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The figure below shows a disk of radius $R$ with a uniform surface charge density $\sigma$. We are interested in the electric field at a distance $d$ along a perpendicular line from the center of the disk of charge.

Which of the expressions below correctly describes the integral that would need to be evaluated to determine the vector electric field a distance $d$ from the charged disk of radius $R$?

A

$\vec E = 0$

B

$\vec E = \int_{0}^{2 \pi} {\: d \theta} {\int_{0}^{R} {\cfrac {\sigma \: r \: d \: dr}{4 \pi \epsilon_0 (r^2 + d^2)^{1/2}}}}\widehat z$

C

$\vec E = \int_{0}^{2 \pi} {\: d \theta} {\int_{0}^{R} {\cfrac {\sigma \: r \: d \: dr}{4 \pi \epsilon_0 (r^2 + d^2)^{3/2}}}}\widehat z$

D

$\vec E = \int_{0}^{2 \pi} {\: d \theta} {\int_{0}^{R} {\cfrac {\sigma \: r \: d \: dr}{4 \pi \epsilon_0 (r^2 + d^2)}}}\widehat z$

E

$\vec E = \int_{0}^{2 \pi} {\: d \theta} {\int_{0}^{R} {\cfrac {\sigma \: r \: d \: dr}{4 \pi \epsilon_0 (r^2 + d^2)^{3/2}}}}$

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