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AP® Physics C: E & M

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Moderate

$Q_{enc}$ Cylinder 8

APPHEM-YXROH5

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

The diagram also shows the cross-section of a Gaussian cylinder of radius $a$ and length $L$ that is coaxial with the charged cylinder.

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

A

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

B

$2\,\pi \,C\,L\,\sqrt{a}$

C

$\dfrac{2}{5}\,\pi \,C\,L\,{a}^{\frac{5}{2}}$

D

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

E

$\pi \,C\,L\,{a}^{\frac{3}{2}}$