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Equal charges q are fixed at the four corners A, B, C, D of a square of side a. What is the magnitude of the net electrostatic force on the charge located at corner B?

1. 3q²/(4*pi*e0*a²)
2. 4q²/(4*pi*e0*a²)
3. ((1 + 2*sqrt(2))/2) * q²/(4*pi*e0*a²)
4. (2 + 1/sqrt(2)) * q²/(4*pi*e0*a²)

Correct answer: (2 + 1/sqrt(2)) * q²/(4*pi*e0*a²)

Solution

Let k = 1/(4*pi*e0). Take B at a corner. The two charges at adjacent corners (A and C) are each at distance a and exert forces F1 = k q²/a² directed along the two sides away from B. These two perpendicular forces have resultant sqrt(2)*F1 directed along the diagonal pointing away from the center. The charge at the opposite corner D is at distance a*sqrt(2) and exerts F2 = k q²/(2 a²) along the same diagonal direction. Total = sqrt(2)*k q²/a² + k q²/(2 a²) = k q²/a² (sqrt(2) + 1/2) = (2 + 1/sqrt(2)) k q²/a².

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