An electrostatic field in a region is radially outward with magnitude E = α r , where α is a

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1. Electric Charges and Fields

medium

An electrostatic field in a region is radially outward with magnitude $E$ = $\alpha r$ , where $\alpha $ is a constant and $r$ is radial distance. The charge contained in a sphere of radius $R$ in this region (centred at the origin) is

A

$\frac{2}{3}\pi {\varepsilon _0}\alpha {R^3}$

B

$4\pi {\varepsilon _0}\alpha {R^3}$

C

$8\pi {\varepsilon _0}\alpha {R^3}$

D

$0$

Solution

$\mathrm{E} \times 4 \pi \mathrm{r}^{2}=\frac{\int \rho \times 4 \pi \mathrm{r}^{2} \mathrm{d} \mathrm{r}}{\epsilon_{0}}$

$4 \pi \alpha r^{3}=4 \pi \int \frac{\rho r^{2}}{\epsilon_{0}} d r=\frac{Q}{\epsilon_{0}}$

$Q=4 \pi \epsilon_{0} \alpha r^{3}$

Standard 12

Physics

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