Question

A resistor is made from a hollow cylinder of length, l, inner radius a, and outer radius b. The region a<r<b is filled with material of resistivity ρ. The resistance R of the resistor is ?​

Answer 1

Answer:

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We recall that J=IAJ=IA, E=dVdrE=dVdr. The area, A will be the surface area of the cylinder. Since

JIAI2πrLdV=1ρE=1ρE=1ρdVdr=Iρ2πrLdr

J=1ρEIA=1ρEI2πrL=1ρdVdrdV=Iρ2πrLdr

Since dR=dVIdR=dVI,

dR=ρ2πrLdr

dR=ρ2πrLdr

Integrating from a to b,

R=∫abdR=∫abρ12πrLdr=ρ2πL∫ab1rdr=ρ2πLlnba

R=∫abdR=∫abρ12πrLdr=ρ2πL∫ab1rdr=ρ2πLlnba

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Answer 2

Given

Length of resistor = L

inner radius = a

outer radius = b

so

inner area = πa^2 *l

outer area = πb^2*l

change in area = π(b^2 - a^2)

region resistivity= a<r<b

since we know that

Resistance = resistivity *Length /area

so

R = resistivity *L /π(a+b)(a - b)L

R =resistivity/π(a+b)(a -b)