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