A capacitor with circular plates of radius a separated by distance d is being ch

Question

A capacitor with circular plates of radius a separated by distance d is being charged by a battery with emf e m f. A vacuum exists between the plates of the capacitor. The constructed circuit has resistance R in series with the capacitor, and the capacitor is initially uncharged when a switch is closed, completing the circuit and causing a current to flow (an RC series circuit). Find an expression for the magnitude of the magnetic field between the plates at a distance r from the axis through the center of the plates. Assume r < a. (Use any variable or symbol stated above along with the following as necessary: t, C, 0, and 0. To represent e m f, use E. Do not substitute numerical values; use variables only.)

Explanation / Answer

B.ds = u0 e0 d(electric flux)/dt = u0 e0 d(AE)dt

B ds = u0 e0 A dE/dt

capacitance of capacitor, C = e0 pi a^2 / d

Voltage across capacitor. Vc = Emf [1 - e^(-t/RC)]

E = Vc/d = Emf [1 - e^(-t/RC)] / d

and A = pi a^2

B (2 pi r) = u0 e0 pi a^2 Eo [ e^(-t/RC)] / (d RC)

B = u0 e0 a^2 E e^(-/RC) / (2 d r R C )

putting C

B = {u0 Eo e^(-t d / e0 pi a^2 R ) } / { 2 pi r R }

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