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NEET PHYSICSELECTROSTATIC POTENTIAL AND CAPACITANCEMedium

Question

Two metal spheres, one of radius RR and the other of radius 2R2R respectively have the same surface charge density σ\sigma. They are brought in contact and separated. What will be the new surface charge densities on them?

A

σ1=56σ,σ2=56σ\sigma_1 = \frac{5}{6}\sigma, \sigma_2 = \frac{5}{6}\sigma

B

σ1=52σ,σ2=56σ\sigma_1 = \frac{5}{2}\sigma, \sigma_2 = \frac{5}{6}\sigma

C

σ1=52σ,σ2=53σ\sigma_1 = \frac{5}{2}\sigma, \sigma_2 = \frac{5}{3}\sigma

D

σ1=53σ,σ2=56σ\sigma_1 = \frac{5}{3}\sigma, \sigma_2 = \frac{5}{6}\sigma

Step-by-Step Solution

  1. Initial Charge Calculation: Surface charge density σ=QA\sigma = \frac{Q}{A} [NCERT Class 12, Sec 1.12]. Charge on first sphere (RR): Q1=σ(4πR2)Q_1 = \sigma (4\pi R^2). Charge on second sphere (2R2R): Q2=σ(4π(2R)2)=4σ(4πR2)=4Q1Q_2 = \sigma (4\pi (2R)^2) = 4\sigma (4\pi R^2) = 4Q_1. Total charge Qtotal=Q1+Q2=5Q1=5σ(4πR2)Q_{\text{total}} = Q_1 + Q_2 = 5Q_1 = 5\sigma(4\pi R^2).
  2. Redistribution of Charge: When brought in contact, charge flows until they reach a common potential VV. The potential of a conducting sphere is V=14πϵ0QrV = \frac{1}{4\pi\epsilon_0}\frac{Q}{r} [NCERT Class 12, Sec 2.3]. Condition for equilibrium: V1′=V2′⇒Q1′R=Q2′2R⇒Q2′=2Q1′V_1' = V_2' \Rightarrow \frac{Q_1'}{R} = \frac{Q_2'}{2R} \Rightarrow Q_2' = 2Q_1'. Conservation of charge: Q1′+Q2′=Qtotal⇒Q1′+2Q1′=5Q1⇒3Q1′=5Q1Q_1' + Q_2' = Q_{\text{total}} \Rightarrow Q_1' + 2Q_1' = 5Q_1 \Rightarrow 3Q_1' = 5Q_1. New charges: Q1′=53Q1Q_1' = \frac{5}{3}Q_1 and Q2′=103Q1Q_2' = \frac{10}{3}Q_1.
  3. Final Surface Charge Densities: σ1′=Q1′4πR2=53σ(4πR2)4πR2=53σ\sigma_1' = \frac{Q_1'}{4\pi R^2} = \frac{\frac{5}{3} \sigma (4\pi R^2)}{4\pi R^2} = \frac{5}{3}\sigma. σ2′=Q2′4π(2R)2=103σ(4πR2)4(4πR2)=1012σ=56σ\sigma_2' = \frac{Q_2'}{4\pi (2R)^2} = \frac{\frac{10}{3} \sigma (4\pi R^2)}{4 (4\pi R^2)} = \frac{10}{12}\sigma = \frac{5}{6}\sigma.

Exam Context & Concepts Covered

This question aligns with the NEET PHYSICS syllabus, specifically targeting concepts from ELECTROSTATIC POTENTIAL AND CAPACITANCE. Mastering this topic is crucial for scoring well in the upcoming medical entrance examinations. Solving conceptually related problems will help you understand the nuances of these concepts and improve your problem-solving speed.

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