Ampère’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 Ampère-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.

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199
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2.
For some cautionary remarks about neglecting the fringing field in this situation, see
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and
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(
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D.F.
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,”
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4.
R.K.
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, 2nd ed. (
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,
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353
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5.
It is interesting to see whether students can explain why the rate of change of electric flux is zero across all parts of the curved surface of S3. It is zero across portion D because there is no field outside an ideal capacitor. It is zero across portion B because the electric field lines skim along that part of the surface. Finally but least obviously, it is zero across portion C because a constant current density is driven by a constant electric field.
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