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Three point charges A = -4q, B = +2q and C = -2q lie on the circumference of a circle of radius d centred at O. Charges A and C are placed symmetrically (one above and one below the x-axis) such that A, C and the centre O form an equilateral triangle of side d, while B lies on the y-axis so that it contributes nothing to the x-component of the field. Find the magnitude of the electric field at O along the x-direction.

1. 2*sqrt(3) q / (pi*e0*d²)
2. sqrt(3) q / (4*pi*e0*d²)
3. 3*sqrt(3) q / (4*pi*e0*d²)
4. sqrt(3) q / (pi*e0*d²)

Correct answer: 3*sqrt(3) q / (4*pi*e0*d²)

Solution

Since A, C and O make an equilateral triangle of side d, OA = OC = d and angle AOC = 60deg. Taking the x-axis as the bisector, A sits at +30deg and C at -30deg. The field from a negative charge at O points toward the charge, so both A's and C's fields have positive x-components (cos30deg = sqrt(3)/2) while their y-components are equal and opposite and cancel. B lies on the y-axis, contributing nothing along x. Adding the x-components: Ex = k(4q)/d²*(sqrt(3)/2) + k(2q)/d²*(sqrt(3)/2) = (sqrt(3)/2)*k*6q/d² = 3*sqrt(3)*kq/d² = 3*sqrt(3) q/(4*pi*e0*d²).

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