?

AP® Physics C: E & M

Free Version

Upgrade subject to access all content

Moderate

$Q_{enc}$ Sphere 5

APPHEM-KRX4SY

The diagram shows a sphere of radius $R$ that carries a charge density that is not uniform. The charge density varies according to radial distance $R$ from the center of the sphere according to the equation $\rho = \dfrac{C}{r}$.

The diagram also shows the cross-section of a Gaussian sphere of radius $a$ that is concentric with the charged sphere.

Which of the following expresses the charge enclosed in the Gaussian sphere?

A

$2\,\pi \,C\,{a}^{2}$

B

$\dfrac{4}{3}\,\pi \,C\,{a}^{2}$

C

$\dfrac{1}{2}\pi \,C\,{a}^{2}$

D

$\dfrac{4}{9}\,\pi \,C\,{a}^{3}$

E

$2\,\pi \,C\,a$