22.4 Magnetic Field Strength: Force on a Moving Charge in a Magnetic Field

Magnetic Field Strength and Force on a Moving Charge

Effects of magnetic fields on charges

A magnetic field only exerts a force on a charge that is moving. If the charge is sitting still, or moving exactly parallel to the field, the magnetic force on it is zero. This is a key difference from electric fields, which push on charges whether they're moving or not.

When a charge does experience a magnetic force, that force acts perpendicular to both the velocity of the charge and the magnetic field. Because the force is always perpendicular to the velocity, it changes the charge's direction but never speeds it up or slows it down.

The magnitude of the force depends on four things:

• $B$ (magnetic field strength): stronger field means more force
• $q$ (charge): larger charge means more force
• $v$ (speed): faster charge means more force
• $\theta$ (angle between velocity and field): force is maximum when the charge moves perpendicular to the field ($\theta = 90°$) and zero when it moves parallel ($\theta = 0°$)

The direction of the force also depends on the sign of the charge. A positive charge gets pushed one way; a negative charge moving with the same velocity in the same field gets pushed the opposite way.

Right-hand rule for magnetic forces

The right-hand rule tells you which direction the magnetic force points. Here's how to use it:

1. Point your fingers in the direction of the charge's velocity ($\vec{v}$).
2. Curl your fingers toward the direction of the magnetic field ($\vec{B}$).
3. Your thumb now points in the direction of the force ($\vec{F}$) on a positive charge.

For a negative charge (like an electron), the force points opposite to your thumb.

A few things that flip the force direction:

• Reversing the velocity reverses the force
• Flipping the magnetic field direction reverses the force
• Switching the sign of the charge (positive to negative or vice versa) reverses the force

Calculation of magnetic force

The magnetic force on a moving charge is given by:

$F = qvB\sin(\theta)$

where:

• $F$ = magnitude of the magnetic force (newtons, N)
• $q$ = magnitude of the electric charge (coulombs, C)
• $v$ = speed of the charge (meters per second, m/s)
• $B$ = magnetic field strength (teslas, T)
• $\theta$ = angle between the velocity and the magnetic field

Two special cases to remember:

• When $\vec{v}$ is perpendicular to $\vec{B}$, $\sin(90°) = 1$, so the equation simplifies to $F = qvB$. This gives the maximum possible force.
• When $\vec{v}$ is parallel to $\vec{B}$, $\sin(0°) = 0$, so $F = 0$. No force at all.

Example: A proton ($q = 1.6 \times 10^{-19}$ C) moves at $3.0 \times 10^{6}$ m/s perpendicular to a 0.50 T magnetic field. The force on it is:

$F = (1.6 \times 10^{-19})(3.0 \times 10^{6})(0.50)\sin(90°) = 2.4 \times 10^{-13} \text{ N}$

That's a tiny force, but the proton has very little mass, so it produces a large acceleration.

The units work out as: $\text{N} = \text{C} \cdot \frac{\text{m}}{\text{s}} \cdot \text{T}$

In vector form, this equation is written using the cross product: $\vec{F} = q(\vec{v} \times \vec{B})$. The cross product automatically encodes both the magnitude ($qvB\sin\theta$) and the direction (perpendicular to both $\vec{v}$ and $\vec{B}$, following the right-hand rule).

Magnetic Flux and Charged Particle Motion

Because the magnetic force is always perpendicular to a charge's velocity, it acts like a centripetal force. A charge moving perpendicular to a uniform magnetic field follows a circular path. If the charge also has a velocity component parallel to the field, that component is unaffected, and the result is a helical (spiral) trajectory.

In many real situations, both electric and magnetic fields are present at the same time. The electric field can speed up or slow down the charge, while the magnetic field curves its path. Combining these effects is how devices like velocity selectors and mass spectrometers work.