Define magnetic flux density, calculate the force on a current-carrying conductor and on a moving charge, and analyse the circular motion of a charge in a magnetic field

What this dot point is asking

SEAB wants you to define magnetic flux density, calculate the force on a current-carrying conductor and on a moving charge, use Fleming's left-hand rule for direction, and analyse the circular motion of a charged particle in a uniform field. This is the motor-effect half of electromagnetism.

The answer

Magnetic flux density

The magnetic flux density $B$ (the strength of a magnetic field) is defined through the force it exerts. For a straight wire of length $L$ carrying current $I$ at angle $\theta$ to a uniform field:

When the current is perpendicular to the field ($\theta = 90^\circ$), the force is maximum, $F = BIL$. When the current is parallel to the field ($\theta = 0$), the force is zero. The unit of $B$ is the tesla (T), where $1\ \text{T} = 1\ \text{N A}^{-1}\text{m}^{-1}$.

The direction: Fleming's left-hand rule

The force is always perpendicular to both the current and the field. Fleming's left-hand rule gives the direction: with the thumb, first finger and second finger mutually perpendicular, the First finger points along the Field, the seCond finger along the Current, and the thuMb gives the Motion (force).

Force on a moving charge

A single charge $q$ moving at velocity $v$ through a field $B$ experiences:

maximum when the velocity is perpendicular to the field. Because the force is always perpendicular to the velocity, it does no work: it changes the direction of motion but not the speed.

Circular motion in a uniform field

A charge moving perpendicular to a uniform field follows a circular path, because the constant-magnitude force is always perpendicular to the velocity. The magnetic force provides the centripetal force:

So heavier or faster particles follow larger circles, while stronger fields or larger charges tighten the path. This is the principle behind mass spectrometers and cyclotrons.

Examples in context

Example 1. The electric motor. In a motor, a current-carrying coil sits in a magnetic field. The force $F = BIL$ on the two sides of the coil acts in opposite directions, producing a turning couple that rotates the coil. A commutator reverses the current every half turn to keep the rotation going, the direct application of the motor effect.

Example 2. The mass spectrometer. Ions accelerated to a known speed enter a magnetic field and follow circular paths with radius $r = \dfrac{mv}{Bq}$. Since heavier ions curve less, measuring the radius separates ions by mass-to-charge ratio, letting chemists identify isotopes and compounds.

Try this

Q1. State the factors that determine the size of the force on a current-carrying conductor in a magnetic field. [2 marks]

• Cue. $F = BIL\sin\theta$: flux density $B$, current $I$, length $L$ in the field, and the angle $\theta$ between current and field.

Q2. A wire carries $6.0\ \text{A}$ over a length $0.20\ \text{m}$ perpendicular to a $0.40\ \text{T}$ field. Find the force on it. [2 marks]

• Cue. $F = BIL = 0.40 \times 6.0 \times 0.20 = 0.48\ \text{N}$.

Q3. Explain why a charged particle moving in a uniform magnetic field travels in a circle at constant speed. [3 marks]

• Cue. The force $Bqv$ is always perpendicular to the velocity, so it does no work (constant speed) and provides a constant-magnitude centripetal force, giving a circular path.