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The current density across a cylindrical conductor of radius R varies in magnitude according to the equation J = J0r/R where r is the distance from the central axis. Thus, the current density is a maximum J0 at the cylinder's surface and decreases linearly to zero at the axis. Calculate the current in terms of J0 and the conductor's cross-sectional area A = πR2 .

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The total current is defined as the current density integrated over the cross-sectional area:

I = ∫JdA

I = total current, J = current density, dA = area element

We already have I as a given quantity:

J = (Jâ‚€/R)r

We need to find dA in terms of another variable so we can actually calculate the integral:

A = πr²

Differentiate both sides with respect to r:

dA/dr = 2Ï€r

dA = 2Ï€rdr

The conductor has a radius R, so set the integration bounds to [0, R]. Set the integrand to (2Ï€Jâ‚€/R)[tex]r^{2}[/tex] with r being the variable of integration.

I = (2Ï€Jâ‚€/3R)[tex]r^{3}[/tex] evaluated between r = 0 and r = R

I = (2Ï€Jâ‚€/3R)[tex]R^{3}[/tex]

I = (2Ï€Jâ‚€/3)[tex]R^{2}[/tex]

I = 2Ï€[tex]R^{2}[/tex]Jâ‚€/3

Substitute π[tex]R^{2}[/tex] with area A:

I = 2AJâ‚€/3

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