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Surface Currents on the Plates of a Charging Capacitor

Ampere’s law is presented in introductory physics as a relation between the line integral of the magnetic field around a closed loop and the net current crossing any open surface spanning that loop. By allowing the surface to pass between the plates of a charging parallel-plate capacitor, Maxwell realized that this law is incomplete and introduced a term called the displacement current Id (due to changing electric flux), which needs to be combined with the conduction current Ic (due to charge flow) to give the effective net current I. Textbooks state that the way these currents are to be combined is by simple addition, I = Ic+Id. They verify that the resulting Ampere-Maxwell law gives the correct expression for the magnetic field at an arbitrary point outside of both plates for two different choices of spanning surfaces. However, those traditional two surfaces are special and do not prove that the total current must be the simple addition of the displacement and conduction currents. A third, more general spanning surface is used here to establish it. This third approach draws attention to the surface currents on the capacitor plates.