Force on a moving charge: $F = BQv \sin \theta$

Magnetic Fields & Forces — AQA A-Level Physics

A moving chargeA property of matter that causes it to experience a force in an electromagnetic field. Measured in coulombs (C). in an external magnetic field experiences a force. Same physics as $F = BIL$, but for a single particle instead of a wire full of electrons.

F = BQv \sin \theta

$F$: magnetic force on the particle (N)
$B$: magnetic flux densityMass per unit volume of a material. Measured in kg m⁻³.The strength of a magnetic field. The force per unit length per unit currentThe rate of flow of chargeA property of matter that causes it to experience a force in an electromagnetic field. Measured in coulombs (C).. Measured in amperes (A). on a currentThe rate of flow of chargeA property of matter that causes it to experience a force in an electromagnetic field. Measured in coulombs (C).. Measured in amperes (A).-carrying conductor perpendicular to the field. Measured in teslaThe SI unit of magnetic flux density. One tesla is the flux density when a force of 1 N acts on a 1 m conductor carrying 1 A perpendicular to the field. (T). (T)
$Q$: charge of the particle (C)
$v$: speed of the particle (m s⁻¹)
$θ$: angle between velocityThe rate of change of displacement. A vector quantity. Measured in m s⁻¹. and B-field (°)

Maximum force at θ = 90° (perpendicular): $F = BQv$.
Zero force at θ = 0° (parallel).
The force is perpendicular to both velocityThe rate of change of displacement. A vector quantity. Measured in m s⁻¹. and field. It changes direction, not speed. No work is done.

Positive vs negative charges

CurrentThe rate of flow of charge. Measured in amperes (A). is defined as the flow of positive charge.
For a positive charge, conventional currentThe direction of current flow defined as from positive to negative. Opposite to the direction of electron flow. is in the same direction as velocityThe rate of change of displacement. A vector quantity. Measured in m s⁻¹..
For an electron, conventional currentThe direction of current flow defined as from positive to negative. Opposite to the direction of electron flow. is opposite to its motion. Flip the second finger when using Fleming's LHR for electrons.

Worked Example

An electron moves at 5.3 × 10⁷ m s⁻¹ perpendicular to a uniform magnetic field of flux density 0.2 T. Calculate the force on the electron.

Show Solution

List known values

Speed: $v = 5.3 \times 10^{7} \text{ m s}^{-1}$
Charge: $Q = 1.60 \times 10^{-19} \text{ C}$ (data sheet)
Flux densityMass per unit volume of a material. Measured in kg m⁻³.: $B = 0.2 \text{ T}$
Angle: $\theta = 90°$ (perpendicular)

Simplify the equation

Since $\sin 90° = 1$:

$$F = BQv$$

Substitute and calculate

$$F = 0.2 \times (1.60 \times 10^{-19}) \times (5.3 \times 10^{7})$$

Evaluate

$$F = 1.696 \times 10^{-12}$$ $$= 1.7 \times 10^{-12} \text{ N (2 s.f.)}$$

Answer

$F = 1.7 \times 10^{-12}$ N

Common Mistake MEDIUM

Students often: Mixing up $F = BIL$ and $F = BQv$.
Instead: $F = BIL$ is for a current-carrying conductor (wire with many charges). $F = BQv$ is for a single isolated moving charge. Same physics, different scale.

Examiner Tips and Tricks

For electrons: conventional currentThe direction of current flow defined as from positive to negative. Opposite to the direction of electron flow. is opposite to the electron's velocity.
Point your second finger opposite to the electron's motion when using Fleming's LHR.