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Three infinitely long straight parallel wires are placed parallel to the z-axis at x = -a , x = a , and x = 3a . Their linear charge densities are - , -2 , and respectively. The magnitude of the electric field at the point x = 2a on the x-axis is N 3 ₀ a . The value of N is _______.

Correct answer

5

Step-by-step solution

Electric field due to an infinitely long line charge is E = 2 ₀ r . At the point x = 2a : The wire at x = -a (distance r = 3a , charge - ) produces field E ₁ = - 2 ₀ (3a) i . The wire at x = a (distance r = a , charge -2 ) produces field E ₂ = - 2 2 ₀ (a) i . The wire at x = 3a (distance r = a , charge ) produces field E ₃ = - 2 ₀ (a) i . Net electric field E = E ₁ + E ₂ + E ₃ E = - ( 1 3 + 2 + 1 ) 2 ₀ a i = - 10 3 2 ₀ a i = - 5 3 ₀ a i . Magnitude is 5 3 ₀ a . Comparing with N 3 ₀ a , we get N = 5 . Answer: 5

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