An iron rod of length $\mathbf{L}$and square cross section (side a) is given a uniform longitudinal magnetization $\mathbf{M}$, and then bent around into a circle with a narrow gap (width $\mathbf{w}$), as shown in Fig. 6.14. Find the magnetic field at the center of the gap, assuming $\mathbf{w}\mathbf{\ll }\mathbf{a}\mathbf{\ll }\mathbf{L}$.

Consider an iron rod of length $\mathrm{L}$and square cross section (side a) is given a uniform longitudinal magnetization $\mathrm{M}$, and then bent around into a circle with a narrow gap (width $\mathrm{w}$).

Assume $\text{w}\ll \text{a}\ll \text{L}$.

Write the formula ofmagnetic field at the center of the gap.

$\overrightarrow{\mathbf{B}}={\overrightarrow{\mathbf{B}}}_{\mathrm{torus}}-{\overrightarrow{\mathbf{B}}}_{\mathrm{loop}}$ …… (1)

Here, ${\overrightarrow{\mathbf{B}}}_{\mathrm{torus}}$is the field of this solenoid and${\overrightarrow{\mathbf{B}}}_{\mathrm{loop}}$ is magnetic field of a square loop at its center.

First we determine the bound currents:

We can (locally) regard the torus as an indefinitely long solenoid if $\text{L}\gg \text{a}$. Similar to the preceding issue, the field of this solenoid is as follows at the location of the gap:

Determine the field of this solenoid.

$\begin{array}{c}{\overrightarrow{B}}_{torus}={\mu }_0\overrightarrow{M}\\ ={\mu }_{0}M\widehat{\varphi }\end{array}$

The (Problem 5.8) a) revealed the magnetic field of a square loop at its centre, which is:

$B_{loop}=\frac{\sqrt{2}{\mu }_{0}l}{\pi R}$ …… (2)

Here, $\text{R}=\text{a}/\text{2}$, the current is, and the field will point in the direction of $\mathrm{M}$.

$\begin{array}{c}I=-K_{b}w\\ =-Mw\end{array}$

Determine the magnetic field of a square loop.

Substitute $-\text{Mw}$for $I$into equation (2).

${\overrightarrow{B}}_{loop}=-\frac{2\sqrt{2}{\mu }_{0}w}{\pi a}\overrightarrow{M}$

Determine the total magnetic field at the center of the gap is then:

Substitute ${\mu }_{0}M\widehat{\varphi }$for ${\overrightarrow{B}}_{torus}$ and $-\frac{2\sqrt{2}{\mu }_{0}w}{\pi a}\overrightarrow{M}$for ${\overrightarrow{B}}_{loop}$into equation (1).

role="math" localid="1657711290769" $\begin{array}{c}\overrightarrow{B}={\mu }_0\overrightarrow{M}-\frac{2\sqrt{2}{\mu }_{0}w}{\pi a}\overrightarrow{M}\\ ={\mu }_0\overrightarrow{M}\left[1-\frac{2\sqrt{2}w}{\pi a}\right]\end{array}$

Therefore, the value of magnetic field at the center of the gap is$\overrightarrow{B}={\mu }_0\overrightarrow{M}\left[1-\frac{2\sqrt{2}w}{\pi a}\right]$ .