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## Homework Statement

This is an EM waves class, but we are given the following problem, which seems deceptively simple. I'd like some feedback to see if this is done correctly.

## Homework Equations

Consider a parallel plate capacitor connected with a resistor with resistance R, as shown in the figure (it's a simple circuit of a resistor R in series with a switch and parallel plate capacitor, I can recreate if necessary; there is no source). Suppose the plate area is A, the distance between the two plates is d, and between the two plates is a uniform, lossless dielectric material with permittivity ε and permeability μ

_{0}. Assume the plate dimension is much larger than the distance d, so you can assume the E field is uniform between the plates and vanishes outside.

A) Calculate the capacitance. Assume the initial voltage across the cap is V

_{0}, and the switch is closed at time t=0. (Voltage polarity is given in the figure.) Calculate the voltage across the capacitor as a function of time. Also calculate the power consumption of the resistor as a function of time.

B) Calculate the total EM energy stored within the cap as a function of time. What conclusion can you draw from your calculations?

## The Attempt at a Solution

A) [itex]V = \frac{Qd}{\varepsilon A}[/itex], where Q is the charge on the plate. As C = Q/V, I have

[tex]C= \frac{\varepsilon A}{d}[/tex]

And

[tex]V = V_0 e^{-\frac{t}{RC}} = V_0 e^{-\frac{td}{R\varepsilon A}}[/tex]

Finally, for power,

[tex]P = \frac{V^2}{R} = \frac{V_0^2}{R}e^{-2 \frac{td}{R\varepsilon A}}[/tex]

B) [tex] u_E = \frac{1}{2} \varepsilon E^2

= \frac{1}{2} \varepsilon \left(\frac{V_0}{d} e^{-\frac{td}{R \varepsilon A}}\right)^2

= \frac{1}{2} \varepsilon \left(\frac{V_0^2}{d^2}e^{-2\frac{td}{R \varepsilon A}}\right)[/tex]

And the 'conclusion' I would reach is that the energy is entirely stored within the electric field, and the permeability does not affect the parameters of the circuit.

Is all of this accurate or have I made any errors?

(Sorry if this should be in the basic board, please feel free to move it there if so.)

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