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Consider a sphere of radius $\mathrm{R}$ which carries a uniform charge density $\rho .$ If a sphere of radius $\frac{\mathrm{R}}{2}$ is carved out of it, as shown, the ratio $\frac{\left|\overrightarrow{\mathrm{E}}_{\mathrm{A}}\right|}{\left|\overrightarrow{\mathrm{E}}_{\mathrm{B}}\right|}$ of magnitude of electric field $\overrightarrow{\mathrm{E}}_{\mathrm{A}}$ and $\overrightarrow{\mathrm{E}}_{\mathrm{B}}$ respectively, at points $\mathrm{A}$ and $\mathrm{B}$ due to the remaining portion is
$\frac{18}{54}$
$\frac{21}{34}$
$\frac{17}{54}$
$\frac{18}{34}$
Solution
Fill the empty space with $+\rho$ and $-\rho$ charge density.
$\left|\mathrm{E}_{\mathrm{A}}\right|=0+\frac{\operatorname{k\rho} \cdot \frac{4}{3} \pi\left(\frac{\mathrm{R}}{2}\right)^{3}}{\left(\frac{\mathrm{R}}{2}\right)^{2}}=\operatorname{k\rho} \frac{4}{3} \pi\left(\frac{\mathrm{R}}{2}\right)$
$\left|\mathrm{E}_{\mathrm{B}}\right|=\frac{\mathrm{k} \rho \cdot \frac{4}{3} \pi \mathrm{R}^{3}}{\mathrm{R}^{2}}-\frac{\mathrm{k} \rho \cdot \frac{4}{3} \pi\left(\frac{\mathrm{R}}{2}\right)^{3}}{\left(\frac{3 \mathrm{R}}{2}\right)^{2}}$
$=\operatorname{k\rho} \frac{4}{3} \pi \mathrm{R}-\mathrm{k} \rho \frac{4}{3} \pi \frac{\mathrm{R}}{18}=\mathrm{k} \rho \cdot \frac{4}{3} \pi\left(\frac{17 \mathrm{R}}{18}\right)$
$\frac{E_{A}}{E_{B}}=\frac{9}{17}=\frac{18}{34}$
Similar Questions
The electric field $E$ is measured at a point $P (0,0, d )$ generated due to various charge distributions and the dependence of $E$ on $d$ is found to be different for different charge distributions. List-$I$ contains different relations between $E$ and $d$. List-$II$ describes different electric charge distributions, along with their locations. Match the functions in List-$I$ with the related charge distributions in List-$II$.
| List-$I$ | List-$II$ |
| $E$ is independent of $d$ | A point charge $Q$ at the origin |
| $E \propto \frac{1}{d}$ | A small dipole with point charges $Q$ at $(0,0, l)$ and $- Q$ at $(0,0,-l)$. Take $2 l \ll d$. |
| $E \propto \frac{1}{d^2}$ | An infinite line charge coincident with the x-axis, with uniform linear charge density $\lambda$ |
| $E \propto \frac{1}{d^3}$ | Two infinite wires carrying uniform linear charge density parallel to the $x$-axis. The one along ( $y=0$, $z =l$ ) has a charge density $+\lambda$ and the one along $( y =0, z =-l)$ has a charge density $-\lambda$. Take $2 l \ll d$ |
| plane with uniform surface charge density |
