Lorentz Magnetic Force

Force on a point charge q moving with velocity v in field B. Always perpendicular to v — magnetic force does no work and cannot change kinetic energy.

Statement

A charge $q$ moving with velocity $\vec{v}$ in magnetic field $\vec{B}$ experiences:

$$\boxed{\vec{F} = q\,(\vec{v} \times \vec{B})}$$

Direction

The cross product $\vec{v} \times \vec{B}$ gives a vector perpendicular to both $\vec{v}$ and $\vec{B}$. For positive $q$, $\vec{F}$ is in the direction of $\vec{v} \times \vec{B}$. For negative $q$, it is opposite.

Use the right-hand rule: point fingers along $\vec{v}$, curl toward $\vec{B}$ — thumb points along $\vec{F}$ for positive charge.

Magnitude

$$F = qvB\sin\theta$$

where $\theta$ is the angle between $\vec{v}$ and $\vec{B}$.

• $F = 0$ when $\vec{v} \parallel \vec{B}$ ($\theta = 0$): no force.
• $F = qvB$ (maximum) when $\vec{v} \perp \vec{B}$ ($\theta = 90°$).

Magnetic Force Does No Work

Since $\vec{F} \perp \vec{v}$ always, the power delivered:

$$P = \vec{F} \cdot \vec{v} = 0$$

Magnetic force can change the direction of motion but never the speed or kinetic energy.