Question

Electromagnetics HW help!!

Quick Response please! (within 45 min)

For the vector field ? = $\hat{R} \frac{1}{R^2}$
1
??2 , evaluate both sides of the divergence theorem
for the region enclosed between the spherical shells defined by R = 1 and R = 4.
(a) Setup equations (30 points)
(b) Show work (20 points)
(c) Final answer (10 points)

Answer 1

Given D^bar = 1/R^2 a^bar r (a) From Gauss law Q = contourintegral_s D^bar middot ds From divergence theorem Q = contourintegral_s D^bar middot ds Q = contourintegral_s D^bar middot ds = nabla middot D^bar dv b) Let contourintegral_s D^bar middot ds = doubleintegral D_r^bar middot d^bar S_r = integral^pi_0 integral^2pi_0 1/R^2 a_r^bar middot R^2 sin theta d theta d phi a_r^bar (a_r^bar middot a_r^bar = 1) = (-cos theta)^pi_0 (phi)^2pi_0 = (2) (2 pi) = 4 pi Ans. \r\n Now Consider integral_v nabla middot Ddv = integral^2pi_0 integral^pi_0 integral^4_0 (1/R^2 partial differential/partial differential R (R^2 (1/R^2)))R^2 sin theta dRd theta d phi = integral^2pi_0 integral^pi_0 integral^4_0 partial differential/partial differential R (1) sin theta d R d theta d phi = 1 - (-cos theta)^pi_0 (phi)^2pi_0 = 4 pi Ans. (c) Ans: 4 pi.