Two thin wire rings each having radius R are placed at distance d apart with their axes coinciding. The charges on the two are `+Q` and `-Q`. The potential difference between the centre so the two rings is

Explanation

The arrangment of the rings are as shwon in Figure. Now, potential at the point 1, `varphi_(1)` = potential at 1 due to the ring `1 +` potential at 1 due to the ring 2.
`= (q)/(4pi epsilon_(0) R) + (-q)/(4pi epsilon_(0)(R^(2) + a^(2))^(1//2))`
Similarly, the potential at point 2,
`varphi_(2) = (-q)/(4pi epsilon_(0) R) + (q)/(4pi epsilon_(0)(R^(2) + a^(2))^(1//2))`
Hence, the sought potentail differnce,
`varphi_(1) - varphi_(2) = Delta varphi = 2( (q)/(4pi epsilon_(0) R) + (-q)/(4pi epsilon_(0)(R^(2) + a^(2))^(1//2)))`
`= (q)/(2pi epsilon_(0) R) (1 - (1)/(sqrt(1 + (a//R)^(2))))`