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Two point charges, -Q at the origin (0,0) and +Q/sqrt(3) at (2,0), are placed in the xy-plane. Together they produce a circular equipotential of potential V = 0 (radius R, centred at (b,0)) in the xy-plane. All lengths are in metres. Find the value of R (in metres).

1. sqrt(3)
2. 2*sqrt(3)
3. 3
4. 1.5

Correct answer: sqrt(3)

Solution

Zero potential requires Q/r1 = (Q/sqrt(3))/r2, i.e. r1 = sqrt(3)*r2, where r1 is distance from origin and r2 from (2,0). The locus r1 = sqrt(3)*r2 is an Apollonius circle. Writing r1² = 3*r2²: x² + y² = 3[(x-2)² + y²] => x² + y² = 3x² -12x +12 +3y² => 0 = 2x² + 2y² -12x +12 => x² + y² -6x +6 = 0 => (x-3)² + y² = 3. So centre (3,0) and R = sqrt(3) m.

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