A capacitor with parallel circular plates of radius R=1.20 cm is discharging via a current of 12

step 1 In this problem, we have a capacitor with parallel circular plates, discharging a current, and w

A capacitor with parallel circular plates of radius R=1.20...

A capacitor with parallel circular plates of radius $R=1.20$ $\mathrm{cm}$ is discharging via a current of 12.0 $\mathrm{A}$ . Consider a loop of radius $R / 3$ that is centered on the central axis between the plates.(a) How much displacement current is encircled by the loop? The maximum induced magnetic field has a magnitude of 12.0 $\mathrm{mT}$ . At what radius (b) inside and (c) outside the capacitor gap is the magnitude of the induced magnetic field 3.00 $\mathrm{mT}$ ?

A capacitor with parallel circular plates of radius $R=1.20$ $\mathrm{cm}$ is discharging via a current of 12.0 $\mathrm{A}$ . Consider a loop of radius
$R / 3$ that is centered on the central axis between the plates.(a) How
much displacement current is encircled by the loop? The maximum induced magnetic field has a magnitude of 12.0 $\mathrm{mT}$ . At what radius
(b) inside and (c) outside the capacitor gap is the magnitude of the
induced magnetic field 3.00 $\mathrm{mT}$ ?

Key Concepts

Displacement Current

Displacement current is the term introduced by Maxwell to account for the effect of a time-varying electric field in regions where there is no conduction current. It is defined as the rate of change of the electric flux through a given surface and allows the continuity of current in scenarios such as charging or discharging a capacitor, where no physical current flows between the plates. This concept is crucial when analyzing situations with time-dependent electric fields and their magnetic consequences.

Ampère–Maxwell Law

The Ampère–Maxwell law is a generalization of Ampère’s circuital law that includes the displacement current. It relates the curl of the magnetic field to the sum of the conduction current and the displacement current through a surface bounded by a closed loop. This law is essential for calculating magnetic fields in scenarios with both steady currents and time-varying electric fields, such as those encountered in capacitor circuits during charging or discharging.

Magnetic Field Distribution in Capacitor Geometries

In capacitor geometries, the symmetry of the system is exploited to determine the magnetic field induced by both conduction and displacement currents. The field's magnitude can vary with distance from the central axis of the capacitor. By applying Ampère–Maxwell law to loops of different radii, one can determine how the induced magnetic field changes both inside the gap between the capacitor plates and in regions outside the capacitor, which is key to solving problems that involve non-uniform distributions of the induced magnetic field.

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