Question

Figure 29-88 shows a cross section of a long conducting coaxial cable and gives its radii (a,b,c). Equal but opposite currents iare uniformly distributed in the two conductors. Derive expressions for B (r) with radial distance rin the ranges (a) r < c, (b) c< r <b , (c) b < r < a, and (d) r > a . (e) Test these expressions for all the special cases that occur to you. (f) Assume that a = 2.0 cm, b = 1.8 cm, c = 0.40 cm, and i = 120 A and plot the function B (r) over the range 0 < r < 3 cm .

Short answer

1. $B=\frac{{\mu }_{0}ir}{2\pi c^2}$.
2. $B=\frac{{\mu }_{0}i}{2\pi r}$
   
3. $B=\frac{{\mu }_{0}i}{2\pi r}·\frac{a^2-r^2}{a^2-b^2}$
   
4. $B=0$
   
5. Tested the expressions for all the special cases.
6. The graph is drawn below.

Step-by-step solution

Given

• The radius of the central solid cylindrical part of the cable$c=0.40\text{cm}=0.40\times {10}^{-2}\text{m}$ .
• The inner radius of the outer hollow cylindrical part of the cable $b=1.8\text{cm}=1.8\times {10}^{-2}\text{m}$.
• The outer radius of the outer hollow cylindrical part of the cable$a=2\text{cm}=2.0\times {10}^{-2}\text{m}$ .
• Current through inner and outer cylindrical part i = 120 A flowing in the opposite sense.

Understanding the concept

For this problem, we can use Ampere’s law by considering a suitable Ampere loop to find the magnetic field at the given point over the specified region. After that, we can find out the current enclosed in the ampere loop. Finally, we get the expression for the Bfield at the particular point.

Formula:

$\oint \overrightarrow{B}·d\overrightarrow{s}={\mu }_0{I}_{enclosed}$

(a) Derive expressions for B (r) with radial distance r  in the range r < c

For r < c,

$\begin{array}{rcl}\oint \overrightarrow{B}·d\overrightarrow{s}& =& {\mu }_0{I}_{enclosed}\\ B\oint ds& =& {\mu }_0{I}_{enclosed}\\ B·2\pi r& =& {\mu }_0{I}_{enclosed}\\ B& =& \frac{{\mu }_0{I}_{enclosed}}{2\pi r}\\ & & \end{array}$

Now let’s findthecurrent enclosed in the ampere loop of radius r:

$\begin{array}{rcl}{I}_{enclosed}& =& \frac{\pi r^2}{\pi c^2}i\\ & =& \frac{r^2}{c^2}i\\ & & \end{array}$

So we get,

$B=\frac{{\mu }_0}{2\pi r}\frac{r^2}{c^2}=\frac{{\mu }_{0}ir}{2\pi c^2}$

Hence,$B=\frac{{\mu }_{0}ir}{2\pi c^2}.$

(b) Derive expressions for B (r)  with radial distance r in the range  c < r < b

Forc < r < b

Now let’s find the current enclosed in the ampere loop of radius r:

$\begin{array}{rcl}\oint \overrightarrow{B}·d\overrightarrow{s}& =& {\mu }_0{I}_{enclosed}\\ B\oint ds& =& {\mu }_0{I}_{enclosed}\\ B·2\pi r& =& {\mu }_0{I}_{enclosed}\\ B& =& \frac{{\mu }_0{I}_{enclosed}}{2\pi r}\\ & & \end{array}$

$I_{enclosed}=i$, so we get$B=\frac{{\mu }_{0}i}{2\pi r}$

Hence,$B=\frac{{\mu }_{0}i}{2\pi r}$

(c) Derive expressions for B (r) with radial distance r in the range b < r < a

Forb < r < a

$\begin{array}{rcl}\oint \overrightarrow{B}·d\overrightarrow{s}& =& {\mu }_0{I}_{enclosed}\\ B\oint ds& =& {\mu }_0{I}_{enclosed}\\ B·2\pi r& =& {\mu }_0{I}_{enclosed}\\ B& =& \frac{{\mu }_0{I}_{enclosed}}{2\pi r}\\ & & \end{array}$

Let’s findthecurrent enclosed in the ampere loop of radiusr:

$$\begin{gathered} I_{enclosed}=i-\frac{i\left(r^2-b^2\right)}{a^2-b^2}=\frac{a^2-r^2}{a^2-b^2}i \\ {I}_{enclosed}=\frac{a^2-r^2}{a^2-b^2}i \end{gathered}$$

Hence using this, we get $B=\frac{{\mu }_{0}i}{2\pi r}·\frac{a^2-r^2}{a^2-b^2}$.

(d) Derive expressions for B (r) with radial distance r in the range r > a

For r > a

Current through the inner and outer cylindrical part is i flowing in the opposite direction, so the current enclosed in the Ampere loop of radius r

$I_{enclosed}=0.0$

Hence $B=0.0$.

(e) Test these expressions for all the special cases

The B (r)for the special cases:

We have

$B\left(r\right)=\frac{{\mu }_{0}i}{2\pi r}·\frac{a^2-r^2}{a^2-b^2}$

When a = 0, we get

$\begin{array}{rcl}B\left(r\right)& =& \frac{{\mu }_{0}i}{2\pi r}·\frac{0-r^2}{0-b^2}\\ & =& \frac{{\mu }_{0}i}{2\pi r}·\frac{r^2}{b^2}\\ & =& \frac{{\mu }_{0}i}{2\pi }·\frac{r}{b^2}\end{array}$

Whenr = b,we get

$\begin{array}{rcl}B\left(r\right)& =& \frac{{\mu }_{0}i}{2\pi r}·\frac{a^2-r^2}{a^2-b^2}\\ & =& \frac{{\mu }_{0}i}{2\pi b}·\frac{a^2-b^2}{a^2-b^2}\\ & =& \frac{{\mu }_{0}i}{2\pi b}\end{array}$

When r = a, we get

$\begin{array}{rcl}B\left(r\right)& =& \frac{{\mu }_{0}i}{2\pi a}·\frac{a^2-a^2}{a^2-b^2}\\ & =& \frac{{\mu }_{0}i}{2\pi a}·0\\ & =& 0.0\\ & & \end{array}$

When b = 0, we get

$\begin{array}{rcl}B\left(r\right)& =& \frac{{\mu }_{0}i}{2\pi r}·\frac{a^2-r^2}{a^2-0}\\ & =& \frac{{\mu }_{0}i}{2\pi r}·\left(1-\frac{r^2}{a^2}\right)\\ & & \end{array}$

Hence, it is tested the expressions for all special cases.

(f) Plot the function B (r) over the range 0 < r < 3 cm

The plot of the function B (r) over the range 0 < r < 3 cm and $i=120\text{A:}$$c=0.40\text{cm}=0.40\times {10}^{-2}\text{m}$, $b=1.8\text{cm}=1.8\times {10}^{-2}\text{m}$,$a=2\text{cm}=2.0\times {10}^{-2}\text{m}$.

Hence the graph is drawn.