Lorentz Force – Definition, Formula, Examples

Lorentz Force is the sum of Magnetic and Electrical Force experienced by a charge. When charges are moving under the influence of a magnetic field. They experience forces on them which cause them to sometimes change their direction, or if they are not able to do that. The forces of these individual charges become a force on the conductor carrying them. There are a lot of applications of this particular phenomenon in real life. All the motors that are employed in many devices around us work on this principle. To understand the working of these devices, it becomes essential to understand this concept. Let’s look at this concept in detail. 

Lorentz force is the combination of the magnetic and electric forces created by an electromagnetic field. The Lorentz force is a fundamental notion in electromagnetism that describes the force experienced by a charged particle moving through an electromagnetic field. This force, named after the Dutch physicist Hendrik Lorentz, is critical to understanding the behavior of charged particles in a variety of physical situations, ranging from fundamental physics to practical technological applications.

Let’s say there is a point charge “q” which is moving with the velocity “v” and located at “r” at a time “t” in the presence of both electric field E(r) and magnetic field B(r). Both of these fields apply some force on the charge under their influence. The force on the charge due to their influence was first given by H.A Lorentz. The formula for this force was derived by Lorentz on the basis of rigorous experimentation that was done by Ampere and others. 

The force on an electric charge “q” due to both of these fields is given by, 

F = q [E(r) + v × B(r)]

F = F_elec + F_mag

This force is called Lorentz Force. 

Lorentz Force Formula for Continuous Charge Distribution

The Lorentz force law for a continuous charge distribution illustrates how an electric and magnetic field affects a charge distribution that is dispersed across space rather than concentrated at discrete spots.

dF = ρ(r)[E(r) + v(r) × B(r)] dv

where,

• ρ(r) is charge density at position r
• v(r) is velocity
• E(r) is electric field
• B(r) is Magnetic Field

Lorentz Force Formula on a Current-Carrying Wire

The force F on a straight segment of wire of length L, carrying a current I, in the presence of a magnetic field B, is given by:

F = I(L × B)

where,

• I is the electric current flowing through the wire
• L represents the length and direction of the wire segment in which the current flows
• B is magnetic field vector, indicating the strength and direction of the magnetic field at the location of the wire.

Relation between Magnetic Field and Lorentz Force

Looking at the formula, the relation between the electric field and the force which charges experience under its influence is known. In case of force experienced under the influence of the magnetic field, the following observations are made: 

1. It depends on q, v, and B (charge, velocity of the particle, and magnetic field). In the case of a negative charge, the direction of the force is reversed.
2. There is a vector product between the velocity and magnetic field. The direction of force is perpendicular to both of the quantities. In case, velocity and magnetic field become parallel. The force on the charge becomes zero.
3. Magnetic force on any charge is zero if it’s not moving that is |v| = 0.

The above figure shows the direction of the magnetic force acting on the particle. The force on the positively charged particle with velocity “v” and making an angle θ with the direction of the magnetic field is given by the right-hand rule.  

It shows a moving charged particle that is deflected from its path due to a magnetic field. Notice that both of the charges get deflected in different directions. 

The direction of the force F can be determined using the right-hand rule:

• Point your fingers in the direction of the current (L).
• Curl your fingers toward the direction of the magnetic field (B).
• Your thumb will point in the direction of the force F on the wire.

Unification of Electric and Magnetic Fields:

• The Lorentz force is central to Maxwell’s equations, which describe how electric and magnetic fields propagate and interact with matter.
• It highlights the interconnected nature of electricity and magnetism.

Foundation of Electrodynamics:

• Understanding the Lorentz force is crucial for studying the behavior of charged particles in fields, which is essential in both classical and modern physics, including quantum mechanics and relativity.

The application of Lorentz Force are mentioned below:

Cathode Ray Tube (CRT): Used in old televisions and oscilloscopes. The Lorentz force is utilized to direct electron paths, resulting in images on the screen.

Mass Spectrometry: Separates ions according to their mass-to-charge ratio. The Lorentz force deflects ions in a magnetic field, allowing their mass to be determined.

Cyclotrons and synchrotrons: The Lorentz force bends the path of charged particles, allowing them to accelerate at high speeds in circular or spiral routes.

Hall Effect Sensors: It Measure magnetic fields. The Lorentz force creates a voltage difference across a conductor, which can be used to calculate the magnetic field intensity.

Electromagnetic induction: It is Fundamental to the operation of electric generators and transformers. The motion of conductors in magnetic fields induces an electromotive force due to the Lorentz force acting on the charge carriers.

Also, Check

• Force on a Current Carrying Conductor in Magnetic Field
• Force between Two Parallel Current Carrying Conductors
• Motion of a Charged Particle in a Magnetic Field

Example 1: Find out the magnitude of the force experienced when the unit charge is kept under the influence of 5 N/C electric fields. 

Solution: 

The for on the charge is given by, 

F = qE

⇒ F = (1)(5) 

⇒ F = 5 N/s. 

Example 2: Find out the magnitude of the force experienced when a 5C charge is kept under the influence of 25 N/C electric fields. 

Solution: 

The for on the charge is given by, 

F = qE

⇒ F = (5)(25) 

⇒ F = 125 N/s. 

Example 3: Find out the magnitude of the force experienced when a 5C charge is moving at 10m/s under the influence of a 25 N/C electric field. The magnetic field of 10 magnitudes is perpendicular to the direction of the electric field and velocity. Find out the magnitude of the force experienced by the charge.  

Solution: 

The for on the charge is given by, 

F = qE + q(v × B)

⇒ F = (5)(25) + 5 (10 × 10 × sin(90))

⇒ F = 125 + 5(100) 

⇒ F = 125 + 500 

⇒ F = 625 N

Example 4: Find out the magnitude of the force experienced when a 10C charge is moving at 10m/s under the influence of 5 N/C electric fields. The magnetic field of 5 magnitudes is perpendicular to the direction of the electric field and velocity. Find out the magnitude of the force experienced by the charge.  

Solution: 

The for on the charge is given by, 

F = qE + q(v × B)

⇒ F = (10)(5) + 10 (10 × 5 × sin(90))

⇒ F = 50 + 500 

⇒ F = 550 N

Example 5: Find out the magnitude of the force experienced when a  -2C charge is moving at 10m/s under the influence of 5 N/C electric fields. The magnetic field of 5 magnitudes is 30° to the direction of the electric field and velocity. Find out the magnitude of the force experienced by the charge.  

Solution: 

The for on the charge is given by, 

F = qE + q(v × B)

⇒ F = (-2)(5) + (-2) (10 × 5 × sin(30))

⇒ F = -10 + -100 × 0.5

⇒ F = -60 N

What is meaning of Lorentz Force?

Lorentz Force is the force equivalent to sum of electric and magnetic force due to an electromagnetic field

What is Expression for Lorentz Force?

The expression for Lorentz Force is given as F = qE + qv × B

How does the Lorentz force act on a stationary charged particle?

For a stationary charged particle (\v=0), the Lorentz force is solely due to the electric field: F = qE

Does the Lorentz force apply to neutral particles?

No, the Lorentz force applies only to charged particles. Neutral particles, which do not have an electric charge, do not experience a Lorentz force in an electric or magnetic field. However, they may interact through other forces depending on the context (e.g., gravitational forces or weak nuclear forces).

What is the right-hand rule for the Lorentz force?

The right-hand rule is a mnemonic for determining the direction of the magnetic component of the Lorentz force. Point your fingers in the direction of the velocity v of the positive charge, curl your fingers in the direction of the magnetic field B, and your thumb will point in the direction of the force F.