Magnetic Flux Density

We are used to hearing about electric fields and magnetic fields. However, we also hear the term electromagnetic field, which suggests that both fields can be combined. Indeed, electric fields and magnetic fields are not independent but are two sides of the same coin, although, in many examples and applications, this dependence can be neglected.

Although the study of Maxwell’s laws, which describe the full behaviour of electromagnetism, is without the scope of this article, we briefly mention one aspect, i.e.:

The magnetic flux density is the measure of the strength of the magnetic field. It is a vector field that indicates the direction of the magnetic field acting on a certain region of space. From now on, it will be useful to consider electric currents as the basic objects of magnetic interactions, just as electric charges are the basic objects for electric interactions.

How is magnetic flux density produced?

Let us consider an infinitely long straight wire carrying a certain electric current with the intensity of I.

In the following, we are going to analyse two different settings, which can be mixed up since wires carrying an electric current can create a magnetic flux density and be affected by electric fields in the same way that electric charges can create an electric field and be affected by other electric fields.

Generation of magnetic flux density with a wire

For our wire, we directly state the formula of the magnetic flux density it creates:

\[\vec{B} = \frac{\mu_0 \cdot I}{2 \cdot \pi \cdot r} \cdot \vec{e}_a\]

Here, vector B is the magnetic flux density, r is the radial distance from the wire, vector e_a is the vector twisting around the wire, and μ₀ is the vacuum permeability with an approximate value of 1.26⋅10^-6 T M/A. A Tesla (T) is a unit defined as kg/s² A, with A being amperes. The image below shows the field lines for a wire.

Figure 1. Magnetic field B created by a current I flowing through a wire. Source: Stannered, Wikimedia Commons (CC BY-SA 3.0).

We are defining this field for an infinitely long wire, so it makes sense to consider quantities such as the magnetic flux density since we are only considering a quantity by area rather than a whole quantity defined on an infinite region.

Experimental definition of magnetic flux density

Consider again the infinitely long wire with a current I. However, at this point, we are not interested in the magnetic flux density it creates. Instead, we are going to consider the presence of a magnetic flux density that is generated by an external source. We only require this magnetic flux density to be constant in space with a fixed value of B.

By placing the wire with the current under the influence of the magnetic flux density, a force will affect the wire in the same way an electric field moves an electric charge. However, the rules for this to happen are more complex.

In general, magnetic fields behave ‘perpendicularly’ to electric fields. This can be seen in Figure 1, where the magnetic field is perfectly perpendicular to the direction of the current. This general feature is translated to how magnetic fields affect currents.

To determine the direction in which a magnetic flux density affects a current, we need to use the rule of the right hand, shown in the image below.

Right-hand rule for currents and magnetic fields.

Essentially, the more perpendicular the magnetic flux density is to the current, the more effectively it will affect the wire. The direction of the force exerted is perpendicular to both the field and the current. This also implies that if the current and the magnetic flux density are in the same direction, the current will not suffer any effect at all.

Supposing the magnetic flux density is perfectly perpendicular to the current, the formula for the force exerted is:

\[F = I \cdot B \cdot L\]

Here, L is the length of the wire. If we consider a wire of finite length, the formula makes sense, whereas it does not make sense for an infinitely long wire. This is why we define the magnetic flux density as the force exerted per unit of length for a current of 1 ampere.