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Q.

Three concentric conducting spherical shells $A, B a n d C$ have radii $r,$ $2 r$ and $3 r$ and possess charges $Q_{1}, Q_{2} a n d Q_{3}$ respectively. The innermost and the outermost shells are earthed as shown in the figure. select the mathematical relations between the charges that are correct.

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a

$Q_{1} = - \frac{Q_{2}}{4}$

b

$Q_{1} + Q_{3} = - Q_{2}$

c

$\frac{Q_{3}}{Q_{1}} = 3$

d

$\frac{Q_{3}}{Q_{1}} = 6$

answer is A, B, C.

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Detailed Solution

Potential of innermost shell is zero, so

$\frac{Q_{1}}{r} + \frac{Q_{2}}{2 r} + \frac{Q_{3}}{3 r} = 0 6 Q_{1} + 3 Q_{2} + 2 Q_{3} = 0$

similarly, potential of the outermost shell is also zero. (in)

$\frac{Q_{1}}{3 r} + \frac{Q_{2}}{3 r} + \frac{Q_{3}}{3 r} = 0$

$\Rightarrow Q_{1} + Q_{3} = - Q_{2}$

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, (1), (2) and (3) are correct.

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Three concentric conducting spherical shells A, B and C have radii r, 2 r and 3r and possess charges Q1, Q2 and Q3 respectively. The innermost and the outermost shells are earthed as shown in the figure. select the mathematical relations between the charges that are correct.