Question

Problem 1: Gauss's Law A circular capacitor of spacing d and radius R is in a circuit carrying the steady current i shown. r → At time t-0 it is uncharged 2. Find the potential at P, V(t), given that the potential at the right hand plate is fixed at 0 3. Find the magnetic field B(t) at P 4. Find the total field energy between the plates U(t) P36-7

1. Find the electric Field E at P Vs. Time
2. Find the potential at P, V(t), given that the potential at the right hand plate is fixed at 0
3. Find the magnetic field B(t) at P
4. Find the total field enery between the plate U(t)

Please help me, especially for 3 problem that i upload.

Answer 1

E = sigma/a_0 = q/pi R^2 epsilon_0_0 Assume a charge q on the left plate(- q on the right) Gauss�s law surfaceintegral E^rightarrow dA^rightarrow = EA = Q_1B/epsilon_0 = sigma A/epsilon_0 since q(t = 0) = 0, q = 1t F^rightarrow (t) = it/pi R^2 epsilon_0 to the right Since the field is uniform V = E * distance V(t) = Vertical-bar E^rightarrow (t) Vertical-bar (d - d�) = it/pi R^2 epsilon_0(d - d�) check: There should be positive since its between a positive plate(left) and zero potential(right) Q_E = E_A = (it/pi R^2)pi r^2 d straightphi_E/dt = r^2i/R^2 Ampere� law contourintegral B^rightarrow dS^rightarrow = mu_0 I_enc + mu_0 epsilon_0 d straightphi_E/d epsilon 2 pi rB = 0 + mu_0 epsilon_0 r^2 i/R^2 B^rightarrow (t) = mu_0 i /2 pi R^2 out of the page \r\n E field Density V_E = E^2/2 = /2(it/pi R^2 epsilon_0)^2 B Field Density: V_B = B^2/2 mu_0 = 1/2mu_0(mu_0 r/2 pi R^2)^2 Total Energy V = triple integral_pi R^2 d(V_E + V_B)dV(integrate over cylinder)