system
Given the following post, respond as if you're commenting on Reddit:
human
Why are equipotential lines perpendicular to electric field lines?

Edit: In electrostatic case of course.
gpt
In electrostatics, **E** is the (negative) gradient of some scalar field *U*(x, y, z). The gradient of *any* scalar field is always perpendicular to level surfaces of *U*. This is just a mathematical property, not a physical one, and it is an immediate consequence of the chain rule.

If **r**(t) is a curve in the level surface *U*(x, y, z) = c, consider the function f(t) = U(**r**(t)), which is identically constant. Differentiating with respect to *t* gives df/dt = 0, or

> grad(U) · d**r**/dt = 0

Now *any* line tangent to the level surface can be described with direction vector d**r**/dt for some curve **r** in the level surface at some parameter value of *t*. (This is more or less the definition of tangent line to a surface in multivariable calculus, but it actually *is* the definition of tangent space in differential geometry.) The above equation shows that grad(U) is perpendicular to all such tangent lines. Hence grad(U) is perpendicular to the level surface U = c at all points of that surface.

The fact that **E** is the *negative* gradient just means that **E** points in the direction in which the the parameter *c* is *decreasing* the fastest, rather than increasing the fastest. Multiplication of **E** by a constant scalar does not affect its perpendicularity with level surfaces.