A magnetic field is a part of space where a force is applied at every point. That force comes from a source that can be of various nature. In the case of a magnetic fields, it is either electrical or magnetic.

Magnetic Field Lines

When you put a magnet under a thin surface and throw metal shards on top, a shape similar to the one in Figure 1 will appear. The reason for this is that, like the needle of a compass, every shard aligns with the lines of the magnetic field generated by the magnet. Those lines represent the force vectors of the field. The denser they are in a region of space, the stronger the field in that region is.

You should note that every line is a closed loop going from one pole to the other, and this is the quality that makes a field solenoidal. This differs from the electric field because of the absence of the magnetic monopole and is strictly connected to it because the magnetic and electric fields generate each other through the movement of the charges. To understand the relationship between these two fields, we must consider the Lorentz force.

The Direction of Magnetic Field: The Lorentz force

Consider an electric field, a magnetic field, and an electrically charged particle passing through them. That particle will be subject to a force that is dependent upon both fields and perpendicular to the magnetic field. These three physical quantities together always constitute an ordered triplet like the cartesian axis. To visualize this relationship, you can use Fleming's rule of the left hand. Going counterclockwise, you have two vectors and their vector product, respectively.

Fleming's left-hand rule tells us the direction of the vectorial product with respect to the direction of the vectors. The Lorentz force F has the direction of the vectorial product between the velocity v and the magnetic field B.

Fig. 2: Fleming's left hand rule. StudySmarter Originals

The expression for the Lorentz force is:

Here q is the charge, E is the electric field, v is the speed of the particle, and B is the magnetic field.

Applying it to the Lorentz force, this has the direction of the thumb when the velocity is along the middle finger, and the magnetic field is parallel to the index finger. There is a also useful rule called the right hand rule to see the magnetic field of a live wire by pointing the thumb in the direction of the current, while the other fingers grabbing the wire represent the magnetic field.

Magnetic Fields: What is the difference between H and B?

When considering the vacuum as the area where the field is present, it is roughly the same to consider the magnetic field strength or the magnetic field flux density because these quantities are proportional. But within a material, we can observe the difference between them. The flux is the portion of the field passing through a surface every second.

Magnetic flux density

The term magnetic field is used to describe two different (but deeply related) quantities. H is the magnetic field's intensity and B is the magnetic flux density These quantities in the vacuum are proportional because of vacuum permeability.

Instead, within a material, the magnetization M is subtracted from the term calculating the field B. This is also called magnetic flux density :

Magnetization takes into account the effect of the magnetic polarization in the material, a concept that is dual to the electric polarization and gives symmetry to the formulas describing the electromagnetic field.

What is electromagnetic induction?

Electric fields and magnetic fields generate each other, but how does that happen? The explanation is electromagnetic induction, which is the phenomenon causing a current to be generated into a conductor thanks to a magnetic field.

Generating an electromotive force with a magnetic field (Faraday's law)

Lenz's law states that the (induced) magnetic field generated by the induction current within a material by an external magnetic field opposes this last one. This makes perfect sense as, otherwise, the magnetic field within the material will increase exponentially.

Faraday studied the effect of a magnetic field applied to an electric circuit and discovered that moving a magnet with the circuit immersed into its field, or moving a magnet into a loop formed by the circuit, causes a current to flow in the circuit itself. Moreover, this current is proportional to the speed of the movement that is represented by the time derivative d / dt of the magnetic field flux ΦB:

EMF here is the electromotive force measured in Volts.

Magnetic Fields: Alternating current

An invention that works on the principle of induction and uses the magnetic field to generate a current is the induction motor that produces an AC current. It consists of a magnetically charged armature called a stator, which contains a moving part called a rotor. The rotor is connected to the circuit through wirings that sense the magnetic field, which always faces the same direction but imposes a current that changes as the rotor moves.

An alternating current is created, which has a sinusoidal waveform and is expressed in the form:

Fig. 3: The movement of a copper coil (in copper color) inside a magnet (left) produces a current flow that alternates in time (right).

It should be noted that by inverting the functions of the stator and the rotor in a magnetic field generating a current, you obtain a circuit that generates mechanical movement.

Magnetic Fields: Inductance and flux linkage

Loops interact with the magnetic field. Thus, the element that inherits this quality into an electrical circuit is the inductor. To study how it works, let us make some definitions. The flux linkage , which is mainly used in engineering applications, is defined as the total flux passing through a coil. This is obtained by multiplying the magnetic flux by the number N of loops:

The inductance, identified by the letter L, is defined as the magnetic linkage of an object divided by the current that causes that flux:

When the current causing the flux linkage is also the one linking the coil (there are no currents external to the coil), this equation becomes simpler, as there is no more need for subscripts. It then takes the name of self-inductance.