D.3.4—Magnetic force on charge

Use the magnitude equation

The angle $\theta$ is measured between the particle velocity and the magnetic field. Use charge magnitude for the force magnitude; determine direction separately with the right-hand rule and reverse it for a negative charge.

Worked example — oblique proton motion

For $|q|=1.60\times10^{-19}\,\mathrm{C}$, $v=3.4\times10^5\,\mathrm{m\,s^{-1}}$, $B=5.3\times10^{-3}\,\mathrm{T}$ and $\theta=32^\circ$, $F=|q|vB\sin\theta=1.5\times10^{-16}\,\mathrm{N}$. Only the velocity component perpendicular to the field contributes.

Find the direction

The magnetic force is perpendicular to both velocity and field. Use the right-hand rule for a positive charge; reverse the result for a negative charge. This force bends the path but does no work.

Connect to circular motion

For perpendicular motion, set $F=|q|vB$ equal to $mv^2/r$ to obtain $r=mv/(|q|B)$. Increasing $|q|$ or $B$ reduces the radius; increasing $m$ or $v$ increases it.

Common trap

Do not use the right-hand rule without reversing for an electron, and do not use $\theta$ as the angle between the field and the force. It is the angle between velocity and field.