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

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

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

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

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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