Force on a Charged Particle in a Magnetic Field: MDCAT Physics notes
Force on a Charged Particle in a Magnetic Field for MDCAT: F = qvB sinθ, direction by the right-hand rule, no work done, F = BIL sinθ and e/m.
The magnetic force
A charge $q$ moving with velocity $\vec{v}$ in a magnetic field $\vec{B}$ experiences a force
$$\vec{F} = q\,\vec{v}\times\vec{B} \qquad F = qvB\sin\theta$$
$\theta$ is the angle between $\vec{v}$ and $\vec{B}$.
| Case | Force |
|---|---|
| Charge at rest ($v = 0$) | Zero |
| Moving parallel or antiparallel to $B$ ($\theta = 0^\circ$ or $180^\circ$) | Zero |
| Moving perpendicular to $B$ ($\theta = 90^\circ$) | Maximum, $qvB$ |
| Neutral particle (neutron, $q = 0$) | Zero at any speed |
So a magnetic field affects only moving charges (and currents), never a stationary charge.
Direction
The force is perpendicular to both $\vec{v}$ and $\vec{B}$. Use the right-hand rule for $\vec{v}\times\vec{B}$ and reverse it for a negative charge. For example, with $\vec{v}$ along $+x$ and $\vec{B}$ along $+z$, $\hat{x}\times\hat{z} = -\hat{y}$: a proton is pushed along $-y$ and an electron along $+y$. Whenever $\vec{v}$ and $\vec{B}$ lie in one plane, the force is along the axis perpendicular to that plane.
No work is done
Because $\vec{F}$ is always perpendicular to $\vec{v}$, the magnetic force does no work. It changes only the direction of velocity, not its magnitude, so the kinetic energy stays constant. An electric force, by contrast, can change both magnitude and direction.
Electron versus proton
An electron and a proton with the same speed in the same field feel forces of equal size but opposite direction. The electron is about 1836 times lighter, so it has a much larger acceleration and is deflected much more (smaller radius).
Force on a current-carrying conductor
A current is a stream of moving charges, so a conductor of length $L$ in a field feels
$$F = BIL\sin\theta$$
- Maximum when the conductor is perpendicular to $B$; zero when parallel.
- If $F$ is the maximum force, at $45^\circ$ the force is $F\sin45^\circ = F/\sqrt2$.
- Doubling $B$, $I$ and $L$ together gives $8F$.
- Maximum force per unit length $= BI$. Example: $B = 2\times10^{-5}$ T and $I = 50$ A give $10^{-3}$ N m$^{-1}$.
Worked examples
1. A charge of 3 C moves at 4 m s$^{-1}$ at $30^\circ$ to a 2 T field. $F = 3\times4\times2\times0.5 = 12$ N.
2. A particle of charge $3e$ moves at 2 m s$^{-1}$ at $90^\circ$ to $B$: $F = 6eB$.
Measuring e/m
In the FSc method, electrons accelerated through $V$ gain $\tfrac12mv^2 = eV$ and move in a circle of radius $r$ in field $B$, with $r = mv/eB$. Eliminating $v$:
$$\frac{e}{m} = \frac{2V}{B^2r^2}$$
In the velocity-selector method, crossed fields balance when $eE = evB$, giving $v = E/B$.
Key formulas
- $F = qvB\sin\theta$
- $F = BIL\sin\theta$
- $e/m = 2V/B^2r^2$
- $v = E/B$ (balanced crossed fields)
Common MDCAT traps
- Magnetic force is maximum at $90^\circ$, not $0^\circ$.
- A neutron has no charge, so no magnetic force regardless of its speed or mass.
- Magnetic force changes direction only, never speed.
- Reverse the direction for a negative charge.
- Conductor parallel to the field: zero force, even with large $I$ and $L$.
Quick revision
- $\vec{F} = q\vec{v}\times\vec{B}$.
- Stationary charges feel no magnetic force.
- Magnetic force does no work.
- The lighter electron is deflected more than the proton.