91Ó°ÊÓ

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 $\text{w}≪\text{a}≪\text{L}$.

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

$\stackrel{\mathbf{→}}{\mathbf{B}}={\stackrel{\mathbf{→}}{\mathbf{B}}}_{\mathrm{torus}}−{\stackrel{\mathbf{→}}{\mathbf{B}}}_{\mathrm{loop}}$ …… (1)

Here, ${\stackrel{\mathbf{→}}{\mathbf{B}}}_{\mathrm{torus}}$is the field of this solenoid and${\stackrel{\mathbf{→}}{\mathbf{B}}}_{\mathrm{loop}}$ 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 $\text{L}≫\text{a}$. 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.

$\begin{array}{c}{\stackrel{→}{B}}_{torus}={\mathrm{μ}}_0\stackrel{→}{M}\\ ={\mathrm{μ}}_{0}M\widehat{\mathrm{ϕ}}\end{array}$

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

$B_{loop}=\frac{\sqrt{2}{\mathrm{μ}}_{0}l}{\mathrm{π}R}$ …… (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).

${\stackrel{→}{B}}_{loop}=−\frac{2\sqrt{2}{\mathrm{μ}}_{0}w}{\mathrm{π}a}\stackrel{→}{M}$

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

Substitute ${\mathrm{μ}}_{0}M\widehat{\mathrm{ϕ}}$for ${\stackrel{→}{B}}_{torus}$ and $−\frac{2\sqrt{2}{\mathrm{μ}}_{0}w}{\mathrm{π}a}\stackrel{→}{M}$for ${\stackrel{→}{B}}_{loop}$into equation (1).

role="math" localid="1657711290769" $\begin{array}{c}\stackrel{→}{B}={\mathrm{μ}}_0\stackrel{→}{M}−\frac{2\sqrt{2}{\mathrm{μ}}_{0}w}{\mathrm{π}a}\stackrel{→}{M}\\ ={\mathrm{μ}}_0\stackrel{→}{M}\left[1−\frac{2\sqrt{2}w}{\mathrm{π}a}\right]\end{array}$

Therefore, the value of magnetic field at the center of the gap is$\stackrel{→}{B}={\mathrm{μ}}_0\stackrel{→}{M}\left[1−\frac{2\sqrt{2}w}{\mathrm{π}a}\right]$ .