The electrostatic potential in a charged spherical region of radius $r$ varies as $V = ar^{3} + b$,

Question

The electrostatic potential in a charged spherical region of radius $r$ varies as $V = ar^{3} + b$, where $a$ and $b$ are constants. The total charge in the sphere of unit radius is $\alpha \times \pi a \in_{0}$. The value of $\alpha$ is $\_\_\_\_$ .

(permittivity of vacuum is $\in_{0}$ )

Accepted Answer

(4) -8Sol. $v = {ar}^{3} + b$$$\begin{matrix} & E = - \frac{dv}{dr} = - 3{ar}^{2} \ & \phi_{	ext{closed~}} = \frac{q_{enc}}{\varepsilon_{0}} \ & q_{enc} = \varepsilon_{0} \cdot E \cdot 	ext{ }A \ & \ = \varepsilon_{0}\left( - 3a \cdot (1)^{2} ight)4\pi(1)^{2} \ & \ = - 12\pi a\varepsilon_{0} \ & \ 	herefore x = - 12 \end{matrix}$$

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