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

Step-by-step solution

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 =