Q.Four charges are arranged at the corners of a square ABCD of side $d$, as shown in Fig. 2.15.

Concept: electrostatic potential energy / potential. The work to assemble charges equals the total potential energy (sum over unique pairs); the extra work to bring $q_0$ to the centre E is $q_0V_E$.

(a) Square of side $d$, alternating charges $+q,-q,+q,-q$. There are 4 adjacent pairs (separation $d$, product $-q^2$) and 2 diagonal pairs (separation $\sqrt{2}\,d$, product $+q^2$):

$W=U=\frac{1}{4\pi\varepsilon_0}\left[4\cdot\frac{-q^2}{d}+2\cdot\frac{+q^2}{\sqrt{2}\,d}\right]=\frac{q^2}{4\pi\varepsilon_0 d}\left(\sqrt{2}-4\right).$ …

Why potential energy is the right tool

The work an external agent does to assemble point charges quasi-statically equals the total electrostatic potential energy of the final configuration (the force is conservative). For part (b), the extra work to place $q_0$ at a point is $q_0$ times the potential there due to the fixed charges.

Part (a) — Work to assemble the four charges

Label the corners A$(+q)$, B$(-q)$, C$(+q)$, D$(-q)$ around a square of side $d$. The potential energy is the sum over the $\binom{4}{2}=6$ distinct pairs:

Four adjacent pairs (AB, BC, CD, DA), separation $d$, each product $(+q)(-q)=-q^2$:

Two diagonal pairs (AC, BD), separation $\sqrt{2}\,d$, each product $(+q)(+q)=(-q)(-q)=+q^2$:

Summing:

$U=\frac{q^2}{4\pi\varepsilon_0 d}\left(-4+\frac{2}{\sqrt{2}}\right)=\frac{q^2}{4\pi\varepsilon_0 d}\left(\sqrt{2}-4\right),$ …

Method: Work to Assemble a Charge Configuration, and Extra Work to Add One More Charge

This method applies whenever several point charges are held at fixed positions and you're asked for (a) the work needed to assemble them from infinity, and/or (b) the extra work to bring one more charge to a specific point with the others held fixed.

Steps

Step 1: List every distinct pair of charges and its separation

For $n$ charges there are $\binom{n}{2}$ distinct pairs. Group pairs that share the same separation (e.g. all "adjacent" pairs in a regular polygon, all "diagonal" pairs) so you can compute each group once instead of pair by pair.

Step 2: Sum the pairwise potential energy over every pair

This total $U$ equals the work an external agent must do to assemble the configuration from infinity, because the electrostatic force is conservative — the work done is stored entirely as potential energy, independent of the order the charges were brought in.

Step 3: For "extra work to add one more charge," find the potential at that point first …