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 .

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 = (J₀/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 = 2πr

dA = 2πrdr

The conductor has a radius R, so set the integration bounds to [0, R]. Set the integrand to (2πJ₀/R) with r being the variable of integration.

I = (2πJ₀/3R) evaluated between r = 0 and r = R

I = (2πJ₀/3R)

I = (2πJ₀/3)

I = 2πJ₀/3

Substitute π with area A:

I = 2AJ₀/3