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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.

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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 $R$?


$\vec E = 0$


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


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


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


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

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