A non-conducting thin disc of radius R rotates about its axis with an angular velocity w . The surface charge density on the disc varies with the distance r from the center as σ r = σ 0 1 + r R β , where σ 0 and β are constants. If the magnetic induction at the center is B = 9 10 μ 0 σ 0 w R , the value of β is

Question

A non-conducting thin disc of radius R rotates about its axis with an angular velocity w . The surface charge density on the disc varies with the distance r from the center as σ r = σ 0 1 + r R β , where σ 0 and β are constants. If the magnetic induction at the center is B = 9 10 μ 0 σ 0 w R , the value of β is, 1 4, 4, 1 2, 2

MARKS App · Accepted Answer

Consider a thin ring of thickness d r at r from the centre of the disc. ∴ d B = μ 0 × d I 2 r but d I = d Q T = d Q 2 π × ω = σ × 2 π r d r 2 π × ω = σ ω r d r B = ∫ 0 R μ 0 2 r × σ ω r d r = ∫ 0 R ω μ 0 2 r × σ 0 1 + r R β r d r ∴ B = ω μ 0 σ 0 2 ∫ 0 R 0 1 + r R β d r = ω μ 0 σ 0 2 2 + β 1 + β R Given B = 9 10 μ 0 σ 0 w R ∴ ω μ 0 σ 0 2 2 + β 1 + β R = 9 10 μ 0 σ 0 w R ∴ 5 β + 10 = 9 β + 9 ⇒ β = 1 4

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