A parallel plate capacitor has circular plates of radius 10.0 cm that are separated by a distan

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A parallel plate capacitor has circular plates of radius...

A parallel plate capacitor has circular plates of radius $10.0 \mathrm{~cm}$ that are separated by a distance of $5.00 \mathrm{~mm}$. The potential across the capacitor is increased at a constant rate of $1.20 \mathrm{kV} / \mathrm{s}$. Determine the magnitude of the magnetic field between the plates at a distance $r=4.00 \mathrm{~cm}$ from the center.

Key Concepts

Parallel Plate Capacitor

A parallel plate capacitor is a configuration consisting of two conductive plates separated by a dielectric (or vacuum) where an electric field is established when a voltage is applied across the plates. Its geometry allows for uniform electric fields in the central region, simplifying the analysis of electrical and magnetic field relationships when the capacitor is used in dynamic situations.

Electric Field

The electric field in a capacitor is defined as the force per unit charge between the plates and is directly related to the voltage difference and the geometry of the plates. In such systems, spatial uniformity of the field allows one to relate the voltage across the plates to the magnitude of the electric field, which is crucial when considering time-dependent changes.

Displacement Current

Displacement current is a concept introduced by Maxwell to account for a time-varying electric field in regions where there is no physical movement of charge. It is mathematically expressed as a term proportional to the rate of change of the electric field, and it contributes to the generation of magnetic fields in situations like charging capacitors.

Ampère-Maxwell Law

The Ampère-Maxwell Law extends Ampère's law by including the contribution of the displacement current in addition to conduction current. It states that the line integral of the magnetic field around a closed loop is proportional to the sum of the conduction and displacement currents passing through the loop. This law is fundamental for calculating magnetic fields in systems with time-varying electric fields.

Symmetry in Field Calculations

Symmetry in field calculations simplifies the integration of the magnetic field around a chosen loop. For a parallel plate capacitor with circular plates, considering a circular path centered on the axis leverages the cylindrical symmetry of the configuration, making it easier to relate the induced magnetic field to the enclosed displacement current.

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