The mobility of charges causes magnetic force, which is a result of magnetic force. Have you ever noticed what type of force acts on the needle of a magnetic compass? Do you know how DC/AC motors rotate? It’s all due to magnetic force. 

The interaction of moving electric charges is the fundamental nature of magnetism. Electric forces act on electric charges whether the charges are moving or not, but magnetic forces act only on moving charges and current-carrying wires.

What is Magnetic Force?

The magnetic force is a consequence of the electromagnetic force and is caused by the motion of charges. An unknown electric field can be determined by the magnitude and direction of the force on a test charge \((q)\) at rest. To explore an unknown magnetic field (denoted by \(\overrightarrow B \)), we must measure the magnitude and direction of the force on a moving test charge.
Magnitude: The magnitude of the magnetic force \((F)\) on a charge \((q)\) moving at a speed \((v)\) in a magnetic field of strength \((B)\) is given by:
\(\overrightarrow F = \;qvB\,\sin \theta \,\widehat n\)
where \(θ\) is the angle between the directions of \(v\) and \(B\).
Direction: The direction of the magnetic force \((F)\) is perpendicular to the plane formed by \(v\) and \(B\), and it can be easily determined by the right-hand rule. According to the right-hand rule, to find the direction of the magnetic force on a positive moving charge, the thumb of the right-hand should point in the direction of \(v\), the fingers should be in the direction of \(B\), and the force \((F)\) is directed perpendicular to the right-hand palm. And, the direction of the force \((F)\) on a negative charge is in the opposite sense to that above (so pointed away from the back of your hand). Two objects containing a charge with the same direction of motion have a magnetic attraction force between them. Similarly, objects with charges moving in opposite directions have a repulsive force between them.

SI Unit: The SI unit for the magnitude of the magnetic field strength is called the tesla \((\rm{T})\), which is equivalent to one Newton per ampere-meter. Sometimes the smaller unit is taken as gauss \((10^{-4}\,\rm{T})\).

Lorentz Force

When the expression for the magnetic force is combined with that for the electric force, the combined expression is known as the Lorentz force.
Consider a point charge \((q)\) (moving with a velocity \(v\) and, located at \(r\) at a given time \(t\)) in the presence of both the electric field \([E(r)]\) and the magnetic field \([B(r)]\). Now, The total force on an electric charge \((q)\) due to both electric and magnetic fields can be written as:
\(\overrightarrow F = \;q\left[ {\overrightarrow E \left( r \right) + \overrightarrow v \times \overrightarrow B \left( r \right)} \right] \equiv {\overrightarrow F _{{\text{electric}}}} + {\overrightarrow F _{{\text{magnetic}}}}\)

This force was given first by H.A. Lorentz based on the extensive experiments of Ampere and others. The following features were observed during the interaction with the magnetic field are:

1. It depends onthe charge \((q)\) of the particle, the velocity \((v)\), and the magnetic field \((B)\), and the force on a negative charge is opposite to that on a positive charge.
2. The magnetic force \(\left[ {q\left( {\overrightarrow v \times \overrightarrow B } \right)} \right]\) includes a vector product of velocity and magnetic field. If velocity and magnetic field are parallel or anti-parallel, then the vector product makes the force due to the magnetic field becomes zero. The force acts in a direction perpendicular to both the velocity and the magnetic field. Its direction is given by the screw rule or right-hand rule for vector (or cross) product as shown in the below figure.
3. The magnetic force is zero if the charge is not moving (as then \(|v |= 0\)).

Magnetic Force on Current-carrying Conductor

When a charged particle is in motion, it experiences a magnetic force in a magnetic field. Similarly, when a current-carrying wire is placed in a magnetic field, it also experiences a force. The current-carrying wire experienced magnetic force due to moving electrons in it. Suppose a conducting wire, carrying a current \(\left( i \right)\), is placed in a magnetic field \(\left( {\overrightarrow B} \right)\).
Consider a small element \((dl)\) of the wire shown in Fig (c). The electrons drift with a speed of \(v_d\) opposite to the direction of the current. The relation between the current \((i)\) and the drift speed \((v_d)\) is,

Fig. (c)

\(i = jA = nev_d A\) …..(i)
Where \(A\) is the area of a cross-section of the wire and \(n\) is the number of electrons per unit volume. Each electron experiences an average magnetic force of
\(\overrightarrow f = – e\overrightarrow {{v_d}} \times \overrightarrow B \)
And the number of electrons in the small element considered is \(nAdl\). Then, the magnetic force on the wire of length \((dl)\) will be
\(d\overrightarrow F = \left( {nAdl} \right)\left( { – e\overrightarrow {{v_d}} \times \overrightarrow B } \right)\)
If we denote the length \((dl)\) along the direction of the current by \(d\overrightarrow {l}\) the above equation becomes as
\(d\overrightarrow F = nAe{v_d}d\overrightarrow {l} \times \overrightarrow B \)
Using equation (i)
\(d\overrightarrow F = i d\overrightarrow {l} \times \overrightarrow B \) …….(ii)
The quantity \(i d\overrightarrow {l}\) is called a current element. If a straight wire of length \((l)\) carrying a current \((i)\) is placed in a uniform magnetic field \(\overrightarrow B \), the force on it is will be
\(\overrightarrow F = i\overrightarrow l \times \overrightarrow B \) …….(iii)

Summary

If a charged particle with velocity \((v)\) is moving in a magnetic field \((B)\), then because of the interaction between the magnetic field produced by moving charge and the magnetic field applied, the charge \((q)\) experiences a magnetic force.
The magnitude of the magnetic force \((F)\) on a charge \((q)\) moving at a speed \((v)\) in a magnetic field of strength \((B)\) is given by \(F = qvB\sin \,\theta \widehat n\). Where \(θ\) is the angle between the directions of \(v\) and \(B\) and the direction of the magnetic force \((F)\) is perpendicular to the plane formed by \(v\) and \(B\), and it can be easily determined by the right-hand rule.
Lorentz force is the total force on a charge \((q)\) moving with velocity \((v)\) in the presence of both magnetic fields and electric fields. It is given by the expression: \(F = q\left[ {E\left( r \right) + v \times B\left( r \right)} \right].\)
If a straight wire of length \((l)\) carrying a current \((i)\) is placed in a uniform magnetic field \((\overrightarrow B)\), the force on it is will be: \(\overrightarrow F = i\overrightarrow l \times \overrightarrow B \)

Solved Problems

Q.1. A charge of \(2.0\,\rm{μC}\) moves with a speed of \(2.0 × 10^6 \,\rm{m}\,\rm{s}^{-1}\) along the positive \(x-\)axis. A magnetic field \((\overrightarrow B)\) of strength \(\left( {0.20\overrightarrow j + 0.40\overrightarrow k } \right){\text{T}}\) exists in space. Then, what is the magnetic force acting on the charge?
Ans: The force on the charge \(= q\overrightarrow v \times \overrightarrow B\)
\(F = q\overrightarrow v \times \overrightarrow B \)
\(\Rightarrow F = \left( {2.0 \times {{10}^{ – 6}}{\text{C}}} \right)\left( {2.0 \times {{10}^6}\;{\text{m}}\;{{\text{s}}^{ – 1}}\;\overrightarrow i \;} \right) \times \left( {0.20\,\overrightarrow j + 0.40\,\overrightarrow k } \right){\text{T}}\)
\(\Rightarrow F = 4.0\left( {0.20\,\overrightarrow i \times \overrightarrow j + 0.40\,\overrightarrow i \times \overrightarrow k } \right){\text{N}}\)
\( \Rightarrow F = \left( {0.8\,\overrightarrow k – 1.6\,\overrightarrow j } \right){\text{N}}.\)

FAQs

Q.1. How magnetic force is created?
Ans: Magnetic force is created by either permanent magnets, by an electric current, or by other moving charges. The magnetic force may be both repulsive and attractive.

Q.2. How to find the direction of a magnetic force using the right-hand rule?
Ans: The right-hand rule states that to find the direction of the magnetic force on a positive moving charge, the thumb of the right-hand point in the direction of \(v\), the fingers lie in the direction of \(B\), and the force \((F)\) is directed perpendicular to the right-hand palm.

Q.3. Write the SI unit for the magnetic field strength?
Ans: The SI unit for the magnitude of the magnetic field strength is called the tesla \((\rm{T})\), which is equivalent to one Newton per ampere-meter. And, the smaller unit is gauss \((10^{-4}\,\rm{T})\) is used instead.

Q.4. What are the common examples of the magnetic force?
Ans: There are many examples of magnetic force such as attraction and repulsion of two magnets, the force which acts on the needle of a compass is a magnetic force, the forces acting in DC/AC motors due to which they rotate are magnetic forces, and magnetic forces are also used in the particle accelerators.

Q.5. What type of force is a magnetic force?
Ans: Magnetic force is an electromagnetic force. It is a field force, magnetic force, attraction, or repulsion that arises between electrically charged particles because of their motion. It is the basic force responsible for such effects as the action of electric motors and the attraction of magnets for iron.

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