SATHEE: Electrostatics 2 Question 18

Electrostatics 2 Question 18

19. A uniformly charged solid sphere of radius $R$ has potential $V_{0}$ (measured with respect to $\infty$ ) on its surface. For this sphere, the equipotential surfaces with potentials $\frac{3 V_{0}}{2}, \frac{5 V_{0}}{4}, \frac{3 V_{0}}{4}$ and $\frac{V_{0}}{4}$ have radius $R_{1}, R_{2}, R_{3}$ , and $R_{4}$ respectively. Then,

(2015 Main)

(a) $R_{1} \neq 0$ and $(R_{2} - R_{1}) > (R_{4} - R_{3})$

(b) $R_{1} = 0$ and $R_{2} > (R_{4} - R_{3})$

(d) $R_{1} = 0$ and $R_{2} < (R_{4} - R_{3})$

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Answer:

Correct Answer: 19. (b,c)

Solution:

$V_{0} =$ potential on the surface $= \frac{K q}{R}$

where, $K = \frac{1}{4 π ε_{0}}$ and $q$ is total charge on sphere.

Potential at centre $= \frac{3}{2} \frac{K q}{R} = \frac{3}{2} V_{0}$

Hence, $R_{1} = 0$

From centre to surface potential varies between $\frac{3}{2} V_{0}$ and $V_{0}$

From surface to infinity, it varies between $V_{0}$ and $0, \frac{5 V_{0}}{4}$ will

be potential at a point between centre and surface. At any point, at a distance $r (r \leq R)$ from centre potential is given by

$\begin{aligned} V & = \frac{K q}{R^{3}} \frac{3}{2} R^{2} - \frac{1}{2} r^{2} \\ = \frac{V_{0}}{R^{2}} \frac{3}{2} R^{2} - \frac{1}{2} r^{2} \end{aligned}$

Putting $V = \frac{5}{4} V_{0}$ and $r = R_{2}$ in this equation, we get

$R_{2} = \frac{R}{\sqrt{2}}$

$\frac{3 V_{0}}{4}$ and $\frac{V_{0}}{4}$ are the potentials lying between $V_{0}$ and zero hence these potentials lie outside the sphere. At a distance $r (\geq R)$ from centre potential is given by $V = \frac{K q}{r} = \frac{V_{0} R}{r}$

Putting $V = \frac{3}{4} V_{0}$ and $r = R_{3}$ in this equation we get, $R_{3} = \frac{4}{3} R$

Further putting $V = \frac{V_{0}}{4}$ and $r = R_{4}$ in above equation,

we get $R_{4} = 4 R$

Thus, $R_{1} = 0, R_{2} = \frac{R}{\sqrt{2}}, R_{3} = \frac{4 R}{3}$ and $R_{4} = 4 R$ with these values, option (b) and (c) are correct.

Electrostatics 2 Question 18

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Electrostatics 2 Question 18

19. A uniformly charged solid sphere of radius R has potential V0 (measured with respect to ∞ ) on its surface. For this sphere, the equipotential surfaces with potentials 3V02,5V04,3V04 and V04 have radius R1,R2,R3, and R4 respectively. Then,

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