Answer to Question #128913 in Electricity and Magnetism for faisal

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 question,

$E=\frac{1}{(4πε_o )}\int \frac{\rho_v R}{R^3}dv$

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

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

$=\frac{\rho_o R\times 4 \pi}{4\pi \epsilon_o} \ln(v)+c$

$=\frac{\rho_o R}{\epsilon_o}\ln (v)+c$

Hence, the required electric field will be calculated from the above.