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Carnot Cycle

Three concent...

Question

Three concentric conducting spherical shells have radii $r, 2 r$ and $3 r$ and charges $q_{1}, q_{2}$ and $q_{3}$ respectively. Innermost and outermost shells are earthed as shown in the figure, then ratio of magnitudes of $q_{1}, q_{2}$ and $q_{3}$ is

3 : 4 : 3

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1 : 3 : 3

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4 : 4 : 3

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1 : 4 : 3

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Solution

The correct option is D $1 : 4 : 3$

Potential of innermost shell is zero.
$∴ \frac{q_{1}}{r} + \frac{q_{2}}{2 r} + \frac{q_{3}}{3 r} = 0$
$6 q_{1} + 3 q_{2} + 2 q_{3} = 0$ ... (1)
Similarly, potential on outermost shell is also zero.
$∴ \frac{q_{1}}{3 r} + \frac{q_{2}}{3 r} + \frac{q_{3}}{3 r} = 0$
$q_{1} + q_{3} = - q_{2}$ ... (2)
Solving equations (1) and (2), we get
$q_{1} = - \frac{q_{2}}{4}, \frac{q_{3}}{q_{1}} = 3$ and $\frac{q_{3}}{q_{2}} = - \frac{3}{4}$
Hence $q_{1} : q_{2} : q_{3} = 1 : - 4 : 3$
Ratio of magnitudes of charges is $1 : 4 : 3$

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