Question

# The conductivity of a cylinder of length L and cross-sectional area A grows linearly with the...

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.

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 so and the other end has conductivity s1

we can write m = (s1 - so)/L

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

s = mx + so

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+so)

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

=> R = (1/mA)[ln(mx+so)]0L

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

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

Using the value of m;

R = (L/(s1 - so)A)ln(s1/so)

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