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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.:
Whenever an electric current is present in our system, it generates a magnetic field. An electrical current is a collective effect achieved by the movement of charges from one point to another.
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 ea 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.
Consider a wire carrying a current of 5·107 A. If we apply the formula for the created field, we find the following radial dependence:
\(\vec{B}_1 = \frac{\mu_0 \cdot I_1}{2 \cdot \pi \cdot r} \cdot \vec{e}_a = \frac{1}{r} \cdot \vec{e}_a [T]\)
This means that at 1 metre, the field will have a value of 1 T, while at 2 metres, it will have a value of 0.5 T.
Now imagine we place another wire parallel to the previous one at a distance of 10 metres. Double the current is flowing through that wire, that is 1⋅108 A. The first wire is 2 metres long, while the second one is 1 metre.
We can first compute the field created by the second wire, which, applying the formula, is:
\(\vec{B}_2 = \frac{\mu_0 \cdot I_2}{2 \cdot \pi \cdot r} \cdot \vec{e}_a = \frac{2}{r} \cdot \vec{e}_a [T]\)
So, if we want to calculate the force exerted by one wire on the other, we simply need to use the formula for the force. In this case, since the wires are parallel, the fields created by them are ensured to be perpendicular to the direction of the current. The application of the formula yields:
\(F_{1 \rightarrow 2} = B_1 \cdot I_2 \cdot L_2 = 9.98 \cdot 10^6 N \)
\(F_{2 \rightarrow 1} = B_2 \cdot I_1 \cdot L_1 = 2.00 \cdot 10^7 N \)
Key takeaways
Currents formed by moving electrical charges create magnetic fields.
Magnetic flux density is the measure of the strength of the magnetic field.
Currents (and the wires through which they travel) are affected by magnetic fields and impacted by magnetic forces.
The magnetic flux density is a vector field that does not exert a force on the direction to which it points but rather in a direction perpendicular to it.
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Frequently Asked Questions about Magnetic Flux Density
What is magnetic flux density?
It is the vector field measuring the strength of the magnetic field.
How do we calculate magnetic flux density?
For a wire carrying a current, one needs to take into account the radial distance to the wire and the intensity of the current. The formula is B = μ0·I/2·π·r.
What is magnetic flux density measured in?
It is measured in Teslas.
Is magnetic flux density a vector?
Yes, it has a spatial direction that does not signal the direction of the associated force.
Why do we use magnetic flux density?
Because it allows us to measure the strength of the magnetic field per unit of length, area, or volume. This is necessary because we usually deal with infinitely large objects for simplicity.
Why is magnetic flux density a vector?
Magnetic flux density is in fact a scalar as it is formed from the scalar product of two vectors, the field vector B and the area vector A.
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