A short electric dipole of dipole moment $\overrightarrow{\mathrm{p}}$ is placed at a distance $\mathrm{r}$…

A short electric dipole of dipole moment $\overrightarrow{\mathrm{p}}$ is placed at a distance $\mathrm{r}$ from the centre of a solid metallic sphere of radius a $(< < \mathbf{r})$ as shown in the figure. The electric field intensity at the centre of sphere $\mathrm{C}$ due to induced charge on the sphere is

1. zero
2. $\frac{1}{4 \pi \varepsilon_{0}} \frac{2 \mathrm{p}}{\mathrm{r}^{3}}$ along CO
3. $\frac{1}{4 \pi \varepsilon_{0}} \frac{2 \mathrm{p}}{\mathrm{r}^{3}}$ along $\mathrm{OC}$
4. $\frac{1}{4 \pi \varepsilon_{0}} \frac{\mathrm{p}}{\mathrm{r}^{3}}$ along $\mathrm{CO}$

Solution

Electric field at $\mathrm{C}$ due to electric dipole
$=\frac{1}{4 \pi \varepsilon_{0}} \frac{2 \mathrm{p}}{\mathrm{r}^{3}}$ along $\mathrm{OC}$
Electric field at $\mathrm{C}$ due to induced charge must be equal and opposite to electric field due to dipole as net field at $\mathrm{C}$ is zero. Therefore, it is along CO.

Asked in: JEE Mains - Electrostatics - Test 5

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