Introduction
Man has been fascinated by magnetism since the very first observations of the properties of naturally occurring magnets, and today, it is used in electric motors, electric generators, and particle accelerators. The essence of this mesmerising phenomenon is the magnetic force notion, which is a primary interaction that guides the behaviour of moving charges (in magnetic fields).
Similar to the action of the electric charges in an electric field, moving charges produce and react to magnetic fields. This manuscript will methodically discuss the character of magnetic force, obtain some important relations, and provide examples to amplify the understanding.
Understanding the concept of fields
Electric field as a starting point
Before going into magnetic fields, it will be wise to refer to the conceptual basis of the electric field. A charge Q, an electric field is produced in the surroundings of a charge, which serves as a source. The electric field is given by
where:
r̂is the unit vector along the line joining the charge to the point of observation,- ε0 is the permittivity of free space.
A test charge q placed in this field experiences a force:
This concept of field is powerful because it allows us to describe interactions at a distance. The source charge sets up a field, and any other charge interacts with that field to experience a force.
From electric to magnetic fields
Sources of magnetic fields
Similarly to the manner in which the stationary charges create electric fields, the moving charges or electric currents create magnetic fields. Such fields were first experimentally confirmed by Oersted and later mathematically by the work of Ampere, Faraday, and Maxwell.
We indicate the magnetic field as a vector field B(r). It is specified at any spatial point and can have time dependence. Although it has certain similarities with the electric field, the magnetic field exists only as a result of the movement of charges or a change in the electric field with time.
One basic concept that can be applied to electric and magnetic fields is the concept of superposition. Based on this, therefore, in the presence of multiple sources, the magnetic field at any point is the vectorial addition of the individual sources generated by the sources.
Lorentz Force: The Combination of Electric and Magnetic Forces
The General Expression
A charge that is in motion undergoes a force of combination, called the Lorentz force, when it is put in an electric field and a magnetic field.
F = q[E(r) + v × B(r)]
The equation shows that the total force on a charge can be broken down into two separate components:
- Electric force (qE) – Acts along the direction of the electric field.
- Magnetic force (qv×B) – Acts perpendicular to both the velocity of the charge and the magnetic field.
Key Characteristics of Lorentz Force
- The force is proportional to the charge, q, the velocity, v, and the magnetic field, B.
- The right-hand rule is used to find the direction of the magnetic component (see Figure 1).
- When the velocity is either parallel or anti-parallel to B with the magnetic field, the magnetic force disappears.
- The force is never parallel to either v or B, and thus does not change the speed of the body but instead changes the direction.
These concepts are shown in Figure 1. In sub-figure (a), a positively charged particle is deflected in the right-hand rule, and in sub-figure (b), a negatively charged particle gets deflected in the reverse direction.
Units of Magnetic Field
SI and Non-SI Units
The magnetic field in the International System of Units is measured in Tesla (T), named after the inventor and scientist, Nikola Tesla (1856-1943).
By definition,
Thus,
A smaller unit often used in practice is the Gauss (G):
1 Tesla = 104 Gauss
Order of Magnitudes of Magnetic Fields
Magnetic fields can vary drastically depending on the situation. Table 4.1 shows the order of magnitudes of magnetic fields in various physical contexts:
| Physical Situation | Magnitude of B (Tesla) |
|---|---|
| Surface of a neutron star | 108 |
| Typical large field in a lab | 1 |
| Near a small bar magnet | 10–2 |
| On the earth's surface | 10–5 |
| Human nerve fibre | 10–10 |
| Interstellar space | 10–12 |
This wide range—from the unimaginably strong fields on neutron stars to the weak fields in interstellar space—shows the diversity of magnetic phenomena.
Magnetic Force on a Current-Carrying Conductor
Force on Mobile Carriers
So far, we have considered the force on a single moving charge. But what happens in a conductor carrying many charges?
Let:
- n = number of mobile carriers per unit volume
- A = cross-sectional area of the conductor
- l = length of conductor
- q = charge of each carrier
- vd = drift velocity
The total number of carriers = nAl.
Thus, the total force on all carriers:
F = (nAl)qvd × B
Since nqvdA = I (current), this simplifies to:
F = I l × B
This is the force on a straight conductor of length l carrying current I in a magnetic field B.
For an arbitrarily shaped conductor:
F = ∑I dl × B
This expression can be generalized using integration.
On Permittivity and Permeability
Comparing Forces
Similar to the law of gravitation that gives Newton the force between two masses, the law of Coulomb gives the force between two charges.
- Gravitational force:
- Electrostatic force:
Here, ε0 (permittivity of free space) is used in a similar way as the gravitational constant, G.
Meaning of Constants
- Electric Permittivity (ε): It is the measure of the degree to which a material opposes the creation of an electric field.
- Magnetic Permeability (μ): This is what determines the ability of a substance to allow the setting up of a magnetic field.
These two constants define the behavior of materials to the electric and magnetic fields, respectively.
Worked-Out Examples
Example 1 – Force on a Current-Carrying Wire
A current of 2A is induced on a straight wire of mass 200g and length 1.5m. The wire is hanging in the air in a homogeneous horizontal magnetic field B (see Fig. 2). Determine the magnitude of B.
Solution:
For suspension, the magnetic force balances the weight:
Substituting values:
Thus, the required magnetic field is 0.65 Tesla.
Example 2 – Direction of Lorentz Force
A charged particle moves along the positive x-axis, while the magnetic field is along the positive y-axis (Fig. 3). Determine the direction of the Lorentz force for:
(a) An electron (negative charge)
(b) A proton (positive charge)
Solution:
F = q(v × B)
- Here, v is along +x, B is along +y.
- Using the right-hand rule, v × B points along +z.
So:
- For an electron (q<0): Force along –z.
- For a proton (q>0): Force along +z.
Conclusion
Magnetic force is a principle in the science of physics, which is used to bring the phenomena of electricity and magnetism into the general picture of electromagnetism. It is therefore fixed that:
- At rest, the charges will produce electric fields, and in motion, the charges will produce magnetic fields.
- The law of the Lorentz force is a single expression that is a combination of the electric and magnetic effects on the net force that a charged particle experiences.
- The principle of magnetic force acting on conductors is the basis of the work of motors, dynamos, and numerous modern technologies.
- Such quantities as permittivity and permeability are fundamental to the understanding of how materials behave in applied fields.
- These principles were applied to real-life, practical problems through the solution of illustrative examples (Fig. 2 and Fig. 3).
To conclude, magnetic force is no longer limited to the plane of a mere theory; it is actively the driver of the technology of civilization, both in everyday appliances and in the avant-garde field of space exploration.