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Equal point charges q are arranged so that the net electric field at point O arises from a square-type geometry of side length l, the charges lying at corners with their position vectors perpendicular to one another. What is the magnitude of the resultant electric field at O?

1. 1/(4*pi*e0) * q/l²
2. 1/(4*pi*e0) * q/(2*l²) * (2*sqrt(2) - 1)
3. q/(4*pi*e0*(2*l)²)
4. 1/(4*pi*e0) * 2q/(2*l²) * sqrt(2)

Correct answer: 1/(4*pi*e0) * q/(2*l²) * (2*sqrt(2) - 1)

Solution

With charges placed on a square-type arrangement of side l, contributions come at distances l and the diagonal sqrt(2)*l. Combining the vector sum of the near contributions (each kq/l²) with the diagonal contribution kq/(2l²) and the geometry of the figure yields a net magnitude of (1/(4*pi*e0)) * q/(2 l²) * (2 sqrt(2) - 1). The factor (2 sqrt(2) - 1)/2 captures the partial cancellation between oppositely directed contributions.

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