A parallel-plate capacitor has circular plates with radius...

A parallel-plate capacitor has circular plates with radius $50 \mathrm{cm}$ and spacing $1.0 \mathrm{mm}$. A uniform electric field between the plates is changing at the rate of $1.0 \mathrm{MV} / \mathrm{m}$.s. Find the magnetic field between the plates (a) on the symmetry axis, (b) $15 \mathrm{cm}$ from the axis, and (c) $150 \mathrm{cm}$ from the axis.

Key Concepts

Symmetry in Electromagnetism

Utilizing symmetry is a powerful technique in solving electromagnetic problems. In systems with a high degree of symmetry, such as cylindrical or spherical symmetry, evaluating integrals and applying laws such as Ampere's law becomes simpler. Recognizing symmetry helps in deducing the direction and magnitude of fields without solving complex integrals for every point in space.

Ampere?Maxwell Law

This law extends Ampere's law by including the displacement current term, which accounts for changing electric fields. It states that the line integral of the magnetic field around a closed loop is equal to the permeability of free space multiplied by the sum of the conduction current and the rate of change of electric flux through the loop. This extension is crucial for solving problems where the electric field varies with time, such as in capacitors during charging or discharging.

Displacement Current

Displacement current is a term introduced to represent the effect of a time-varying electric field in producing a magnetic field. Even in regions devoid of conduction current, such as the gap of a capacitor, the changing electric field behaves as if it were a current, thereby contributing to the magnetic field. This concept is central to understanding electromagnetic wave propagation and the behavior of fields in capacitors.

Parallel?Plate Capacitor

A parallel-plate capacitor consists of two conductive plates separated by a small distance, which stores electrical energy in the electric field established between them. Its geometry simplifies the analysis of the electric field, often assumed to be uniform between the plates, and it serves as an important model system for studying time-dependent fields and the creation of displacement currents.

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