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
s_{1}

we can write m = (s_{1} - 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/(s_{1} -
s_{o})A)ln(s_{1}/s_{o})

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ΔL
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