Two coaxial rings, each of radius R, made of thin wire are separated by a small distance `l(l lt lt R)` and carry the charges `q` and `-q`. Find the electric field potential and strength at the axis of the system as a function of the x coordinate (see figure). Investigate these functions at `|x| gt gt R`

Explanation

The potentail can be calculated by superposition. Choice the plate of the upper ring as `x = l//2` and that of the lower ring as `x = -l//2`.
Then `varphi = (q)/(4pi epsilon_(0) [R^(2) + (x - //2)^(2)]^(1//2)) - (q)/(4pi epsilon_(0) [R^(2) + (x + //2)^(2)]^(1//2))`
`= (q)/(4pi epsilon_(0) [R^(2) + x^(2) - lx]^(1//2)) - (q)/(4pi epsilon_(0) [R^(2) + x^(2) + lx]^(1//2))`
`= (q)/(4pi epsilon_(0) (R^(2) + x^(2))^(1//2)) (1 + (lx)/(2 (R^(2) + x^(2)))) - = (q)/(4pi epsilon_(0) (R^(2) + x^(2))^(1//2)) (1 - (lx)/(2 (R^(2) + x^(2))))`
`= (q lx)/(4pi epsilon_(0) (R^(2) + x^(2))^(3//2))`
For `|x| gt gt r, varphi = (q l)/(4pi x^(2))`
The electric feild is `E = - (del varphi)/(del x)`
`= - (ql)/(4pi epsilon_(0) (R^(2) + x^(2))^(3//2)) + (3)/(2) (ql)/((R^(2) + x^(2))^(5//2) 4pi epsilon_(0)) xx 2x = (ql (2 x^(2) - R^(2)))/(4pi epsilon_(0) (R^(2) + x^(2))^(5//2))`
For `|x| gt gt R, E~~(ql)/(2pi epsilon_(0) x^(3))`. The plot is as given in the book.