Question

A solid nonconducting sphere of radius R carries a uniform charge density throughout its volume. At a radial distance r1 = R/4 from the center, the electric field has a magnitude E0.
What is the magnitude of the electric field at a radial distance r2 = 4R?

A) E0/2

B) 4E0

C) E0

D) E0/4

E) 2E0

Answer 1

Answer:D

Explanation:

If Q is the total charge on the sphere then

charge density $\rho =\frac{Q}{\frac{4}{3}\pi R^3}$

Charge enclosed up to $r=\frac{R}{4}$

$q_{enclosed}=\frac{Q}{\frac{4}{3}\pi R^3}\times \frac{4}{3}\pi (\frac{R}{4})^3$

$q_{enclosed}=\frac{Q}{64}$

Applying Gauss law

$EA=\frac{q_{enclosed}}{\epsilon }$

$E=\frac{\frac{Q}{64}}{4\pi (\frac{R}{4})^2}$

$E=\frac{1}{4}\times \frac{Q}{4\pi \epsilon }$

and Electric field at $r=\frac{R}{4}$

$E_0=\frac{1}{4}\times \frac{Q}{4\pi \epsilon }$

at $r=4 R$

charge enclosed is Q

applying Gauss law

$E\cdot A=\frac{q_{enclosed}}{\epsilon }$

$E=\frac{q_{enclosed}}{4\pi (4R)^2\epsilon }$

$E=\frac{1}{4}\times \frac{Q}{4\pi \epsilon }\times \frac{1}{4}$

$E=\frac{E_0}{4}$