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Two small metal spheres A and B, each of...

Two small metal spheres A and B, each of radius r and supported on insulating stands, located at a distance `a (a gt gt r)` from each other are connected by a thin conducting wire. A point charge q is brought near the spheres at distance `l (l gt gt r)` on the line joining the centers of the spheres. What are the moduli of charge induced on the spheres ?
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Text Solution

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As A and B are connected, `V_A = V_B`.
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Let charges `-q'` and `+q'` be induced on A and B, respectively.
`V_A` = potential due to its own charge + potential due to charge placed at O + potential due to charge placed at B
=`(K) ((-q'))/(r) + (K) q/l + (K) ((q'))/a` ... (i)
Similarly, potential of sphere B is
`V_B` = potential due to its own charge + potential due to q + potential due to -q
=`(K) (q')/(r) + (K) q/((l + a)) + (K) ((-q'))/a` ...(ii)
As `V_A = V_B`,
`K ((-q')/(r)) + K(q/(l)) + K((q')/a)`
=`K((q')/(r)) + K (q/(l + a)) + K ((-q')/a)`
or `(2 q')/( r) - (2 q')/a = q/(l) - q/(l + a) or q' = a/(2) [(-a^2 r)/(l (l + a)(r - a))]`
As `r lt lt a`,
`q' = a/(2) [(-a^2 r)/(l( l + a)(-a))] = -(q ar)/(l(l + a))`.
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