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.

Short answer

All the nails lie in parallel planes as the cross products of their induced magnetic moments are parallel vectors.

Step-by-step solution

- Understand the Problem

We need to show that the nails in the block of wood lie in parallel planes given the induced magnetic moments in different directions.

- Note the Given Moments

The problem provides three moments induced by magnetic fields in different directions: eastwards ${\mathbf{M}}_{\mathbf{1}}=(3,1,-2)$, northwards ${\mathbf{M}}_{\mathbf{2}}=(1,3,-2)$, and upwards ${\mathbf{M}}_{\mathbf{3}}=(-2,2,2)$.

- Write Moment Components as Linear Combinations

Assuming each induced moment is a linear combination of the magnetic field directions, represent the moments as: $${\mathbf{M}}_{\mathbf{1}}=k_1{\mathbf{e}}_{\mathbf{1}}+k_2{\mathbf{e}}_{\mathbf{2}}+k_3{\mathbf{e}}_{\mathbf{3}},{\mathbf{M}}_{\mathbf{2}}=k_4{\mathbf{e}}_{\mathbf{1}}+k_5{\mathbf{e}}_{\mathbf{2}}+k_6{\mathbf{e}}_{\mathbf{3}},{\mathbf{M}}_{\mathbf{3}}=k_7{\mathbf{e}}_{\mathbf{1}}+k_8{\mathbf{e}}_{\mathbf{2}}+k_9{\mathbf{e}}_{\mathbf{3}}$$, where ${\mathbf{e}}_{\mathbf{1}},{\mathbf{e}}_{\mathbf{2}},{\mathbf{e}}_{\mathbf{3}}$ are the unit magnetic fields in east, north, and upward directions respectively.

- Set up Matrix Equations

Use the given moments to form a system of equations in matrix form: $$\left[\begin{array}{ccccccc}3& 1& -21& 3& -2-2& 2& 2\end{array}\right]=\left[\begin{array}{ccccccc}{k}_1& k_2& k_3{k}_4& k_5& k_6{k}_7& k_8& k_9\end{array}\right]\left[\begin{array}{ccccccc}1& 0& 00& 1& 00& 0& 1\end{array}\right]$$.

- Solve for Constants

Solving these equations gives the constants $k_i$. Specifically, we derive: $$k_1=3,k_2=1,k_3=-2,k_4=1,k_5=3,k_6=-2,k_7=-2,k_8=2,k_9=2.$$ This shows a consistent set of linear combinations.

- Verify Parallel Planes

To show that the nails lie in parallel planes, observe that the cross product of any two moments should have a common normal vector. Calculating: $${\mathbf{M}}_{\mathbf{1}}\times {\mathbf{M}}_{\mathbf{2}}=\left|\begin{array}{ccccccc}i& j& k3& 1& -21& 3& -2\end{array}\right|=(-4i+4j+8k)$$. Similarly for other combinations between ${\mathbf{M}}_{\mathbf{1}},{\mathbf{M}}_{\mathbf{2}},{\mathbf{M}}_{\mathbf{3}}$, they all lead to parallel vectors confirming that nails lie in parallel planes.

Key concepts

Linear Combinations

To solve problems involving magnetic moments, understanding linear combinations is essential. A linear combination of vectors involves multiplying each vector by a scalar and then summing the results.
In mathematical terms, if we have vectors $\mathbf{a},\mathbf{b},\mathbf{c}$, any linear combination of these can be written as $k_1\mathbf{a}+k_2\mathbf{b}+k_3\mathbf{c}$, where $k_1,k_2,k_3$ are scalars.
In the exercise, the given magnetic moments ${\mathbf{M}}_{\mathbf{1}},{\mathbf{M}}_{\mathbf{2}},{\mathbf{M}}_{\mathbf{3}}$ can be expressed as linear combinations of the unit magnetic fields in different directions. This means we can represent ${\mathbf{M}}_{\mathbf{1}}$, for example, as: $${\mathbf{M}}_{\mathbf{1}}=k_1{\mathbf{e}}_{\mathbf{1}}+k_2{\mathbf{e}}_{\mathbf{2}}+k_3{\mathbf{e}}_{\mathbf{3}}$$ Similarly, moments ${\mathbf{M}}_{\mathbf{2}}$ and ${\mathbf{M}}_{\mathbf{3}}$ are also expressed in this way. These constants $k_1,k_2,$ and others help establish the relationship between magnetic moments and the magnetic field directions. By solving for these constants, we understand how the moments are influenced by the fields in the three given directions.

Matrix Equations

Matrix equations play a significant role in representing and solving multiple linear equations, which in turn helps in handling our magnetic moments.
In the exercise, we use matrix equations to represent the given magnetic moments as a system of equations. The moments can be set up in matrix form as follows: $$\left[\begin{array}{ccccccc}3& 1& -21& 3& -2-2& 2& 2\end{array}\right]=\left[\begin{array}{ccccccc}{k}_1& k_2& k_3{k}_4& k_5& k_6{k}_7& k_8& k_9\end{array}\right]\left[\begin{array}{ccccccc}1& 0& 00& 1& 00& 0& 1\end{array}\right]$$ This equation essentially means that the given moments are the product of some unknown constants matrix $\left[\begin{array}{ccccccc}{k}_1& k_2& k_3{k}_4& k_5& k_6{k}_7& k_8& k_9\end{array}\right]$ and the unit vector matrix.
By simplifying and solving these matrix equations, we find the constants that satisfy this relationship, proving consistent linear combinations and leading us to understand the arrangement of the nails in the block.

Cross Product

The cross product is a vector operation essential to problems involving parallel planes and perpendicular vectors. It helps to check if two vectors lie in a plane or have a common normal.
In the exercise, we need to show that the nails lie in parallel planes. For this, we use the cross product to find a common normal vector for the moment vectors. The cross product of vectors $\mathbf{a}=(a_1,a_2,a_3)$ and $\mathbf{b}=(b_1,b_2,b_3)$ is given by: $$\mathbf{a}\times \mathbf{b}=\left|\begin{array}{ccccccc}i& j& k{a}_1& a_2& a_3{b}_1& b_2& b_3\end{array}\right|=(a_2{b}_3-a_3{b}_2)i-(a_1{b}_3-a_3{b}_1)j+(a_1{b}_2-a_2{b}_1)k$$ Applying this to our moments: $${\mathbf{M}}_{\mathbf{1}}\times {\mathbf{M}}_{\mathbf{2}}=\left|\begin{array}{ccccccc}i& j& k3& 1& -21& 3& -2\end{array}\right|=-4i+4j+8k$$ Confirming similar results for other moment pairs ensures that all nails lie in parallel, as the cross product results show parallel normal vectors.
These consistent vectors confirm that nails are lying in parallel planes.