A unit positive test charge is slowly moved from point A at distance 2R to point B at distance 3R from the centre of a uniformly positively charged dielectric sphere of radius R and volume charge density ρ. Its path passes through a thin tube in the sphere, as shown. The magnitude of work is ρR²/(nε₀). Find n. Image
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Question

A unit positive test charge is slowly moved from point A at distance 2R to point B at distance 3R from the centre of a uniformly positively charged dielectric sphere of radius R and volume charge density ρ. Its path passes through a thin tube in the sphere, as shown. The magnitude of work is ρR²/(nε₀). Find n. Image

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G1 Physics · Accepted Answer

Step 1: The sphere's total charge is Q = (4/3)πR³ρ. Step 2: Both endpoints lie outside the spherical distribution. Their potentials are V(2R) = Q/(4πε₀ × 2R) and V(3R) = Q/(4πε₀ × 3R). Step 3: The electrostatic field is conservative, so the work magnitude is path-independent even if the charge traverses a tube. Step 4: |W| = |q(V_A − V_B)| with q = +1 C = [Q/(4πε₀)](1/(2R) − 1/(3R)). Step 5: Substitute Q: |W| = [(4πR³ρ/3)/(4πε₀)] × [1/(6R)] = ρR²/(18ε₀). Final answer: n = 18 (Option D).