Question

A wire of radius $R$has a current density that increases linearly with distance from the center of the wire: $J\left(r\right)=kr$, where $k$is a constant. Find an expression for $k$in terms of $R$and the total current$I$carried by the wire.

Short answer

An expression for $k$in terms of $R$and the total current $I$carried by the wire is $k=\frac{2I}{R^2}$.

Step-by-step solution

Given information

A wire of radius is $R$, has a current density that increases linearly with distance from the center of the wire: $J\left(r\right)=kr$, where $k$ stands as a constant.

Explanation

The cross-sectional diagram of the wire is the thickness of the shaded region is infinitesimal and $dr.$

So, the area of the shaded region is,$d{a}_1=\left(2\pi r\right)dr$

$d{a}_1$means the area is perpendicular to the flow of current.

Hence, the total current in the wire is,

$I={\int }_0^R J\left(r\right)\left(2\pi r\right)dr$

Find an expression in terms of$k$is,

$=2\pi k{\int }_0^R r^{2}dr$

role="math" localid="1649159756177" $=\frac{2\pi }{3}k{\left[r^3\right]}_0^R$

$I=\frac{2\pi }{3}k{R}^3$

role="math" localid="1649159879711" $k=\frac{3I}{2\pi R^3}$

Final Answer

Hence, the expression for $k$in terms of$R$and the total current$I$carried by the wire is$\frac{3I}{2\pi R^3}.$