Magnetic Fields and Forces

Magnetic fields (AQA specification 3.7.5) describe the forces experienced by moving charges and current-carrying conductors. This topic introduces magnetic flux density, the motor effect, and the motion of charged particles in magnetic fields.

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Magnetic Flux Density

Key Definition: The magnetic flux density (B) at a point is the force per unit length per unit current on a conductor carrying current perpendicular to the field at that point.

The SI unit of magnetic flux density is the tesla (T), where 1 T = 1 kg s⁻² A⁻¹.

Alternatively, 1 T is the flux density that produces a force of 1 N on a 1 m length of wire carrying a current of 1 A perpendicular to the field.

Typical values of B:

Source	Flux density (T)
Earth's magnetic field	5 × 10⁻⁵
Bar magnet (at pole)	0.01–0.1
Laboratory electromagnet	0.1–2
MRI scanner	1.5–7
Strong research magnet	10–45

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Force on a Current-Carrying Conductor

When a straight conductor of length L carrying current I is placed in a uniform magnetic field of flux density B, the force on the conductor is:

F = BIL sin θ

where θ is the angle between the current direction and the magnetic field direction.

When the current is perpendicular to the field (θ = 90°, sin θ = 1):

F = BIL

When the current is parallel to the field (θ = 0°), there is no force (F = 0).

Fleming's Left-Hand Rule

Fleming's left-hand rule gives the direction of the force:

• First finger → Field direction (N to S)
• Second finger → Current direction (conventional current, positive to negative)
• Thumb → Thrust (force direction)

All three are mutually perpendicular.

Worked Example 1 — Force on a Wire

Question: A straight wire of length 0.25 m carries a current of 4.0 A in a uniform magnetic field of flux density 0.30 T. The wire is perpendicular to the field. Calculate the force on the wire.

Solution:

F = BIL = 0.30 × 4.0 × 0.25 = 0.30 N

Worked Example 2 — Wire at an Angle

Question: The same wire is now oriented at 30° to the magnetic field. Calculate the new force.

Solution:

F = BIL sin θ = 0.30 × 4.0 × 0.25 × sin 30° = 0.30 × 0.50 = 0.15 N

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Force on a Moving Charged Particle

A charged particle moving through a magnetic field experiences a force given by:

F = BQv sin θ

where B is the magnetic flux density (T), Q is the charge (C), v is the speed (m s⁻¹), and θ is the angle between the velocity and the field.

When the velocity is perpendicular to the field (θ = 90°):

F = BQv

Key properties of this force:

• The force is always perpendicular to both the velocity and the field (it acts as a centripetal force).
• Because the force is perpendicular to the velocity, it does no work on the particle — the speed remains constant, only the direction changes.
• A positive charge moving to the right in a field pointing into the page experiences an upward force (using Fleming's left-hand rule, with the second finger pointing in the direction of positive charge motion).

Circular Motion in a Magnetic Field

When a charged particle enters a uniform magnetic field perpendicular to its velocity, the magnetic force provides centripetal acceleration, causing the particle to move in a circle.

Setting the magnetic force equal to the centripetal force:

BQv = mv²/r

Solving for the radius:

r = mv/(BQ)

This shows:

• Faster particles move in larger circles (r ∝ v)
• Heavier particles move in larger circles (r ∝ m)
• Stronger fields produce tighter circles (r ∝ 1/B)
• More highly charged particles move in tighter circles (r ∝ 1/Q)

Worked Example 3 — Proton in a Magnetic Field