Four charges are arranged at the corners of the square PQRS of side a as shown in the figure . (a) Find the work required to assemble these charges in the given configuration . (b) Suppose a charge q is brought to the center of the square by keeping the four charges fixed at the corners how much extra work is required for this ?

Explanation

(a) The work done to arrange the charges in the corners of the square is independent of the way they are arranged . We can follow any order.
(i) First the charge +q is brought to the corner P . This requires no work since no charge is already present `W_(p)` = 0
(ii) Work required to bring the charge `-q` to the corner Q `=(-q)xx` potential at a point Q due to `+q` located at a point P . `W_(Q)=-qxx(1)/(4piepsilon_(0))(q)/(q)=-(1)/(4piepsilon_(0)) (q^(2))/(a)`
(iii) Work required tpo bring the charge +q to the corner R q`xx` potential at the point R due to charges at the point P and Q .
`W_(r)=qxx(1)/(4piepsilon_(0))(-(q)/(a)+(q)/(sqrt(2a)))=(1)/(4piepsilon_(0))(q^(2))/(a)(-1+(1)/(sqrt(2)))`
(iv) Work required to bring the fourth charge -q at the position S = q`xx` potential at the point S due the all the three charges at the three charges at the point P,Q and R .
`W_(s)=-qxx(1)/(4piepsilon_(0))((q)/(a)+(q)/(a)-(q)/(sqrt(a)))=-(1)/(4piepsilon)(q)/(a)(2-(1)/(sqrt(2)))`
(b) Work required to bring the charge q to the center of the square =q `xx` potential at the center point O due to tall the four charges in the four corners . The potential created by the two +q charges are canceled by the potential created by the -q charges which are located in the opposite corners . Therefore the net electric potential at the center O due to all the charges in the corners is zero . Hence no work no work is required to bring any charge to the point O . Physically this implies that if any charge q when brought close to O then it moves to the point O without any external force.