The charge of a parallel plate capacitor is varying as. q=q0 sin 2 πf t. The plates are very lar

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

The charge of a parallel plate capacitor is varying as. $q=q_{0}$ sin $2 \pi f t$. The plates are very large and close together (area $=A$, separation $=d) .$ Neglecting edge effects, the displacement current through the capacitor is (a) $\frac{d}{A \varepsilon_{0}}$ (b) $\frac{d}{\varepsilon_{0}} \sin 2 \pi f$ (c) $2 \pi f q_{0} \cos 2 \pi f$ (d) $\frac{2 \pi f q_{0}}{\varepsilon_{Q}} \cos 2 n f$

The charge of a parallel plate capacitor is varying as. $q=q_{0}$ sin $2 \pi f t$. The plates are very large and close together (area $=A$, separation $=d) .$ Neglecting edge effects, the displacement current through the capacitor is
(a) $\frac{d}{A \varepsilon_{0}}$
(b) $\frac{d}{\varepsilon_{0}} \sin 2 \pi f$
(c) $2 \pi f q_{0} \cos 2 \pi f$
(d) $\frac{2 \pi f q_{0}}{\varepsilon_{Q}} \cos 2 n f$

Key Concepts

Electric Field in a Parallel Plate Capacitor

For a parallel plate capacitor with large plates and negligible edge effects, the electric field between the plates is uniform and is given by the charge on the plates divided by the product of the permittivity of free space (??) and the plate area (A). This relationship is key in linking the change in charge to the change in the electric field, which in turn is related to the displacement current via Maxwell's equations.

Differentiation of Sinusoidal Functions

When the charge on the capacitor varies sinusoidally, such as in a sine function, differentiating this time-varying function involves applying the rules of differentiation for trigonometric functions. The derivative of a sine function leads to a cosine function with an additional factor from the chain rule. This is important for determining the explicit expression for the displacement current as a function of time.

Displacement Current

Displacement current is a concept introduced by Maxwell to account for the effect of a changing electric field in regions where there is no physical charge flow. It is defined as the time derivative of the electric flux multiplied by the permittivity of free space (??). This current term is crucial in modifying Ampere's law so that it holds true in situations involving time-varying fields, such as between the plates of a capacitor with a changing voltage.

Time-Varying Charge and its Derivative

In circuits where the charge on a component such as a capacitor varies with time, the current can be determined by taking the derivative of the charge with respect to time. This is particularly pertinent when the charge varies sinusoidally, since differentiating a sine function results in a cosine function. This relationship is essential for understanding how the changing charge translates into a displacement current in the capacitor.

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