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A charge q sits at the centroid of an equilateral triangle, and three equal charges Q are placed at the triangle's vertices. For the entire system of four charges to be in equilibrium, q must equal:

1. -Q/sqrt(3)
2. -Q/3
3. -Q*sqrt(3)
4. Q/sqrt(3)

Correct answer: -Q/sqrt(3)

Solution

Consider a corner charge Q at distance a (side length) from the other two corner charges and at distance a/sqrt(3) from the centroid. The two equal corner charges each repel it; their resultant along the line to the centroid has magnitude 2*kQ²/a² * cos(30deg) = sqrt(3)*kQ²/a². The central charge q must attract Q with force k|q|Q/(a/sqrt(3))² = 3k|q|Q/a² directed toward the centre. Setting these equal: 3k|q|Q/a² = sqrt(3)*kQ²/a² => |q| = Q/sqrt(3), and q must be negative (attractive). Thus q = -Q/sqrt(3).

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