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D.3.4 Charged Particles in Magnetic Fields — IB Physics HL Revision Notes & Topical Questions

5 min read 10 parts Lesson 31 of 39 1 quiz

📚 IB Physics HL Hub / Topical Revision
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D.3.4 Charged Particles in Magnetic Fields — IB PHYSICS SL notes: Derive and use the radius and period of circular motion of a charged particle in a uniform magnetic field.Why a circle — the magnetic force F = qvB is always perpendicular to the velocity and of constant sizeForce,…

  • Centripetal acceleration
  • Derivation
  • Charge-to-mass ratio
  • Independence of speed
  • Force direction
  • Opposite charges

Part 1 of 8Circular motion from a perpendicular force

Why a circle
  • Why a circle the magnetic force F = qvB is always perpendicular to the velocity and of constant size
  • Force, velocity and field are all mutually perpendicular.
  • The force is therefore a centripetal force directed to the centre.
Path
  • Path the particle moves in a circle in the plane perpendicular to the field at constant speed
Centripetal acceleration
  • Centripetal acceleration it points in the same direction as the force and follows from F = ma

Part 2 of 8Radius of the path

Derivation
  • Derivation set the magnetic force equal to the centripetal force, qvB = mv2/r, and cancel one v
Radius
  • Radius r = mv / (qB)
  • m is the mass in kg and v the speed in m s−1.
  • q is the charge magnitude in C and B the field in T.
Dependence
  • Faster particles move in larger circles, r ∝ v.
  • Heavier particles move in larger circles, r ∝ m.
  • Greater charge gives smaller circles, r ∝ 1/q.
  • A stronger field gives smaller circles, r ∝ 1/B.
Momentum
  • Momentum a larger momentum mv gives a bigger circle
Charge-to-mass ratio
  • Charge-to-mass ratio for an electron e/me = 1.76 × 1011 C kg−1, so r = v/((e/m)B)

Part 3 of 8Period and frequency

Period
  • T = 2πr/v = 2πm/(qB)
Frequency
  • Frequency f = qB/(2πm)
Independence of speed
  • Independence of speed the speed cancels, so the period does not depend on speed or radius
  • A faster particle travels a bigger circle in the same time.
Cyclotron
  • Cyclotron particles are accelerated across a gap by an alternating electric field of fixed frequency while spiralling outward
  • The electric field gives the energy and the magnetic field only bends the path.

Part 4 of 8Direction and sense of rotation

Force direction
  • Force direction use the rule from D.3.3 to find where the particle curves
Opposite charges
  • Opposite charges positive and negative particles with the same speed circle in opposite senses
Bubble-chamber tracks
  • Bubble-chamber tracks a pair of tracks curving oppositely from a common point is the signature of an electron–positron pair
Spiral inwards
  • Spiral inwards tracks shrinking in radius mean the particle is losing kinetic energy, for example through ionisation

Part 5 of 8General angle: helical motion

General angle: helical motion
Components
  • Components split the velocity into v sin θ perpendicular to B and v cos θ parallel to B
Perpendicular component
  • Perpendicular component it gives circular motion of radius r = mv sin θ/(qB)
Parallel component
  • Parallel component it is unaffected, so the particle moves along the field at constant speed
Path
  • Path a helix whose axis lies along B
Example
  • Example charged particles from the Sun spiral along the Earth's field lines towards the poles

Part 6 of 8Mass spectrometer

1Principle
  • Principle particles of the same speed and charge but different masses follow circles of different radii
2Acceleration first
  • Acceleration first ions pass through a p.d.
  • V, so qV = ½mv2
3Radius
  • Radius combining the equations gives r = (1/B)√(2mV/q)
4Measurement
  • Measurement r gives m/q, so isotopes land at different points on a detector and their relative abundance can be measured
5Electron beams
  • Electron beams accelerated through V, r = √(2mV/e)/B, which allows e/m to be found

Part 7 of 8Using the equations

Derive
  • Derive write qvB = mv2/r and cancel one v rather than memorising
Data booklet values
  • Data booklet values e = 1.60 × 10−19 C, me = 9.11 × 10−31 kg and mp = 1.67 × 10−27 kg
Sketching tracks
  • Sketching tracks a larger radius means a higher momentum or a weaker field

Part 8 of 8More from the specification

1Lorentz magnetic force on moving charges
  • A charged particle moving across a magnetic field experiences a deflecting force: F = q v B sin(theta), acting perpendicular to both velocity and field.
2Circular trajectory in uniform magnetic fields
  • The magnetic force acts as a centripetal force, bending the particle into a circular path of radius r = mv / (qB).
    • (qvB = m v2 / r)
3Applications in mass spectrometry
  • Measuring trajectory curvature in known magnetic fields allows precise determination of isotopic masses and charge-to-mass ratios.
4Motor rule deflection geometry
  • Motor rule deflection geometry three orthogonal fingers in Fleming's left-hand convention determine magnetic force on a moving current-carrying conductor.
5Parallel current-carrying conductors
  • Parallel current-carrying conductors parallel wires carrying current in identical directions exert mutual attractive magnetic forces along their conductor length.
6Deflection between charged plates
  • Deflection between charged plates electric fields between parallel plates exert constant perpendicular electrostatic forces on passing charged particles.
7F = Bqv
What the symbols mean: F = force on the moving charge (N) · B = magnetic flux density (T) · q = charge (C) · v = speed (m s−1)
8Current
  • Flow of electrons in a circuit.
9Conductor
  • Material that allows current to flow.
10Length
  • Resistance is directly proportional to the length of a wire.
11Proton
  • Positive particle in the nucleus, relative charge +1 and relative mass 1.
12Density
  • Mass per unit volume.
13Field strength
  • G = Fg/m.
14F = BIL
What the symbols mean: F = force on the wire (N) · B = magnetic flux density (T) · I = current (A) · L = length of wire in the field (m) · θ = angle between wire and field

Key terms

TermMeaning
Centripetal forceNet force toward the centre of a circular path, here provided by qvB.
Cyclotron frequencyf = qB/(2πm), independent of speed.
HelixSpiral path when the velocity has a component along B.
Mass spectrometerDevice that separates ions by r = mv/(qB) to find m/q.
Charge-to-mass ratioq/m, determining how tightly a particle is deflected.

Avoid these mistakes

1
Thinking the period depends on speed — T = 2πm/(qB).
2
Using the full speed in r = mv/(qB) when the velocity is at an angle — only the perpendicular component counts.
3
Believing the magnetic field speeds the particle up — in a cyclotron the electric field gives the energy.
4
Mixing up mass and charge when comparing particles — an alpha particle has twice the charge and about four times the mass of a proton.
5
Forgetting that opposite charges curve in opposite directions.

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