Question

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 .

Answer 1

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)$r^{2}$ with r being the variable of integration.

I = (2πJ₀/3R)$r^{3}$ evaluated between r = 0 and r = R

I = (2πJ₀/3R)$R^{3}$

I = (2πJ₀/3)$R^{2}$

I = 2π$R^{2}$J₀/3

Substitute π$R^{2}$ with area A:

I = 2AJ₀/3