
Expression
|
{"$E$": "energy", "$E_{\\text{max}}$": "maximum energy near top of valence band", "$k$": "wave vector", "$k_{x}$": "wave vector component in x-direction", "$k_{y}$": "wave vector component in y-direction", "$k_{z}$": "wave vector component in z-direction", "$m_{\\mathrm{n}}^{*}$": "effective mass of electron", "$m_{\\mathrm{p}}^{*}$": "effective mass of hole", "$v_{x}$": "velocity component in x-direction", "$v_{y}$": "velocity component in y-direction", "$v_{z}$": "velocity component in z-direction", "$h$": "Planck's constant"}
|
Electronic states in semiconductors
|
Electronic states in semiconductors
|
|
195 |
Semiconductors
|
In a one-dimensional periodic potential, the wavefunctions of electrons take the following form:
$\psi_{k}(x)=\sin \frac{\pi}{a} x$:
Try to use Bloch's theorem to point out the wave vector $\mathbf{k}$ values within the reduced Brillouin zone.
|
[] |
Expression
|
{"$\\psi_{k}$": "wavefunction of electrons with wave vector k", "$k$": "wave vector", "$a$": "lattice constant", "$x$": "position"}
|
Electronic states in semiconductors
|
Electronic states in semiconductors
|
|
196 |
Semiconductors
|
In a one-dimensional periodic potential, the wavefunctions of electrons take the following form:
$\psi_{k}(x)=i \cos \frac{3 \pi}{a} x$ ；
Try to use Bloch's theorem to point out the wave vector $\mathbf{k}$ values within the reduced Brillouin zone.
|
[] |
Expression
|
{"$\\psi_{k}$": "wavefunction of electrons with wave vector $\\mathbf{k}$", "$x$": "position variable", "$a$": "lattice constant", "$\\mathbf{k}$": "wave vector"}
|
Electronic states in semiconductors
|
Electronic states in semiconductors
|
|
197 |
Semiconductors
|
The electron wave function moving in a one-dimensional periodic potential field has the following form: $\psi_{k}(x)=\sum_{l=-\infty}^{\infty}(-1)^{l} f(x-l a)$ . Here $a$ is the lattice constant of the one-dimensional lattice, $f(x)$ is a certain function, try using Bloch's theorem to indicate the values of the wave vector $\boldsymbol{k}$ within the reduced Brillouin zone.
|
[] |
Expression
|
{"$\\psi_{k}$": "electron wave function", "$x$": "position", "$a$": "lattice constant", "$f(x)$": "certain function", "$\\boldsymbol{k}$": "wave vector", "$k$": "wave vector"}
|
Electronic states in semiconductors
|
Electronic states in semiconductors
|
|
198 |
Semiconductors
|
It is known that the electronic energy band of a one-dimensional crystal can be expressed as
$$
E(k)=\frac{h^{2}}{m_{0} a^{2}}(\frac{7}{8}-\cos 2 \pi k a+\frac{1}{8} \cos 6 \pi k a)
$$
Where $a$ is the lattice constant. Try to find: the width of the energy band;
|
[] |
Expression
|
{"$E$": "energy", "$k$": "wave number", "$h$": "Planck's constant", "$m_{0}$": "rest mass", "$a$": "lattice constant", "$\\Delta E$": "energy band width"}
|
Electronic states in semiconductors
|
Electronic states in semiconductors
|
|
199 |
Semiconductors
|
It is known that the electron energy band of a one-dimensional crystal can be expressed as
$$
E(k)=\frac{h^{2}}{m_{0} a^{2}}(\frac{7}{8}-\cos 2 \pi k a+\frac{1}{8} \cos 6 \pi k a)
$$
where $a$ is the lattice constant. Try to find the velocity of the electron at wave vector $k$;
|
[] |
Expression
|
{"$k$": "wave vector", "$E$": "energy", "$v$": "velocity", "$h$": "Planck's constant", "$m_{0}$": "rest mass of the electron", "$a$": "lattice constant"}
|
Electronic states in semiconductors
|
Electronic states in semiconductors
|
|
200 |
Semiconductors
|
It is known that the electronic energy band of a one-dimensional crystal can be expressed as
$$
E(k)=\frac{h^{2}}{m_{0} a^{2}}(\frac{7}{8}-\cos 2 \pi k a+\frac{1}{8} \cos 6 \pi k a)
$$
where $a$ is the lattice constant. Find: Find the effective mass of electrons at the bottom of the band.
|
[] |
Expression
|
{"$m_{\\mathrm{n}}^{*}$": "effective mass of electrons", "$h$": "Planck's constant", "$E$": "energy", "$k$": "wave vector", "$m$": "mass", "$m_{0}$": "free electron rest mass"}
|
Electronic states in semiconductors
|
Electronic states in semiconductors
|
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