Magnetic Force

Magnetic Force: The Lorentz Force

Magnetism is a force that acts on moving charges. A stationary charge experiences no magnetic force. The magnetic force on a charge depends on three things: the charge itself, its velocity, and the magnetic field it moves through.

The Lorentz Force Law

The total force on a charged particle in an electric field $\vec{E}$ and a magnetic field $\vec{B}$ is the Lorentz force:

Here:

• $q$ is the charge of the particle.
• $\vec{v}$ is its velocity.
• $\vec{E}$ is the electric field.
• $\vec{B}$ is the magnetic field.
• $\times$ denotes the vector cross product.

The electric part $q\vec{E}$ is independent of velocity. The magnetic part $q(\vec{v} \times \vec{B})$ depends on velocity and is always perpendicular to both $\vec{v}$ and $\vec{B}$.

Direction of Magnetic Force

The magnetic force $\vec{F}_m$ on a moving charge is given by:

Its direction is given by the right-hand rule:

• Point your fingers in the direction of $\vec{v}$.
• Curl them toward $\vec{B}$.
• Your thumb points in the direction of $\vec{v} \times \vec{B}$.
• For a positive charge ($q > 0$), the force is along the thumb.
• For a negative charge ($q < 0$), the force is opposite to the thumb.

Magnitude of Magnetic Force

The magnitude of the magnetic force is:

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

• If $\vec{v}$ is parallel to $\vec{B}$ ($\theta = 0^\circ$ or $180^\circ$), $\sin\theta = 0$, so $F_m = 0$. No magnetic force acts.
• If $\vec{v}$ is perpendicular to $\vec{B}$ ($\theta = 90^\circ$), $\sin\theta = 1$, so $F_m = |q| v B$ — the maximum possible force.

Key Properties of Magnetic Force

• Always perpendicular to both velocity and magnetic field: $\vec{F}_m \perp \vec{v}$ and $\vec{F}_m \perp \vec{B}$. …