The charge on a parallel plate capacitor is varying as q=q0 sin2 π·n t The plates are very large

step 1 In this problem I can write the value of ID is equal to DQ by DT which can be further written as

The charge on a parallel plate capacitor is varying as q=q0...

The charge on a parallel plate capacitor is varying as $q=q_{0} \sin 2 \pi \cdot n t$ The plates are very large and close together. Neglecting the edge effects, the displacement current through the capacitor is (A) $\frac{q}{\varepsilon_{0} A}$ (B) $\frac{q_{0}}{\varepsilon_{0}} \sin 2 \pi n t$ (C) $2 \pi n q_{0} \cos \pi n t$ (D) $\frac{2 \pi n q_{0}}{\varepsilon_{0}} \cos 2 \pi n t$

The charge on a parallel plate capacitor is varying as $q=q_{0} \sin 2 \pi \cdot n t$ The plates are very large and close together. Neglecting the edge effects, the displacement current through the capacitor is
(A) $\frac{q}{\varepsilon_{0} A}$
(B) $\frac{q_{0}}{\varepsilon_{0}} \sin 2 \pi n t$
(C) $2 \pi n q_{0} \cos \pi n t$
(D) $\frac{2 \pi n q_{0}}{\varepsilon_{0}} \cos 2 \pi n t$

Key Concepts

Parallel Plate Capacitor

A parallel plate capacitor is an electrical component consisting of two large, closely spaced conducting plates separated by a small distance. When a charge is placed on the plates, an approximately uniform electric field is created between them (neglecting edge effects). The relationship between the charge q on the capacitor and the electric field E between the plates is given by E = q/(??A), where A is the area of the plate and ?? is the permittivity of free space.

Displacement Current

Displacement current is a concept introduced by Maxwell to ensure the continuity of current in circuits with time?varying electric fields. It is defined as I_D = ??(d?_E/dt), where ?_E is the electric flux. In the case of a parallel plate capacitor, since the electric flux is directly related to the charge on the plates (?_E = q/??), the displacement current becomes equal to the time rate of change of the charge, i.e. I_D = dq/dt.

Time-varying Charge and Sinusoidal Functions

When the charge on a capacitor varies sinusoidally with time, such as in the form q = q?sin(?t), its time derivative results in a cosine function. Since the displacement current is given by the derivative dq/dt, the process of differentiating a sinusoidal function (with respect to time) introduces both an amplitude factor (?q?) and a phase change from sine to cosine.

Maxwell’s Correction to Ampère’s Law

Maxwell’s correction, which introduces the displacement current term into Ampère’s law, is fundamental in unifying the laws of electricity and magnetism, particularly when dealing with time-varying fields. This correction allows for the proper accounting of magnetic fields in situations where there is no actual charge motion (conduction current) across a gap, as in a charging or discharging capacitor, by incorporating the effect of a changing electric field.

Please give Ace some feedback