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5.6.2. Lorentz Force

Table of Contents

Charged Particles in Magnetic and Electric Fields

The Lorentz force is the force experienced by a charged particle when it moves through electric and magnetic fields. It is one of the central ideas of electricity and magnetism because it tells us how fields affect motion.

If a particle has charge $q$, velocity $\vec{v}$, electric field $\vec{E}$, and magnetic field $\vec{B}$, then the Lorentz force is

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

This equation combines two effects. The electric field produces a force in the direction of the field for a positive charge, and in the opposite direction for a negative charge. The magnetic field produces a force only if the particle is moving, and that force depends on the cross product $\vec{v} \times \vec{B}$.

The Lorentz force law is
$$
\vec{F} = q\left(\vec{E} + \vec{v} \times \vec{B}\right)
$$
A magnetic field alone exerts no force on a particle at rest.

Magnetic Part of the Lorentz Force

If only a magnetic field is present, then the force becomes

$$
\vec{F}_B = q\,\vec{v} \times \vec{B}
$$

The magnitude of this magnetic force is

$$
F_B = |q|vB\sin\theta
$$

where $\theta$ is the angle between $\vec{v}$ and $\vec{B}$.

This formula shows several important facts. If the particle moves parallel or antiparallel to the magnetic field, then $\theta = 0$ or $\theta = 180^\circ$, so $\sin\theta = 0$ and the magnetic force is zero. If the particle moves perpendicular to the field, then $\theta = 90^\circ$ and the force is maximum.

For the magnetic force,
$$
F_B = |q|vB\sin\theta
$$
The force is maximum when $\vec{v} \perp \vec{B}$ and zero when $\vec{v}$ is parallel to $\vec{B}$.

Direction of the Force

The direction of the magnetic force is perpendicular to both the velocity and the magnetic field. For a positive charge, the direction is given by the right hand rule applied to $\vec{v} \times \vec{B}$. For a negative charge, the force is in the opposite direction.

A simple way to use the right hand rule is this. Point your fingers in the direction of $\vec{v}$. Turn them toward $\vec{B}$. Your thumb then points in the direction of $\vec{v} \times \vec{B}$ for a positive charge.

Right hand rule for magnetic force

What the Magnetic Force Does

Because the magnetic force is always perpendicular to the velocity, it changes the direction of motion but not the speed. This is a very important result. A magnetic field can bend the path of a charged particle, but by itself it does not increase or decrease the particle's kinetic energy.

To understand this, recall that work depends on the component of force along the displacement. Since the magnetic force is perpendicular to the motion, the work done by a purely magnetic force is zero.

A magnetic field does no work on a charged particle because the magnetic force is perpendicular to the particle's velocity.
So, a magnetic field can change direction, but not speed.

Special Cases of Motion

When a charged particle enters a uniform magnetic field, its path depends on the angle between its velocity and the field.

If the velocity is perpendicular to the magnetic field, the particle moves in a circle. The magnetic force acts as the centripetal force. Then

$$
|q|vB = \frac{mv^2}{r}
$$

so the radius is

$$
r = \frac{mv}{|q|B}
$$

The angular speed is

$$
\omega = \frac{v}{r} = \frac{|q|B}{m}
$$

and the period is

$$
T = \frac{2\pi r}{v} = \frac{2\pi m}{|q|B}
$$

If the velocity is parallel to the magnetic field, there is no magnetic force, so the particle continues in a straight line.

If the velocity has both parallel and perpendicular components, the particle follows a helical path. The perpendicular part produces circular motion, and the parallel part carries the particle forward along the field direction.

Circular motion in a magnetic field

Helical Motion

A helical path appears when the velocity is split into two components, one parallel to the field and one perpendicular to the field.

Let

$$
v_\parallel = v\cos\theta, \qquad v_\perp = v\sin\theta
$$

Then the circular part of the motion comes from $v_\perp$, and its radius is

$$
r = \frac{mv_\perp}{|q|B}
$$

The forward motion comes from $v_\parallel$. During one full turn, the particle advances a distance called the pitch:

$$
p = v_\parallel T
$$

Using $T = \frac{2\pi m}{|q|B}$, we get

$$
p = v_\parallel \frac{2\pi m}{|q|B}
$$

Helical motion along a magnetic field

Electric and Magnetic Forces Together

When both electric and magnetic fields are present, the total force is the sum of the electric force and the magnetic force:

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

The electric part can change the speed of the particle, because it can act along the direction of motion. The magnetic part bends the path.

In some situations, the electric and magnetic forces can balance each other. If they point in opposite directions and have equal magnitude, then the net force is zero. For perpendicular fields and suitable direction of motion,

$$
qE = qvB
$$

so the speed for zero net force is

$$
v = \frac{E}{B}
$$

This idea is used in velocity selectors.

If electric and magnetic forces balance, then
$$
qE = qvB \quad \Rightarrow \quad v = \frac{E}{B}
$$
This is valid when the electric and magnetic forces are opposite in direction.

Comparing Electric and Magnetic Effects

The electric force depends only on charge and field,

$$
\vec{F}_E = q\vec{E}
$$

The magnetic force depends on charge, velocity, magnetic field, and angle,

$$
\vec{F}_B = q\vec{v} \times \vec{B}
$$

The table below summarizes the main differences.

FeatureElectric forceMagnetic force
Formula$\vec{F}_E = q\vec{E}$$\vec{F}_B = q\vec{v} \times \vec{B}$
Acts on charge at restYesNo
Depends on velocityNoYes
Can change speedYesNo, if magnetic field acts alone
DirectionAlong or opposite $\vec{E}$Perpendicular to $\vec{v}$ and $\vec{B}$

Units

From the formula for the magnetic force,

$$
F = |q|vB\sin\theta
$$

the SI unit of magnetic field can be written as

$$
1\ \text{tesla} = 1\ \text{T} = \frac{1\ \text{N}}{1\ \text{C} \cdot 1\ \text{m/s}}
$$

which is also equivalent to

$$
1\ \text{T} = \frac{1\ \text{N}}{1\ \text{A} \cdot 1\ \text{m}}
$$

because $1\ \text{C/s} = 1\ \text{A}$.

Physical Meaning

The Lorentz force connects fields to motion. It explains why charged particles curve in magnetic fields, why beams of particles can be steered, and why electric and magnetic fields together can control trajectories very precisely. In later applications, this force is the starting point for understanding particle motion in laboratories, devices, and natural environments.

Key ideas to remember:
$$
\vec{F} = q\left(\vec{E} + \vec{v} \times \vec{B}\right)
$$
$$
F_B = |q|vB\sin\theta
$$
A magnetic field changes direction of motion, not speed, when acting alone.

Example

Consider a particle of charge $q$ moving with speed $v$ perpendicular to a uniform magnetic field $B$. The magnetic force has magnitude

$$
F = |q|vB
$$

This force provides the centripetal force:

$$
|q|vB = \frac{mv^2}{r}
$$

Solving for the radius gives

$$
r = \frac{mv}{|q|B}
$$

So a larger mass or larger speed gives a larger circular path, while a larger charge magnitude or stronger magnetic field gives a smaller path.

This result is very useful because it allows the motion of charged particles to reveal their physical properties.

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