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Four charges q(1)=q(2)=-q(3)=-q(4)=q a...

Four charges `q_(1)=q_(2)=-q_(3)=-q_(4)=q` are arranged in the y-plane forming an elastic quadrupole. Bringing these charges from infinity, what is the needed work to compose such an arrangement?

A

`k(q)/(a)(1-2sqrt(2))`

B

`k(q^(2))/(a) (2sqrt(2))`

C

`k(q^(2))/(2a)(1-2sqrt(2))`

D

`k(q^(2))/(a) (1-2sqrt(2))`

Text Solution

Verified by Experts

The correct Answer is:
D
The electrostatic energy required to compose such a system is given as follows
`U=(k)/(2) sum_(i ne j) (q_(i)q_(j))/(r_(ij))`
Where `r_(ij)=|r_(i)-r_(j)|`
Inserting data from the figure to equation (i), we get,
`U=((q_(1)q_(2))/(r_(12))+(q_(1)q_(3))/(r_(13))+(q_(1)q_(4))/(r_(14))+(q_(2)q_(4))/(24)+(q_(2)q_(3))/(r_(23))+(q_(3)q_(4))/(r_(34)))k`
`=k(q^(2))/(a)((1)/(2)-(1)/(sqrt(2))-(1)/(sqrt(2))-(1)/(sqrt(2))-(1)/(sqrt(2))+(1)/(sqrt(2)))=k(q^(2))/(a)(1-2sqrt(2))`
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