Step 1: Distance of Central charge from charges at corners[Refer Fig.]From Figure:
O is center of cube
Since YZ is face diagonal
∴ YZ=√a2+a2=√2a
Since XZ is body diagonal
∴ XZ=√(√2a)2+a2=√3a
Distance between center and corner r=XZ2
∴ r=√3a2
Step 2: Total Potential Energy
Potential energy of two charges U=Kq1q2r
Here 8(−q) charges are placed at corners of the cube.
And all these charges are at same distance from central charge q.
Since, Potential energy is scalar quantity, so it will be added due all the corner charges.
Therefore, UTotal=8×U1
⇒ UT=8×Kq1q2r
=8K(+q)(−q)(√3a2) =−16Kq2√3a
So, UT=−4q2√3π∈0a
Hence Option ′C′ is correct.