Two point like charges `q_(1)` and `q_(2)` are fixed in free space. At every point on the curve shown, the net electrostatic potential created by these charges is V. Find the separation r between the charges.
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Explanation

The electric field is zero where equipotential surface cuts. At P
`|E_1| = |E_2|`
or `kq_(1)/(r_1^2) = kq_(2)/(r_2^2)`
or `(q_(1)/q_(2))^(1//2) = r_(1)/r_(2)` …(i)
Also `V = kq_(1)/r_(1) + kq_(2)/r_(2)` ...(ii)
and `r = r_1 + r_2` ...(iii)
From (i) and (iii),
`r_1 = (r sqrtq_(1))/(sqrtq_(1)+ sqrtq_(2))` and `r_2 = (r sqrtq_(2))/(sqrtq_(1) +sqrtq_(2))` ...(iv)
From (ii) and (iv),
`V = ((sqrt(q_(1))+sqrt(q_(2))))/(4piepsilon_(0))[q_(1)/(rsqrt(q_(1))+rsqrt(q_(2)))]`
`=((sqrt(q_(1))+sqrt(q_(2))))/(4piepsilon_(0)r)[sqrt(q_(1))+sqrt(q_(2))]`
or `r=((sqrt(q_(1))+sqrt(q_(2)))^(2))/(4piepsilon_(0)V)`