Magnetic Lorentz Force

The magnetic Lorentz force on a charge q moving with velocity v in a magnetic field B is given by:

where:

• F is the magnetic force (vector),
• q is the charge (scalar),
• v is the velocity of the charge (vector),
• B is the magnetic field (vector),
• × denotes the cross product.

Key Points:

• The force is perpendicular to both v and B, following the right-hand rule.
• The magnitude of the force is F = q v B sinθ, where θ is the angle between v and B.
• If v is parallel to B (θ = 0° or 180°), the force is zero.
• The force does no work on the charge since it is always perpendicular to the direction of motion.

Direction of Magnetic Force Using Fleming’s Left-Hand Rule

When a charge moves with velocity $\overrightarrow{v}$ in a region where there is a magnetic field $\overrightarrow{B}$, it experiences a magnetic Lorentz force $\overrightarrow{F}=\; q\; (\overrightarrow{v}\times \overrightarrow{B})$ $\overrightarrow{F}$, given by:

$\overrightarrow{F}=\; q\; (\overrightarrow{v}\times \overrightarrow{B})$

Where:

• q = charge on the particle
• $\overrightarrow{v}$ = velocity vector of the charge
• $\overrightarrow{B}$ = magnetic field vector
• $\overrightarrow{F}$ = magnetic force vector

Fleming’s Left-Hand Rule:

To determine the direction of magnetic force on a positive charge, stretch your left hand as follows:

• Index Finger → direction of Magnetic Field ($\overrightarrow{B}$) — from North to South
• Middle Finger → direction of Velocity or Current ($\overrightarrow{v}$)
• Thumb → direction of Force ($\overrightarrow{F}$)

Example:

If a positive charge moves to the right (→) in a magnetic field that is directed into the page (⊗):

• Middle Finger points to the right (velocity)
• Index Finger points into the page (magnetic field)
• Thumb points upward → this is the direction of the force