Shown in the figure is a semicircular metallic strip that has thickness t...
Shown in the figure is a semicircular metallic strip that has thickness t and resistivity ρ. Its inner radius is R1 and outer radius is R2. If a voltage V0 is applied between its two ends, a current I flows in it. In addition, it is observed that a transverse voltage ΔV develops between its inner and outer surfaces due to purely kinetic effects of moving electrons (ignore any role of the magnetic field due to the current). Then (figure is schematic and not drawn to scale)
Resistance of the strip =ρπrtdr
R=net resistance
1R=∫tdrρπrtπρ∫R2R1drr
R=πρtln(R2R1)
Hence, current i =V0R=V0tπρln(R2R1)
for elementary dr,
di=V0ρ(πrtdr)=V0tdrρπr
i=nAev
V0tdrρπr=nt(dr)ev
eE=mv2r=mV20n2ℓ2ρ2π2r2×1r
mv2r=mV20n2ℓ2ρ2π2r2×1r
E∝1r3
E=k1r3
ΔV=∫Edr=k∫R2R1drr2∝V20
∴ ∝ i2
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