Question

Figure 29-67 shows a cross section across a diameter of a long cylindrical conductor of radius $\mathit{a}\mathbf{=}\mathbf{}\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{}\mathit{c}\mathit{m}$carrying uniform current $\mathbf{170}\mathbf{}\mathit{A}$. What is the magnitude of the current’s magnetic field at radial distance (a) 0 (b) 1.00 cm, (c) 2.00 cm (wire’s surface), and (d) 4.00 cm?

Short answer

(a) The magnitude of the current’s magnetic field at a radial distance 0 cm is zero.

(b)The magnitude of the current’s magnetic field at a radial distance 1.00cm is$8.50\times {10}^{-4}T$

(c) The magnitude of the current’s magnetic field at a radial distance 2.00 cm is$1.70\times {10}^{-3}T$

(d) The magnitude of the current’s magnetic field at a radial distance 4.00 cm is$8.50\times {10}^{-4}T$

Step-by-step solution

Listing the given quantities

• The radius of the conductor is$a=2.00cm=0.0200m$
• Current through the conductor is$i=170A$

Understanding the concept of the magnetic field at a point

Ampere’s law states that,

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

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.

Using Ampere’s law, we can derive the magnetic fieldinside the wire$\mathbf{(}\mathit{r}\mathbf{}\mathbf{<}\mathbf{}\mathit{a}\mathbf{)}$as,

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

Here,$\mathit{i}$is the current,$\mathit{a}$is area of cross-section of the wire,$\mathit{r}$is the radius.

For the first two conditions $\mathit{r}\mathbf{}\mathbf{<}\mathbf{}\mathit{a}$, therefore, using, we can find the magnitude of the current’s magnetic field at the corresponding radii. We can also use the same equation for the third condition $\mathit{a}\mathbf{}\mathbf{=}\mathbf{}\mathit{r}$. For the fourth condition,$\mathit{r}\mathbf{}\mathbf{>}\mathbf{}\mathit{a}$, therefore, we can use equation (i) to find the corresponding magnitude of the current’s magnetic field.

The magnetic field outside the wire $\mathbf{(}\mathit{r}\mathbf{}\mathbf{>}\mathbf{}\mathit{a}\mathbf{)}$is given by,

$\mathit{B}\mathbf{=}\frac{{\mathbf{\mu }}_{\mathbf{0}}\mathbf{i}}{\mathbf{2}\mathbf{\pi }\mathbf{r}}$ (ii)

(a) Calculations of the magnitude of the current’s magnetic field at a radial distance 0 cm

For$r=0$,

Since$a>r$, therefore, the magnetic field is given by,

$$\begin{gathered} B=\left(\frac{{\mu }_{0}i}{2\pi a^2}\right)r \\ =\left(\frac{{\mu }_{0}i}{2\pi a^2}\right)\times \left(0\right) \\ =0 \end{gathered}$$

The magnitude of the current’s magnetic field at a radial distance 0 cm is zero.

(b) Calculations of the magnitude of the current’s magnetic field at a radial distance 1.00 cm

For$r=0.0100m$

Since,$a>r$, therefore,fromequation (i), the magnetic field is given by,

$$\begin{gathered} B=\left(\frac{{\mu }_{0}i}{2\pi a^2}\right)r \\ =\left(\frac{\left(4\pi \times {10}^{-7}T.m/A\right)\left(170A\right)}{2\pi {\left(0.0200m\right)}^2}\right)\left(0.0100m\right) \\ =8.50\times {10}^{-4}T \end{gathered}$$

The magnitude of the current’s magnetic field at a radial distance 1.00 cm is $8.50\times {10}^{-4}T$.

(c) Calculations of the magnitude of the current’s magnetic field at a radial distance 2.00 cm

For$r=0.0200m$

Since,$a=r$, therefore,fromequation (ii), the magnetic field is given by,

$$\begin{gathered} B=\left(\frac{{\mu }_{0}i}{2\pi a^2}\right)r \\ =\left(\frac{\left(4\pi \times {10}^{-7}T.m/A\right)\left(170A\right)}{2\pi {\left(0.0200m\right)}^2}\right)\left(0.0200m\right) \\ =1.70\times {10}^{-3}T \end{gathered}$$

The magnitude of the current’s magnetic field at a radial distance 2.00 cm is$1.70\times {10}^{-3}T$.

(d) Calculations of the magnitude of the current’s magnetic field at a radial distance 4.00 cm

For$r=0.0400m$,

Since,$a<r$, therefore,fromequation (ii), the magnetic field is given by

$$\begin{gathered} B=\left(\frac{{\mu }_{0}i}{2\pi r}\right) \\ =\left(\frac{\left(4\pi \times {10}^{-7}T.m/A\right)\left(170A\right)}{2\pi \left(0.0400m\right)}\right) \\ =8.50\times {10}^{-4}T \end{gathered}$$

The magnitude of the current’s magnetic field at a radial distance 4.00 cm is$8.50\times {10}^{-4}T$.