A parallel-plate capacitor with circular plates of radius R=16 mm and gap width d=5.0 mm has a u

step 1 In this question, we have a parallel plate capacitor with circular plates with a uniform electri

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

A parallel-plate capacitor with circular plates of radius $R=16 \mathrm{mm}$ and gap width $d=5.0 \mathrm{mm}$ has a uniform electric field between the plates. Starting at time $t=0,$ the potential difference between the two plates is $V=(100 \mathrm{V}) e^{-t / t}$ , where the time con- stant $\tau=12 \mathrm{ms} .$ At radial distance $r=0.80 R$ from the central axis, what is the magnetic field magnitude (a) as a function of time for $t \geq 0$ and $(\mathrm{b})$ at time $t=3 \tau$ ?

A parallel-plate capacitor with circular plates of radius $R=16 \mathrm{mm}$ and gap width $d=5.0 \mathrm{mm}$ has a uniform electric field
between the plates. Starting at time $t=0,$ the potential difference
between the two plates is $V=(100 \mathrm{V}) e^{-t / t}$ , where the time con-
stant $\tau=12 \mathrm{ms} .$ At radial distance $r=0.80 R$ from the central axis, what is the magnetic field magnitude (a) as a
function of time for $t \geq 0$ and $(\mathrm{b})$ at time $t=3 \tau$ ?

Key Concepts

Parallel-Plate Capacitor

A parallel-plate capacitor consists of two conductive plates separated by a gap, with an electric field established between them when a potential difference is applied. This concept is fundamental in electrostatics and circuit analysis, providing insight into energy storage, field distribution, and the effects of plate geometry on the behavior of the system.

Time-Varying Electric Field

A time-varying electric field occurs when the voltage applied to a capacitor changes with time. Such fields are crucial in understanding transient responses in capacitors, particularly during charging or discharging processes, and they play a key role in inducing currents and consequent magnetic effects in nearby regions.

Displacement Current

Displacement current is a term introduced by Maxwell to account for the effect of a changing electric field in regions where no conduction current exists. This concept bridges the gap in Ampère’s law for time-dependent fields, ensuring the continuity of current in scenarios such as the changing fields within a capacitor, and is essential for correctly predicting the associated magnetic field.

Maxwell’s Correction to Ampère’s Law

Maxwell’s correction to Ampère’s law incorporates the displacement current, thus extending the law to apply to time-dependent situations. This modification allows for the accurate calculation of magnetic fields in systems where the electric field varies with time, making it integral for analyzing electromagnetic phenomena in circuits like a discharging capacitor.

Magnetic Field Induction by Displacement Current

The magnetic field induced by a displacement current arises in response to a changing electric field and is obtained by applying the modified Ampère’s law. This concept explains how magnetic fields can be generated in regions devoid of free conduction current, such as between the plates of a capacitor during transient processes, and is key to linking electric field dynamics with magnetic field production.

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