Magnetism and Magnetic Effects: Conceptual Doubts Cleared (8)

Question

A charged particle moves through a region with both electric and magnetic fields. It travels in a straight line at constant speed. What can we say about the field configuration? And why does this happen even though both fields exert forces?

Solution — Step by Step

Why This Works

The magnetic force is always perpendicular to velocity, so it can never do work — it only changes direction. To cancel it, we need an equal and opposite force, also perpendicular to velocity. The electric force fits this role only if $\vec{E}$ is perpendicular to $\vec{v}$.

This is the principle behind the velocity selector used in mass spectrometers: only particles with exactly the right speed pass through undeflected. Faster or slower particles get pushed off-axis.

Alternative Method — Energy Argument

If the particle’s speed is constant, kinetic energy is constant, so net work is zero. Magnetic force never does work. So the electric force must also not do work — meaning $\vec{E}$ is always perpendicular to $\vec{v}$.

This rules out any component of $\vec{E}$ along the motion. Combined with force balance, we get the same configuration.

Common Mistake

Students sometimes think that “both forces cancel” means the fields are parallel or anti-parallel. Wrong — magnetic force on a charged particle moving parallel to $\vec{B}$ is zero, so there is nothing for the electric force to cancel.