Moving charges in a magnetic fieldAQA A-Level Physics: Revision notes
Section 1
The force on a moving charge
A charged particle moving through a magnetic field experiences a force. When the velocity is at right angles to the field:
F = BQv
where B is the flux density (T), Q the charge (C) and v the speed (m s⁻¹). This follows from F = BIl, since a current I = Q/t and l = vt. There is no force if the particle moves parallel to the field or is stationary.
Section 2
Direction of the force
The force is perpendicular to both the velocity and the field, so use Fleming's left-hand rule with the second finger along the conventional current (the direction a positive charge moves).
- Positive particle: second finger along its velocity.
- Negative particle: second finger opposite its velocity, so the force is in the opposite direction to that on a positive particle of the same velocity.
Because the force is always perpendicular to the velocity, it does no work: the speed (and kinetic energy) is constant and only the direction changes.
For an electron, the current direction is opposite to its velocity: forgetting this gives a deflection the wrong way.
Section 3
Circular paths
Because the force is perpendicular to the velocity and constant in size, it provides a centripetal force and the particle moves in a circle (if it enters at right angles to a uniform field):
BQv = mv²/r, so r = mv/(BQ) = p/(BQ)
The radius is larger for larger mass, speed or momentum, and smaller for a stronger field or larger charge. The time for one revolution is T = 2πr/v = 2πm/(BQ), which does not depend on speed.
Start every circular-motion problem with BQv = mv²/r and cancel one v.
Section 4
Worked example
An electron travels at 3.0 × 10⁶ m s⁻¹ perpendicular to a 1.5 mT field.
r = mv/(BQ) = (9.11 × 10⁻³¹ × 3.0 × 10⁶) / (1.5 × 10⁻³ × 1.60 × 10⁻¹⁹) = 1.1 × 10⁻² m.
The magnetic force is F = BQv = 1.5 × 10⁻³ × 1.60 × 10⁻¹⁹ × 3.0 × 10⁶ = 7.2 × 10⁻¹⁶ N.
Section 5
The cyclotron
A cyclotron accelerates charged particles using a uniform magnetic field and an alternating electric field.
- Two hollow D-shaped chambers sit in a uniform field; particles move in semicircles inside them (no electric field in the chambers).
- Each time particles cross the gap between the chambers, an alternating p.d. accelerates them.
- As speed increases, the radius r = mv/(BQ) grows, so the particles spiral outwards.
- The time for a semicircle, πm/(BQ), is independent of speed, so the p.d. alternates at a constant frequency f = BQ/(2πm).
- Maximum speed is set by the outer radius: v = BQr/m.
That's the notes covered.
Carry on to the next subtopic.
Exam questions on Moving charges in a magnetic field
- A beam of protons travels in a vacuum at a speed of 2.0 × 10⁶ m s⁻¹ and enters a region of uniform magnetic field of flux density 0.15 T. The velocity of the protons is at right angles to the field. The charge of a proton is 1.60 × 10⁻¹⁹ C.Explain why the magnetic force does not change the kinetic energy of a proton.2 marks
- In a vacuum chamber, a beam of protons and then a beam of electrons are fired in turn, each at a speed of 4.0 × 10⁶ m s⁻¹, horizontally due east into a region of uniform magnetic field of flux density 2.0 mT directed vertically upwards. The mass of an electron is 9.11 × 10⁻³¹ kg, the mass of a proton is 1.67 × 10⁻²⁷ kg and the magnitude of the charge on each is 1.60 × 10⁻¹⁹ C.Calculate the radius of the circular path of the electrons.2 marks
- A cyclotron accelerates protons. They move in semicircles inside two hollow, D-shaped metal chambers that lie in a uniform magnetic field of flux density 0.80 T, perpendicular to the plane of the semicircles. Each time a proton crosses the gap between the two chambers it is accelerated by an alternating potential difference. The protons spiral outwards and leave the cyclotron at an outer radius of 0.30 m. Proton mass = 1.67 × 10⁻²⁷ kg and charge = 1.60 × 10⁻¹⁹ C.Explain why the protons move in semicircles inside the chambers, and why the frequency of the alternating potential difference does not need to change as the protons gain speed.3 marks
Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).