Animated Solution for Physics - Electrostatics: Eight point charges are placed at the corners of a cube of edge $a$ as shown in figure. Find the work done in disassembling this system of charges.

Imagine a perfect cube, suspended in space, with a point charge anchored at each of its eight corners. The charges alternate in sign: positive, negative, positive, negative, forming a highly symmetric, crystalline structure. Our mission? To completely disassemble this structure, taking every single charge and moving it infinitely far away from all the others. The question asks for the work required to achieve this.

Work and Potential Energy In electrostatics, the work done by an external agent to assemble or disassemble a configuration of charges is directly tied to the system's electrostatic potential energy ($U$). When we disassemble the system, we are moving the charges to infinity, where the potential energy is defined to be zero ($U_{final}=0$).

The work done by the external force is the change in potential energy:

So, our entire problem boils down to calculating the initial potential energy of this alternating charge cube. If the total energy is negative, the system is bound, and we must do positive work to tear it apart.

Combinatorics of a Cube The total potential energy of a system of discrete charges is the sum of the potential energies of every possible pair of charges. Since we have 8 charges, the number of unique pairs is given by the combination formula ${}^8{C}_2$:

We need to calculate the energy for all 28 pairs. Instead of doing this one by one, we can group them by the distance separating them. In a cube, there are exactly three possible distances between any two corners: the edges, the face diagonals, and the body diagonals.

The Edges A cube has 12 edges. The distance between the charges at the ends of an edge is simply $a$. Because the charges alternate, every single edge connects a $+q$ charge to a $-q$ charge. Therefore, the potential energy for each edge pair is negative:

The Face Diagonals A cube has 6 faces, and each square face has 2 diagonals, giving us $6\times 2=12$ face diagonals. The length of a face diagonal is $\sqrt{2}a$. What about the signs? To travel along a face diagonal, you effectively move along two edges. Since the sign flips on each edge, it flips twice ($+\to -\to +$), meaning the charges at the ends of any face diagonal are identical (either both positive or both negative). Thus, the potential energy for these pairs is positive (repulsive):

The Body Diagonals Finally, we have the body diagonals that cut straight through the center of the cube. There are 4 such diagonals, and their length is $\sqrt{3}a$. To traverse a body diagonal, you must move along three edges (length, width, and height). The sign flips three times ($+\to -\to +\to -$), which guarantees that the opposite corners of the cube always hold opposite charges. This gives us a negative potential energy for these pairs:

Notice that $12+12+4=28$. We have successfully accounted for all 28 pairs!

Now, we simply sum the energies of the three groups to find the total initial potential energy:

Let's evaluate the numerical value inside the bracket:

The negative sign confirms that the attractive forces dominate; the cube is a bound system. To disassemble it, we must supply an amount of work equal to the magnitude of this energy:

And there we have it! By breaking down the complex 3D geometry into simple, logical groups, we transformed a daunting 28-term summation into an elegant and satisfying calculation.