Four charges are arranged at the comers of a square ABCD of side d, as shown in Fig.
a. Find the work required to put together this arrangement
b. A charge `q_(0)` is brought to the centre E of the square, the four charges being held fixed at its corners. How much extra work is needed to do this?n

Explanation

Since the work done depends on the final arrangement of the charges, and not on how they are put togeth calculate work needed for one way of putting the charges at A, B, C and D. Suppose, first the charge +q is brought to A, and then the charges -q, +q, and -q are brought to B, C and D, respectively.
The total work needled oan be caloulated in steps:
i. Work needed to bring charge +q to A when no charge is present elsewhere, this is zero. i. Work nooded to bring - q to B when + q is at A. This is given by (charge at B) `xx` (eleectrostatic potential at B due to charge +q at A) `=-q xx q/(4pi epsi_(0) d)=-(q^(2))/(4pi epsi_(0)d)`
iii. Work needed to bring charge +q to C when +q is at A and - is at B. This is given by `"(charge at C) "xx` (potential at C due to charges at A and B)
`=+q ((+q)/(4pi epsi_(0)dsqrt2) +(-q)/(4pi epsi_(0) d))= (-q^(2))/(4pi epsi_(0)d) (1-1/sqrt2)`
iv. Work needed to bring -q to D when +q at A, - q at B, and + q at C. This is given by (charge at D) `xx` (potential at D due to charges at A, B and C)
`=-q ((+q)/(4pi epsi_(0)d) +(-q)/(4pi epsi_(0) dsqrt2) +(q)/(4pi epsi_(0) d))=(-q^(2))/(4pi epsi_(0) d) (2-1/sqrt2)`
Add the work done in steps (i), (ii), (iii) and (iv). The total work required is `=(-q^(2))/(4pi epsi_(0) d) {(0)+(1)(1-1/sqrt2) +(2-1/sqrt2)}=(-q^(2))/(4pi epsi_(0) d) (4-sqrt2)`
The work done depends only on the arrangement of the charges, and not how they are assembled. By definition, this is the total electrostatic energy of the charges.
b. The extra work necessary to bring a charge `q_(0)` to the point E when the four charges are at A, B, C and D is `q_(0) xx` (electrostatic potential at E due to the charges at A, B, C and D). The electrostatic potential at E is clearly zero since potential due to A and C is cancelled by that due to B and D. Hence no work is required to bring any charge to point E.