7.5.2 Moving charges in a magnetic field

Force on a Moving Charge in a Magnetic Field

A force acts on charged particles when they move through a magnetic field. This is the basis for why a force is exerted on a current-carrying wire, as current consists of moving charges, primarily electrons. Since these electrons are negatively charged particles, they are influenced by the magnetic field.

The magnitude of the force $F$ exerted on a charged particle with charge $Q$, moving at a velocity $v$ and perpendicular to a magnetic field of flux density $B$, is given by the formula:

$$F = BQv$$

Determining the Direction of Force

The direction of the force experienced by a moving charge in a magnetic field can be determined using Fleming's Left-Hand Rule:

• Thumb represents the direction of Motion (or force).
• First finger represents the direction of the Field.
• Second finger represents the direction of Conventional Current (opposite to electron flow). Note: If the charge is negative, reverse the direction indicated by the second finger, as it represents conventional current (positive to negative).

Circular Motion of Charged Particles in a Magnetic Field

When a charged particle enters a magnetic field, the force exerted on it is always perpendicular to its motion. This causes the particle to move in a circular path, as the magnetic force acts as a centripetal force. For a charged particle in a magnetic field:

$$F = BQv$$

Since $F$ also provides the centripetal force $\frac{mv^2}{r}$ , equating these gives:

$$BQv = \frac{mv^2}{r}$$

Rearranging to solve for the radius of the circular path $r$ yields:

$$r = \frac{mv}{BQ}$$

where:

• $m$ is the particle's mass,
• $v$ is its velocity,
• $Q$ is its charge,
• $B$ is the magnetic flux density.