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.

Homework Answers

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 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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