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Two electric dipoles A and B lie on the x-axis with dipole moments p_A = -4*q*a (in x-direction) and p_B = -2*q*a (in x-direction), separated by a distance R. At what distance from A (along the line) do the two dipoles produce equal potential?

1. sqrt(2)*R/(sqrt(2) + 1)
2. R/(sqrt(2) + 1)
3. sqrt(2)*R/(sqrt(2) - 1)
4. R/(sqrt(2) - 1)

Correct answer: sqrt(2)*R/(sqrt(2) + 1)

Solution

Both dipoles point the same way; for the potentials to be equal in magnitude at a point between them, take the point at distance x from A and R-x from B. V_A = k*(4qa)/x², V_B = k*(2qa)/(R-x)². Setting equal: 4/x² = 2/(R-x)², so (R-x)²/x² = 1/2, giving (R-x)/x = 1/sqrt(2), R-x = x/sqrt(2), R = x(1 + 1/sqrt(2)) = x(sqrt(2)+1)/sqrt(2). Thus x = sqrt(2)*R/(sqrt(2)+1).

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