20.3.5Magnetic fields and alternating currents (A Level)

Motion of a Charged Particle in a Magnetic Field

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Lesson summary

20.3.5 Motion of a Charged Particle in a Magnetic Field — AS PHYSICS 9702 notes: Derive and use the radius and period of the circular path of a charged particle in a uniform magnetic field.Entry — a particle of mass m and charge q enters a uniform field with velocity v perpendicu…

Circular motion
Centripetal acceleration
Magnetic force provides the centripetal force
Dependence
Faster particles
Heavier particles

Section 1 of 10Why the path is a circle

Entry

Entry a particle of mass m and charge q enters a uniform field with velocity v perpendicular to B

Force

Force the magnetic force Bqv is perpendicular to the velocity at every instant, so the speed is constant and only the direction changes

Circular motion

Circular motion a constant-size force always perpendicular to the velocity gives uniform circular motion in the plane perpendicular to the field

Centripetal acceleration

Centripetal acceleration found from F = ma, in the same direction as the force

Section 2 of 10The radius

1Magnetic force provides the centripetal force

Bqv = mv²/r

2Rearrange

Rearrange r = mv/(Bq)

The radius

Dependence

Dependence r is proportional to the momentum mv, and inversely proportional to B and q

Faster particles

Faster particles they move in larger circles

Heavier particles

Heavier particles they move in larger circles

Larger charge or stronger field

Larger charge or stronger field tighter circles

Section 3 of 10Period and frequency

Period

T = 2πr/v = 2πm/(Bq)

Frequency

Frequency f = Bq/(2πm)

Independence

Independence the period does not depend on speed or radius, as faster particles travel a longer path in the same time

Cyclotron

Cyclotron particles are given extra speed twice per revolution by an alternating voltage of fixed frequency while spiralling outwards

Section 4 of 10Direction and general paths

Direction of rotation

Direction of rotation from Fleming's left-hand rule
Positive and negative particles of the same speed circulate opposite ways

Velocity at an angle

Velocity at an angle the component perpendicular to B gives circular motion and the component parallel to B is unaffected

Helix

Helix the combined path is a spiral along the field lines

Parallel to B

Parallel to B the path is straight

Section 5 of 10Applications

Mass spectrometer

Mass spectrometer the radius reveals the mass-to-charge ratio

Bubble-chamber tracks

Bubble-chamber tracks the curvature identifies the particle

Pair production

Pair production the two tracks curl in opposite directions

Section 6 of 10Useful forms of the radius

Useful forms of the radius

Momentum form

Momentum form r = p/(Bq), so particles with equal momentum and charge follow circles of equal radius

Identical particles

Identical particles radius is proportional to speed, but all return to the start in the same time

After acceleration through V

After acceleration through V qV = ½mv², so v = √(2qV/m) and r ∝ √(mV/q)/B

Checking the sense of rotation

Checking the sense of rotation apply Fleming's rule at the start and again a quarter of the way round

Section 7 of 10Limited field regions

1Crossing a field region

Crossing a field region the particle follows an arc and emerges with the same speed but a changed direction

2Entering at right angles to the edge

Entering at right angles to the edge the deflection angle satisfies sin θ = x/r for a field width x, if x is at most r

3Wider region

Wider region if x is greater than r the particle makes a semicircle and returns out of the entry edge

4Method

Method sketch the circle and mark the centre on the line perpendicular to the velocity

Section 8 of 10More from the specification

1Magnetic forces in uniform fields

Magnetic forces in uniform fields perpendicular magnetic fields exert centripetal magnetic forces that deflect moving charges into circular trajectories.

2Magnetic Force on a Moving Charge

A charged particle of charge q moving with velocity v at an angle theta to a magnetic field of flux density B experiences a magnetic deflecting force given by F = B * q * v * sin(theta), directed perpendicularly to both velocity and field vectors.

3Fleming's Left-Hand Rule

The direction of force on a moving conductor or charged particle is determined using Fleming's left-hand rule: first finger points in the direction of magnetic field (B), second finger indicates conventional current direction,
- (opposite to electron flow)
and thumb points in the direction of mechanical motion.

4Hall Effect in a Current-Carrying Conductor

When electric current flows through a thin conductor placed in an external magnetic field, moving conduction electrons are deflected to one side, generating a transverse potential difference called the Hall voltage: V_H = (B * I) / (n * t * q).

5Hall Probe for Magnetic Flux Density Measurement

A calibrated Hall probe utilizes a thin semiconductor slice whose measured Hall voltage is directly proportional to the applied magnetic flux density B perpendicular to the probe face.

6Field strength

G = F_g/m.

7Current

Flow of electrons in a circuit.

8Density

Mass per unit volume.

9Conductor

Material that allows current to flow.

10Electron

Negatively charged particle in an atom.

Section 9 of 10Key terms

Term	Meaning
Centripetal force	the force towards the centre that causes circular motion
Radius of path	r = mv/(Bq)
Cyclotron	an accelerator using a fixed-frequency alternating voltage
Helix	a spiral path along the field lines
Mass spectrometer	an instrument using circular paths to find mass-to-charge ratios
Momentum	p = mv

Section 10 of 10Exam technique

Saying the magnetic force speeds up the particle — it only changes direction.

Taking a larger charge to mean a larger circle — larger charge gives a smaller radius.

Forgetting that period is independent of speed.

Using the velocity in the direction of B for the circle — only the perpendicular component counts.

Transformer Golden Equations Physics Standard

Transformer Turns Ratio & Efficiency

Ratio: V_p / V_s = N_p / N_s = I_s / I_p

Step-up reduces grid thermal loss (P = I²R).
Laminated core prevents eddy currents.