The charge on a parallel plate capacitor is varying as q=qo sin2 πn t . the plates are very larg

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

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

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

Key Concepts

Displacement Current

Displacement current is a term introduced by Maxwell to extend Ampere’s law to time-varying electric fields. It is defined as the time rate of change of electric flux through a surface, multiplied by the permittivity of the medium. This concept provides a way to account for the effects of changing electric fields in regions where no physical charge carriers are moving, such as the gap between capacitor plates in a capacitor with a varying charge.

Electric Field in a Parallel Plate Capacitor

In a parallel plate capacitor, the electric field between the plates is uniform and directly related to the charge on the plates. This relation is given by E = q/(??A), where q is the charge, ?? is the permittivity of free space, and A is the area of a plate. This formula is crucial in understanding how the electric field varies when the charge is time-dependent, thus affecting the displacement current through the capacitor.

Time Derivative of Sinusoidal Functions

When a physical quantity varies sinusoidally with time, its time derivative involves a phase shift and multiplication by the angular frequency. For instance, if a quantity is given by a sine function, its derivative becomes a cosine function multiplied by the frequency factor. This mathematical relationship is essential in translating a time-varying charge into a corresponding current, such as the displacement current, by differentiating the sinusoidal charge expression with respect to time.

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