Lorentz Force

The Lorentz force is a fundamental concept in electromagnetism, describing the force exerted on a charged particle moving through an electric and magnetic field. This force is central to many technological applications, including electric motors, particle accelerators, and the understanding of plasma behavior in both laboratory and cosmic conditions.

Definition

The Lorentz force is given by the following equation:

$$ \vec{F} = q(\vec{E} + \vec{v} \times \vec{B}) $$

where:

• $\vec{F}$ is the Lorentz force acting on the particle (in newtons, N)
• $q$ is the electric charge of the particle (in coulombs, C)
• $\vec{E}$ is the electric field (in volts per meter, V/m)
• $\vec{v}$ is the velocity of the particle (in meters per second, m/s)
• $\vec{B}$ is the magnetic field (in teslas, T)
• $\times$ denotes the cross product between two vectors

Components of the Lorentz Force

The Lorentz force consists of two components:

1. Electric Force: $q\vec{E}$
2. Magnetic Force: $q(\vec{v} \times \vec{B})$

The electric force acts in the direction of the electric field, while the magnetic force is perpendicular to both the velocity of the particle and the magnetic field.

Table of Differences and Important Points

Property	Electric Force	Magnetic Force
Direction	Parallel to $\vec{E}$	Perpendicular to both $\vec{v}$ and $\vec{B}$
Formula	$q\vec{E}$	$q(\vec{v} \times \vec{B})$
Depends on	Charge and electric field	Charge, velocity, and magnetic field
Work done	Can do work	Does no work (changes direction, not speed)
Units	Newtons (N)	Newtons (N)

Examples

Example 1: Charged Particle in a Uniform Electric Field

Consider a particle with charge $q = +1 \text{ C}$ moving with a velocity $\vec{v} = 0$ (stationary) in an electric field $\vec{E} = 5 \text{ V/m}$ directed to the right.

The Lorentz force on the particle is:

$$ \vec{F} = q(\vec{E} + \vec{v} \times \vec{B}) = q\vec{E} = (1 \text{ C})(5 \text{ V/m}) = 5 \text{ N} $$

Since the particle is stationary, there is no magnetic force component, and the electric force acts to the right.

Example 2: Charged Particle in a Uniform Magnetic Field

Consider a particle with charge $q = +1 \text{ C}$ moving with a velocity $\vec{v} = 10 \text{ m/s}$ to the right in a magnetic field $\vec{B} = 2 \text{ T}$ directed out of the page.

The Lorentz force on the particle is:

$$ \vec{F} = q(\vec{E} + \vec{v} \times \vec{B}) = q(\vec{v} \times \vec{B}) = (1 \text{ C})((10 \text{ m/s}) \times (2 \text{ T})) $$

To find the direction of the magnetic force, we use the right-hand rule. Point your fingers in the direction of $\vec{v}$ (right), curl them toward $\vec{B}$ (out of the page), and your thumb will point in the direction of the force, which is upward.

The magnitude of the force is:

$$ |\vec{F}| = qvB\sin(\theta) = (1 \text{ C})(10 \text{ m/s})(2 \text{ T})\sin(90^\circ) = 20 \text{ N} $$

The magnetic force acts upward, perpendicular to both the velocity and the magnetic field.

Example 3: Charged Particle in Combined Electric and Magnetic Fields

Consider a particle with charge $q = +1 \text{ C}$ moving with a velocity $\vec{v} = 10 \text{ m/s}$ to the right in a combined field with $\vec{E} = 5 \text{ V/m}$ directed to the right and $\vec{B} = 2 \text{ T}$ directed out of the page.

The Lorentz force on the particle is:

$$ \vec{F} = q(\vec{E} + \vec{v} \times \vec{B}) = q\vec{E} + q(\vec{v} \times \vec{B}) $$

The electric force is $5 \text{ N}$ to the right, and the magnetic force is $20 \text{ N}$ upward (as calculated in Example 2). The total force is the vector sum of these two forces, resulting in a force that has components both to the right and upward.

Conclusion

The Lorentz force is a critical concept in understanding how charged particles interact with electric and magnetic fields. It has both electric and magnetic components, which can be calculated and analyzed separately. The force plays a vital role in many areas of physics and engineering, and understanding its principles is essential for students preparing for exams in these fields.