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Question

Two identical thin rings, each of radius R metres are coaxially placed at distance R metres apart. If Q1 coulumb and Q2 coulumb are respectively, the charges uniformly spread on the two rings, the work done in moving a charge q from the centre of one ring to that of the other

A
zero
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B
q(Q1Q2)(21)42πϵ0r
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C
q2(Q1+Q2)42πϵ0r
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D
q(Q1Q2)(2+1)42πϵ0r
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Solution

The correct option is B q(Q1Q2)(21)42πϵ0r

Potential due to an elemental charge dQ on a ring of radius r at a distance x on its axis is given by

V0dV=Q0kdQr2+x2
V=kQr2+x2



Work done in moving charge q from the centre of one ring to the other = Change in potential energy of the charge

i.e W=ΔU=UfUi=UAUB

UA=kQ1qR+kQ2q2R
=kqR(Q1+Q22)

UB=kQ2qR+kQ1q2R
=kqR(Q2+Q12)

W=UAUB=kqR(Q1Q2)(112)
=q(Q1Q2)(21)42πϵ0r

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