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Illustration Current Depends on the Choice of the Enclosing Loop in the Plate Capacitor

The choice of the enclosing loop for the Ampere law is not unique
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An AC voltage \(U(t)\) is applied to a plate capacitor. This causes a time-dependent current \(I(t)\) to flow through the wire, charging and discharging the capacitor. This charging and discharging process causes the charge \(Q\) on the electrodes to change with time and thus also the electric field \(E\) between the electrodes.

By choosing a closed loop \(S\) enclosing the current, Ampere's law gives either the current \(I\) (in the case of \(A_1\)) or zero (in the case of \(A_2\)). Here \(A_1\) and \(A_2\) are possible surfaces with the edge \(S\).

To resolve this contradiction, Ampere's law (fourth Maxwell equation of electrostatics) is extended with the so-called displacement current\(I_{\text e}\) , which represents the temporal change of the E-field.