Shown in the figure is a semicircular metallic strip that has thickness t and resistivity ρ . Its inner radius is R 1 and outer radius is R 2 . If a voltage V 0 is applied between its two ends, a current I flows in it. In addition, it is observed that a transverse voltage Δ V develops between its inner and outer surfaces due to purely kinetic effects of moving electrons (ignore any role of the magnetic field due to t

Step-by-step solution

Resistance of the strip = ρ π r tdr R = net resistance 1 R = ∫ t d r ρ π r t π ρ ∫ R 1 R 2 d r r R = π ρ t ln R 2 R 1 Hence, current i = V 0 R = V 0 t π ρ ln R 2 R 1 for elementary dr, di = V 0 ρ π r tdr = V 0 tdr ρ π r i = nAe v V 0 t d r ρ π r = n t d r e v eE = m v 2 r = m V 0 2 n 2 ℓ 2 ρ 2 π 2 r 2 × 1 r m v 2 r = m V 0 2 n 2 ℓ 2 ρ 2 π 2 r 2 × 1 r E ∝ 1 r 3 E = k 1 r 3 Δ V = ∫ E d r = k ∫ R 1 R 2 d r r 2 ∝ V 0 2 ∴ ∝ i 2