What Happens to a Charged Particle in Crossed (Perpendicular) Electric and Magnetic Fields?

SL and HL: A charged particle in crossed electric and magnetic fields experiences both an electric force and a magnetic force. If these forces are equal in magnitude and opposite in direction, the particle travels undeflected; otherwise, it accelerates in the direction of the resultant force.

How the Forces Combine

The electric force has magnitude

$$F_E = qE$$

For a positive charge, it acts in the direction of the electric field. For a negative charge, it acts in the opposite direction.

The magnetic force on a particle moving perpendicular to the magnetic field has magnitude

$$F_B = qvB$$

Its direction is perpendicular to both the particle's velocity and the magnetic field. Use the appropriate hand rule for a positive charge, then reverse the direction for a negative charge. Because the magnetic force is perpendicular to velocity, it does no work and cannot directly change the particle's speed.

In a velocity selector, the electric field, magnetic field, and particle velocity are mutually perpendicular, with the two forces arranged to oppose each other. For no deflection:

$$qE = qvB$$

Therefore,

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

The charge cancels, so the selected speed is independent of the magnitude or sign of the charge.

For example, if $E=3.0\times10^4\ \text{N C}^{-1}$ and $B=0.20\ \text{T}$, then

$$v=\frac{3.0\times10^4}{0.20}=1.5\times10^5\ \text{m s}^{-1}.$$

For a negative charge, both force directions reverse, so the cancellation condition is unchanged but any deflection direction reverses.

Exam Technique

This is syllabus topic D.3 Motion in electromagnetic fields for both SL and HL. Draw and label both force vectors, determine their directions, and compare (qE) with (qvB). Avoid the common misconception that an undeflected particle experiences no forces: it experiences two balanced forces, giving zero resultant force.