A cube of side b has a charge q at each of its vertices. Determine the potential and electric field due to this charge array at the center of the cube.

Options

A. zero
B. `(4q)/(sqrt(3)pi epsilon_(0)b)`
C. `(q)/(pi epsilon_(0)b)`
D. `(4q)/(pi epsilon_(0)b)`

Answer

B

Explanation

All the eight charges are symmetrically situated at the eight vertices of the cube, each at a distance r equal to half the longest diagonal from the centre.
Length of the longest diagonal of the cube,
`d = sqrt((b^(2) + b^(2) + b^(2))) = sqrt(3)b` Hence, `r = (d)/(2) = (sqrt(3))/(2)b`
Potential at the centre of the cube due to all the eight charges is,
`V = 8 xx (1)/(4pi epsilon_(0))(q)/(r ) = 8 xx (1)/(4pi epsilon_(0)) xx (q)/((sqrt(3))/(2)b) = (4q)/(sqrt(3)pi epsilon_(0)b)`