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In the electric field of a positive point charge, the variation of electric potential phi with distance r from the charge is plotted. Points a, b, c, d lie on the curve with coordinates (rₐ, phiₐ), (r_b, phi_b), (r_c, phi_c), (r_d, phi_d) such that rₐ: r_b: r_c: r_d = 1: 2: 3: 4 (equal spacing on r-axis with phi values kq/r). The electric field magnitudes at these points are Eₐ, E_b, E_c, E_d. A positive test charge is moved successively from a to b to c to d. The work done by the electric field during successive displacements is W_ab, W_bc, W_cd. Which of the following statements is/are correct?
- Eₐ: E_b = 4: 1
- E_c: E_d = 2: 1
- W_ab: W_bc = 3: 1
- W_bc: W_cd = 1: 3
Correct answer: Eₐ: E_b = 4: 1
Solution
Since E = kq/r², the ratio Eₐ:E_b = r_b²:rₐ² = 4:1 (correct). E_c:E_d = r_d²:r_c² = 16:9, not 2:1 (incorrect). Work W proportional to (1/r₁ - 1/r₂): W_ab proportional to (1/1 - 1/2) = 1/2, W_bc proportional to (1/2 - 1/3) = 1/6, so W_ab:W_bc = 3:1 (correct). W_bc:W_cd = (1/2 - 1/3):(1/3 - 1/4) = (1/6):(1/12) = 2:1, not 1:3 (incorrect).
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