Variable Current Density

Electricity and Magnetism Level 3

A long straight hollow cylindrical conductor of inner radius $5$ and outer radius $10$ carries current of magnitude $2\pi\times \ln(2)$ along its length. The current density varies as $J=\dfrac{J_{0}}{r^{2}}$ within the conductor, where $r$ is the radial distance from axis of symmetry of the conductor and $J_{0}$ is a constant. Assume that the relative permeability of the conductor material is essentially unity. Find the value of $J_{0}$

Details: Current density is represented by $\dfrac{I}{Area}$

The answer is 1.

1 solution

Amal Hari
Mar 9, 2019

Since current density is varying with respect to r We use differential Area dA to represent the region with some value of J, given by the equation in question.

We have $J\times dA=dI$ , total current flowing through that area.

$\dfrac{dA}{dr}=2\pi r$

$dA=2\pi r dr$

$J=\dfrac{J_{o}}{r^{2}}$

$dI=\dfrac{J_{o}}{r^{2}}\times 2\pi r dr$

$dI=\dfrac{J_{o}}{r}\times 2\pi dr$

$\displaystyle \int_5 ^{10} dI=J_{o} \times 2\pi ln(\dfrac{10}{5})$

$I=J_{o} \times 2\pi ln(2)$

$2\pi ln(2)=J_{o} \times 2\pi ln(2)$

$J_{o}=1$

Variable Current Density

The answer is 1.

1 solution

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