Two coaxial conducting cylinders of equal length ℓ have radii √2R and 2R. The inner cylinder carries charge Q, the outer cylinder is grounded, and the dielectric between them has κ = 5. An imaginary plane of length ℓ is at distance R from the common axis and parallel to it. Ignoring edge effects, find the electric flux through the plane. Image
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Question

Two coaxial conducting cylinders of equal length ℓ have radii √2R and 2R. The inner cylinder carries charge Q, the outer cylinder is grounded, and the dielectric between them has κ = 5. An imaginary plane of length ℓ is at distance R from the common axis and parallel to it. Ignoring edge effects, find the electric flux through the plane. Image

Options

G1 Physics · Accepted Answer

Between the cylinders, E(r) = Q/(2πκε₀ℓr). Take the plane at perpendicular distance R from the axis. In cross-section, let y run along the plane. Then r² = R² + y² and the normal component is Eₙ = E(R/r). Only the portions inside the dielectric contribute, with y from R to √3R on each side. Thus Φ = 2 ∫[R to √3R] [Q/(2πκε₀ℓr)](R/r)ℓ dy = QR/(πκε₀) ∫[R to √3R] dy/(R² + y²) = Q/(πκε₀)[tan⁻¹(y/R)] from R to √3R = Q/(πκε₀)(π/3 − π/4) = Q/(12κε₀). With κ = 5, Φ = Q/(60ε₀). Hence option C is correct.