Answer to Question #128730 in Electricity and Magnetism for Faisal Khan

Question

In eq. E=1/(4πε_o )∫ρ_v R" " /R^3 dv', let ρ_v to be known. Explain in detail how you would numerically evaluate the vector integral to obtain E

Answer 1

As per the given question,

$E=\frac{1}{4\pi \epsilon_o}\int \frac{\rho_v R}{R^3}dv$

$=\frac{1}{4\pi \epsilon_o}\int \frac{\frac{4 \pi}{3}\rho_v R}{\frac{4 \pi}{3}R^3}dv$

$=\frac{1}{4\pi \epsilon_o}\int \frac{\frac{4 \pi}{3}\rho_v R}{V}dv$

$=\frac{1}{4\pi \epsilon_o}\times\frac{4 \pi}{3}\times\int \frac{\rho_v R}{V}dv$

$=\frac{1}{3 \epsilon_o}\times \rho_v R\int \frac{1}{V}dv$

$=\frac{\rho_v R}{3 \epsilon_o}\times \ln V+c$