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Three identical point charges Q are plac...

Three identical point charges Q are placed at the three corners of an equilateral triangle of side length a. Find the electric potential at points O and M due to the three charges.

Text Solution

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`cos30^(@)=(MC)/(OC)=((a)/(2))/(OC)`
`rArr OC=(a)/(2)(2)/(sqrt(3))=(a)/(sqrt(3))=OA=OB`
The potential at O is same for all charges, so we can multiply the potential due to any of the charges by three to get the net potential at point O. Hence, potential at point O is as follows:
`V_(O)=3(1)/(4pi epsilon_(0))(Q)/(OC)= (3Q)/(4pi epsilon_(0)((a)/(sqrt(3))))`
`rArr V_(O)=(3sqrt(3)Q)/(4pi epsilon_(0)a)`
At point M, the potential due to B and C are same, and the potential due to charge on A has different magnitude. Hence, the total potential at M is as follows:
`V_(M)=2(1)/(4pi epsilon_(0)) (Q)/(BM)+(1)/(4pi epsilon_(0)) (Q)/(AM)`
`rArr V_(M)=2(1)/(4pi epsilon_(0)) (Q)/((a)/(2))+(1)/(4pi epsilon_(0)) (Q)/(sqrt(a^(2)-(a^(2))/(4)))`
`rArr V_(M)=(1)/(4pi epsilon_(0)) (4Q)/(a) +(1)/(4pi epsilon_(0))(2Q)/(a sqrt(3))`
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