Question

The current density $\overrightarrow{\mathbf{J}}\mathbf{}$ inside a long, solid, cylindrical wire of radius a=3.1mm is in the direction of the central axis, and its magnitude varies linearly with radial distance rfrom the axis according to$\mathbf{J}\mathbf{=}\mathbf{}{\mathbf{J}}_{\mathbf{0}}\mathbf{r}\mathbf{/}\mathbf{a}\mathbf{,}$ where ${\mathbf{J}}_{\mathbf{0}}\mathbf{=}\mathbf{310}\mathbf{}\mathbf{A}\mathbf{/}{\mathbf{m}}^{\mathbf{2}}$. (a) Find the magnitude of the magnetic field at role="math" localid="1663132348934" $\mathbf{r}\mathbf{=}\mathbf{}\mathbf{0}$, (b) Find the magnitude of the magnetic field$\mathbf{r}\mathbf{=}\mathbf{a}\mathbf{/}\mathbf{2}$ , and(c) Find the magnitude of the magnetic field $\mathbf{r}\mathbf{=}\mathbf{a}$.

Short answer

1. The magnitude of the magnetic field at$\mathrm{r}=0$is zero.
2. The magnitude of the magnetic field at$\mathrm{r}=\frac{\mathrm{a}}{2}$is$\mathrm{B}=0.10\mathrm{\mu T}$
3. The magnitude of the magnetic field at $\mathrm{r}=\mathrm{a}$ is $\mathrm{B}=0.40\mathrm{\mu T}$

Step-by-step solution

Listing the given quantities

• The radius of the wire$\mathrm{a}=3.1\mathrm{mm}=0.0031\mathrm{m}$
• Current density varies as$\mathrm{J}=\frac{{\mathrm{J}}_0\mathrm{r}}{\mathrm{a}}$

• ${\mathrm{J}}_0=310\mathrm{A}/{\mathrm{m}}^2$

Understanding the concept of magnetic field and Ampere’s law

The relation between current and current density is,

$\mathbf{i}\mathbf{=}\mathbf{\int }\mathbf{J}\mathbf{}\mathbf{dA}$

Ampere’s law states that,

$\mathbf{\oint }\overrightarrow{\mathbf{B}}\mathbf{\cdot }\mathbf{d}\overrightarrow{\mathbf{s}}\mathbf{=}{\mathbf{\mu }}_{\mathbf{0}}\mathbf{i}$

The line integral in this equation is evaluated around a closed-loop called an Amperian loop.The current ion the right side is the net current encircled by the loop.

By using the current density equation in Ampere’s law and integrating it with respect to distance r, we can get the general equation for the magnetic field due to the current-carrying cylindrical wire. By using this, we can find the value of the magnetic field at given distances.

Explanation

According to Ampere’s law,

$\begin{array}{c}\oint \overrightarrow{\mathrm{B}}\cdot \mathrm{d}\overrightarrow{\mathrm{s}}={\mathrm{\mu }}_0{\mathrm{i}}_{\mathrm{enc}}\\ \mathrm{B}\oint \mathrm{ds}={\mathrm{\mu }}_0{\mathrm{i}}_{\mathrm{enc}}\end{array}$

Since$\oint \mathrm{ds}=\mathrm{length}\mathrm{of}\mathrm{the}\mathrm{circular}\mathrm{path}=2\mathrm{\pi r}$

$\therefore \mathrm{B}\left(2\mathrm{\pi r}\right)={\mathrm{\mu }}_0{\mathrm{i}}_{\mathrm{enc}}$ (i)

Current is given by,

${\mathrm{i}}_{\mathrm{enc}}=\int \mathrm{J}\mathrm{dA}$

Using the given current density $\mathrm{J}=\frac{{\mathrm{J}}_0\mathrm{r}}{\mathrm{a}}$

${\mathrm{i}}_{\mathrm{enc}}=\int \frac{{\mathrm{J}}_0\mathrm{r}}{\mathrm{a}}\mathrm{dA}$

Area of a differential element of the circle is$\mathrm{dA}=2\mathrm{\pi r}\mathrm{dr}$

$\begin{array}{c}{\mathrm{i}}_{\mathrm{enc}}=\int \frac{{\mathrm{J}}_0\mathrm{r}}{\mathrm{a}}2\mathrm{\pi r}\mathrm{dr}\\ =\frac{2{\mathrm{\pi J}}_0}{\mathrm{a}}\int {\mathrm{r}}^2\mathrm{dr}\\ =\frac{2{\mathrm{\pi J}}_0}{\mathrm{a}}\frac{{\mathrm{r}}^3}{3}\end{array}$

Using this in equation (i),

$\mathrm{B}\left(2\mathrm{\pi r}\right)={\mathrm{\mu }}_0\frac{2{\mathrm{\pi J}}_0}{\mathrm{a}}\frac{{\mathrm{r}}^3}{3}$

Therefore,

$\mathrm{B}=\frac{{\mathrm{\mu }}_0{\mathrm{J}}_0}{3}\frac{{\mathrm{r}}^2}{\mathrm{a}}$ (ii)

(a) Calculations of the magnitude of magnetic field at r=0

At $\mathrm{r}=0$, Equation (ii) becomes,

$\begin{array}{c}\mathrm{B}=\frac{{\mathrm{\mu }}_0{\mathrm{J}}_0}{3}\times \frac{0}{\mathrm{a}}\\ =0\end{array}$

Thus, the magnetic field at $\mathrm{r}=0$ is zero.

(b) Calculations of the magnitude of magnetic field at r=a/2

At$\mathrm{r}=\frac{\mathrm{a}}{2}$,Equation$\left(2\right)$becomes

$\begin{array}{c}\mathrm{B}=\frac{{\mathrm{\mu }}_0{\mathrm{J}}_0}{3}\times \frac{{\left(\raisebox{1ex}{$\mathrm{a}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)}^2}{\mathrm{a}}\\ \mathrm{B}=\frac{{\mathrm{\mu }}_0{\mathrm{J}}_0\mathrm{a}}{12}\\ =\frac{4\mathrm{\pi }\times {10}^{-7}\times 310\times 0.0031}{12}\\ =\frac{4\times 3.14\times {10}^{-7}\times 310\times 0.0031}{12}\\ =0.1\times {10}^{-6}\mathrm{T}\\ =0.10\mathrm{\mu T}\end{array}$

Thus, The magnetic field at $\mathrm{r}=\frac{\mathrm{a}}{2}$ is $\mathrm{B}=0.10\mathrm{\mu T}$

(c) Calculations of the magnitude of magnetic field at r=a 

At$\mathrm{r}=\mathrm{a}$,Equation$\left(2\right)$becomes

$\begin{array}{c}\mathrm{B}=\frac{{\mathrm{\mu }}_0{\mathrm{J}}_0}{3}\times \frac{{\mathrm{a}}^2}{\mathrm{a}}\\ =\frac{{\mathrm{\mu }}_0{\mathrm{J}}_0\mathrm{a}}{3}\\ =\frac{4\mathrm{\pi }\times {10}^{-7}\times 310\times 0.0031}{3}\\ =\frac{4\times 3.14\times {10}^{-7}\times 310\times 0.0031}{3}\\ =4.02\times {10}^{-7}\mathrm{T}\\ =0.40\mathrm{\mu T}\end{array}$

Thus, the magnetic field at $\mathrm{r}=\mathrm{a}$ is $\mathrm{B}=0.40\mathrm{\mu T}$.