Consider two charged metallic spheres

Question:

Consider two charged metallic spheres $S_{1}$ and $S_{2}$ of radii $R_{1}$ and $R_{2}$, respectively. The electric fields $E_{1}$ (on $S_{1}$ ) and $E_{2}$ (on $S_{2}$ ) on their surfaces are such that $E_{1} / E_{2}=R_{1} / R_{2}$. Then the ratio $V_{1}\left(\right.$ on $\left.S_{1}\right) / V_{2}\left(\right.$ on $\left.S_{2}\right)$ of the electrostatic potentials on each sphere is:

  1. (1) $R_{1} / R_{2}$

  2. (2) $\left(R_{1} / R_{2}\right)^{2}$

  3. (3) $\left(R_{2} / R_{1}\right)$

  4. (4) $\left(\frac{R_{1}}{R_{2}}\right)^{3}$


Correct Option: , 2

Solution:

(2) Electric field at a point outside the sphere is given by

$E=\frac{1 Q}{4 \pi \in_{0} r^{2}}$

But $\rho=\frac{Q}{\frac{4}{3} \pi R^{3}}$

$\therefore E=\frac{\rho R^{3}}{3 \in_{0} r^{2}}$

At surface $r=R$

$\therefore E=\frac{\rho R^{3}}{3 \epsilon_{0}}$

Let $\rho_{1}$ and $\rho_{2}$ are the charge densities of two sphere.

$E_{1}=\frac{\rho R_{1}}{3 \varepsilon_{0}}$ and $E_{2}=\frac{\rho_{2} R_{2}}{3 \varepsilon_{0}}$

$\because \frac{E_{1}}{E_{2}}=\frac{\rho_{1} R_{1}}{\rho_{2} R_{2}}=\frac{R_{1}}{R_{2}}$

This gives $\rho_{1}=\rho_{2}=\rho$

Potential at a point outside the sphere

$V=\frac{1}{4 \pi \varepsilon_{0}} \frac{Q}{r}$

$=\frac{\rho R^{3}}{3 \varepsilon_{0} r}\left(\because \rho=\frac{Q}{\frac{4}{3} \pi R^{3}}\right)$

At surface, $r=R$

$V=\frac{\rho R^{2}}{3 \varepsilon_{0}}$ so, $V_{1}=\frac{\rho R_{1}^{2}}{3 \varepsilon_{0}}$ and $V_{2}=\frac{\rho R_{2}^{2}}{3 \varepsilon_{0}}$

$\therefore \frac{V_{1}}{V_{2}}=\left(\frac{R_{1}}{R_{2}}\right)^{2}$

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