Two electric charges q and -2q are placed 6m apart on a horizontal plane. Find the locus of any point on this plane where the potential has a value zero.

Explanation

The potential at P due to the charges q and -2q,
`V = 1/(4 pi epsilon_(0))(q/(r_1)-(2q)/(r_2))`

But V = 0 (given)
`:. 1/(r_1) = 2/(r_2) or r_(2) = 2r_(1) or, r_(2)^(2) = 4r_(1)^(2)`
or, `y^(2) + (6-x)^(2) = 4(x^(2)+y^(2))`
or, `y^(2) + 36 + x^(2) - 12 x = 4x^(2) + 4y^(2)`
or, `3x^(2)+12x+3y^(2) = 36 or, x^(2) +4x+y^(2) = 12`
or, `(x-2)^(2) +y^(2) = 4^(2) or, [x-(-2)]^(2)+(y-0)^(2) = 4^(2)`
Therefore, the locus of point P is a circle whose centre is at (-2,0) and radius is 4m.