Twelve point charges are placed at the centres of the twelve edges of a cube of side r. With the cube oriented as in the given source figure, the charges on the four top edges are -q, +q, -q, +q; on the four vertical edges they are +q, +q, +q, -q; and on the four bottom edges they are +2q, -2q, +2q, -2q in the corresponding positions. A charge Q is placed at the centre of the cube. The magnitude of the electrostatic force on Q is:
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Question

Twelve point charges are placed at the centres of the twelve edges of a cube of side r. With the cube oriented as in the given source figure, the charges on the four top edges are -q, +q, -q, +q; on the four vertical edges they are +q, +q, +q, -q; and on the four bottom edges they are +2q, -2q, +2q, -2q in the corresponding positions. A charge Q is placed at the centre of the cube. The magnitude of the electrostatic force on Q is:

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G1 Physics · Accepted Answer

Every edge centre is at distance d = r/√2 from the cube centre. Vector addition of the twelve contributions gives a resultant equivalent to a weighted position-vector sum of magnitude |Σ qᵢ r⃗ᵢ| = 2√2 |q|r. Therefore, E = k |Σ qᵢ r⃗ᵢ| / d³ = k(2√2|q|r) / (r³/(2√2)) = 8k|q|/r². Hence, F = |Q|E = 8k|qQ|/r².