An inner solid sphere A of radius r_A = 1 has charge density rho_A = k*r (k 0, r is radial distance). It is surrounded by a spherical shell B of outer radius r_B with charge density rho_B = 2k/r. (Take all quantities in SI units.) Which statement(s) is/are correct? (A) If r_B = sqrt(3/2)

Question

An inner solid sphere A of radius r_A = 1 has charge density rho_A = k*r (k > 0, r is radial distance). It is surrounded by a spherical shell B of outer radius r_B with charge density rho_B = 2k/r. (Take all quantities in SI units.) Which statement(s) is/are correct? (A) If r_B = sqrt(3/2), the field is zero everywhere outside B. (B) If r_B = 3/2, the potential just outside B is k/e0. (C) If r_B = 2, the total charge is 15*pi*k. (D) If r_B = 5/2, the magnitude of the field just outside B is 13*pi*k/e0.

Accepted Answer

Only (B). Q_A = integral 0..1 (k*r)(4*pi*r²)dr = 4*pi*k*[r⁴/4] = pi*k. Q_shell(R) = integral 1..R (2k/r)(4*pi*r²)dr = 8*pi*k*[r²/2] = 4*pi*k*(R² - 1). Both charges are positive. (A) Field zero outside requires total charge = 0, impossible -> FALSE. (B) r_B = 3/2: Q_total = pi*k + 4*pi*k*(9/4 - 1) = 6*pi*k; V = Q/(4*pi*e0*r_B) = 6*pi*k/(4*pi*e0*(3/2)) = k/e0 -> TRUE. (C) r_B = 2: Q_total = pi*k + 4*pi*k*(4 - 1) = 13*pi*k, not 15*pi*k -> FALSE. (D) r_B = 5/2: Q_total = pi*k + 4*pi*k*(25/4 - 1) = 22*pi*k; E = Q/(4*pi*e0*r_B²) = 22*pi*k/(4*pi*e0*(25/4)) = 22*k/(25*e0), not 13*pi*k/e0 -> FALSE. Only (B) is correct.

Solution

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