A block of wood contains a number of thin soft-iron nails (of constant permeability). A unit magnetic field directed eastwards induces a magnetic moment in the block having components (3, 1,−2), and similar fields directed northwards and vertically upwards induce moments (1, 3,−2) and (−2,−2, 2)

Question

A block of wood contains a number of thin soft-iron nails (of constant permeability). A unit magnetic field directed eastwards induces a magnetic moment in the block having components (3, 1, −2), and similar fields directed northwards and vertically upwards induce moments (1, 3, −2) and (−2,−2, 2) respectively. Show that all the nails lie in parallel planes.

Accepted Answer

The magnetic moment M, the permeability µ and the magnetic field H for iron of constant pemeability are connected by [latex]M_i=\mu_{i j} H_j[/latex]. Taking the 1-, 2- and 3-directions as East, North and vertical, µ has the form

[latex]\left(\begin{array}{ccc}3 & 1 & -2 \1 & 3 & -2 \-2 & -2 & 2\end{array} ight)[/latex].

By adding the first two columns to twice the third one, it can be seen that this matrix has zero determinant. The matrix therefore has at least one zero eigenvalue. (The same conclusion can be reached using the routine method for finding eigenvalues; they are 0, 2 and 6.)

Thus, a field parallel to the eigenvector corresponding to this zero eigenvalue will induce no moment in the block. Physically this means that all the nails lie in planes to which this direction is a normal. To find the direction we solve

[latex]\left(\begin{array}{ccc}3 & 1 & -2 \1 & 3 & -2 \-2 & -2 & 2\end{array} ight)\left(\begin{array}{l}x \y \z\end{array} ight)=\left(\begin{array}{l}0 \0 \0\end{array} ight) \Rightarrow\left(\begin{array}{l}x \y \z\end{array} ight)=\left(\begin{array}{l}1 \1 \2\end{array} ight) .[/latex]

We conclude that all the nails lie at right angles to this direction.

Question 26.19: A block of wood contains a number of thin soft-iron nails (o...

This Question has been Answered.

Already have an account?

Login

Related Answered Questions