$$\require{cancel}$$

6.3: Definition of the Magnetic Field

• • Contributed by Jeremy Tatum
• Emeritus Professor (Physics & Astronomy) at University of Victoria

We are going to define the magnitude and direction of the magnetic field entirely by reference to its effect upon an electric current, without reference to magnets or lodestones. We have already noted that, if an electric current flows in a wire in an externally-imposed magnetic field, it experiences a force at right angles to the wire.

I want you to imagine that there is a magnetic field in this room, originating, perhaps, from some source outside the room. This need not entail a great deal of imagination, for there already is such a magnetic field – namely, Earth’s magnetic field. I’ll tell you that the field within the room is uniform, but I shan’t tell you anything about either its magnitude or its direction.

You have a straight wire and you can pass a current through it. You will note that there is a force on the wire. Perhaps we can define the direction of the field as being the direction of this force. But this won’t do at all, because the force is always at right angles to the wire no matter what its orientation! We do notice, however, that the magnitude of the force depends on the orientation of the wire; and there is one unique orientation of the wire in which it experiences no force at all. Since this orientation is unique, we choose to define the direction of the magnetic field as being parallel to the wire when the orientation of the wire is such that it experiences no force.

This leaves a two-fold ambiguity since, even with the wire in its unique orientation, we can cause the current to flow in one direction or in the opposite direction. We still have to resolve this ambiguity. Have patience for a few more lines.

As we move our wire around in the magnetic field, from one orientation to another, we notice that, while the direction of the force on it is always at right angles to the wire, the magnitude of the force depends on the orientation of the wire, being zero (by definition) when it is parallel to the field and greatest when it is perpendicular to it.

Definition. The intensity $$B$$ (also called the flux density, or field strength, or merely “field”) of a magnetic field is equal to the maximum force exerted per unit length on unit current (this maximum force occurring when the current and field are at right angles to each other).

The dimensions of $$B$$ are

$\frac{\text{MLT}^{-2}}{\text{LQT}^{-1}}=\text{MT}^{-1}\text{Q}^{-1}.$

Definition. If the maximum force per unit length on a current of 1 amp (this maximum force occurring, of course, when current and field are perpendicular) is 1 N m-1, the intensity of the field is 1 tesla (T).

By definition, then, when the wire is parallel to the field, the force on it is zero; and, when it is perpendicular to the field, the force per unit length is $$IB$$ newtons per metre.

It will be found that, when the angle between the current and the field is $$\theta$$, the force per unit length, $$F'$$, is

$F'=IB\sin \theta .$

In vector notation, we can write this as

$\textbf{F}'=\textbf{I}\times \textbf{B},\label{6.3.2}$

where, in choosing to write $$\textbf{I}\times \textbf{B}$$ rather than $$\textbf{F}'=\textbf{B}\times \textbf{I}$$ we have removed the two-fold ambiguity in our definition of the direction of $$\textbf{B}$$. Equation \ref{6.3.2} expresses the “right-hand rule” for determining the relation between the directions of the current, field and force.