Question

The conductivity of a cylinder of length L and cross-sectional area A grows linearly with the distance, assuming the value s0 at one end and s1 at the other. Calculate total cylinder resistance.

Answer 1

Since the growth of conductance is linear thus we can assume the rate of change of conductivity with length

ds/dx = m

Now assume one end to be origin or x = 0; Since the end has a conductivity of s_o and the other end has conductivity s₁

we can write m = (s₁ - s_o)/L

Now, the equation for conductivity as a function of x can be written as

s = mx + s_o

Now, the resistance of a wire is given as R = L/sA

Assume for a length element dx at a distance x,resistance is given as

dR = dx/sA

=> R = dx/A(mx+s_o)

=> R = dx/A(mx+s_o)

=> R = (1/mA)[ln(mx+s_o)]₀^L

=> R = (1/mA)ln{(mL+s_o)/s_o}

=> R = (1/mA)ln(1+mL/s_o)

Using the value of m;

R = (L/(s₁ - s_o)A)ln(s₁/s_o)