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Question
Three point charges, + Q + 2Q and – 3Q are placed at the vertices of an equilateral triangle ABC of side l. If these charges are displaced to the mid-point A1, B1 and C1, respectively, find the amount of the work done in shifting the charges to the new locations.
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Solution
q1= +Q
q2= +2Q
q3= -3Q
r = l (for each side)
Intial potential energy of system
`U_1=1/(4piin_0l)[(q_1xxq_2)+(q_2xxq_3)+(q_3xxq_1)]`
`U_1=1/(4piin_0l)[(Qxx2Q)+(2Qxx(-3Q))+((-3Q) xxQ)]`
`U_1=(-7Q^2)/(4piin_0l`
These charges displaced to mid points then final potential energy of system,
`U_2=1/(4piin_0l/2)[(q_1xxq_2)+(q_2xxq_3)+(q_3xxq_1)]`
`U_2=2/(4piin_0l)[(Qxx2Q)+(2Qxx(-3Q))+((-3Q)xxQ)]`
`U_2=(-7Q^2)/(2piin_0l)`
Work done, W = U2 - U1
`W=(-7Q^2)/(2piin_0l)-(-7Q^2)/(4piin_0l)`
`W=(7Q^2)/(piin_0l)[(-1)/2-((-1)/4)]=(7Q^2)/(piin_0l)[(-1)/2+1/4]`
`W=(-7)/4(Q^2/(piin_0l))`
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