Three concentric conducting spherical shells A, B and C have radii r, 2r 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.

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

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

$\frac{Q_{3}}{Q_{2}} = - \frac{1}{3}$

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

answer is A, B, C.

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Detailed Solution

Potential of innermost shell is zero, so
$\frac{Q_{1}}{r} + \frac{Q_{2}}{2 r} + \frac{Q_{3}}{3 r} = 0$
$\Rightarrow 6 Q_{1} + 3 Q_{2} + 2 Q_{3} = 0.......... (1)$
Similarly, Potential of the outermost shell is also zero.so,
$\Rightarrow \frac{Q_{1}}{3 r} + \frac{Q_{2}}{3 r} + \frac{Q_{3}}{3 r} = 0$
$\Rightarrow Q_{1} + Q_{3} = - Q_{2} ................... (2)$
Solving equations (1) and (2), we get
$Q_{1} = - \frac{Q_{2}}{4}, \frac{Q_{3}}{Q_{1}} = 3 a n d \frac{Q_{3}}{Q_{2}} = \frac{- 3}{4}$