Source: PHYS 212 Final Examination, University of Illinois at Urbana-Champaign
Tags: magnetic force, Lorentz force, circular motion, cyclotron radius, solenoid, Ampere's law, electromagnetic wave, force on wire, ILB, right-hand rule, work done by magnetic field
Difficulty: Intermediate | Prerequisites: Newton's second law, centripetal acceleration, cross product basics, Ampere's law introduction.
Magnetic forces on moving charges are fundamentally different from electric forces: they never do work, they always act perpendicular to the velocity, and they curve paths rather than speed things up or slow them down. This topic covers charged particles circling in uniform magnetic fields (the basis of mass spectrometers and cyclotrons), the field inside a solenoid (the basis of electromagnets and MRI machines), forces on current-carrying wires, and electromagnetic wave propagation. If you are not comfortable with the cross product and the right-hand rule, go back to those first; nearly every result here depends on the direction of v × B or I × B.
A charged particle moving perpendicular to a uniform magnetic field follows a circle of radius r = mv / (qB). Reversing the sign of the charge reverses the direction of circulation but does not change the radius or speed. Magnetic force never does work because it is always perpendicular to velocity. Inside a long solenoid the field is uniform at B = μ₀nI, and electromagnetic waves have E and B oscillating perpendicular to each other and to the direction of propagation.
Lorentz force (magnetic component)
The force on a charge q moving with velocity v in a magnetic field B: F = qv × B. The force is perpendicular to both v and B. In simple terms, the field pushes the charge sideways, never speeding it up or slowing it down.
Cyclotron radius (Larmor radius)
The radius of the circular path of a charged particle in a uniform magnetic field: r = mv / (qB). A heavier or faster particle makes a bigger circle; a stronger field or larger charge makes a tighter one. Think of it as: more momentum means harder to bend, more force means easier to bend.
Cyclotron frequency
The angular frequency of the circular motion: ω = qB / m. This does not depend on the speed of the particle, which is why cyclotrons work (the frequency stays the same as the particle speeds up, at least at non-relativistic speeds).
Solenoid
A long coil of wire wound in a helix. When current flows through it, the field inside is nearly uniform and parallel to the axis. Think of it as a way to create a controlled, uniform magnetic field in a defined region.
Ampere's law
The line integral of B around any closed loop equals μ₀ times the enclosed current: ∮ B · dl = μ₀ I_enc. This is the magnetic analogue of Gauss's law and is used to derive the field inside a solenoid.
Electromagnetic wave
A self-propagating disturbance in the electric and magnetic fields. E and B oscillate perpendicular to each other and to the direction of travel, in phase, at the speed of light. In simple terms, a changing electric field creates a magnetic field and vice versa, and the pair travels together through space.
Force on a current-carrying wire
F = ILB sin θ, where I is the current, L is the length of wire in the field, B is the field strength, and θ is the angle between the wire and the field. This is the basis of electric motors.
When a particle with charge q, mass m, and speed v enters a region of uniform B perpendicular to its velocity:
The magnetic force qvB provides the centripetal force: qvB = mv² / r
Solving for the radius: r = mv / (qB)
The particle traces a circle (or helix if there is a velocity component along B).
The magnetic force does no work, so the speed and kinetic energy do not change.
Key point: the radius depends on momentum (mv), not on kinetic energy alone. A common distractor is r = mv² / (qB), which has the wrong dimensions and is incorrect.
If the particle is an electron (negative charge) instead of a positive charge:
The magnitude of the force |qvB| is unchanged (assuming same |q|, v, B).
The direction of the force reverses (because F = qv × B and q has flipped sign).
The particle curves the opposite way: the direction of rotation reverses.
The radius and speed remain the same.
Work is defined as W = F · ds = F · v dt. The magnetic force F = qv × B is always perpendicular to v (because the cross product is perpendicular to both of its inputs). A force perpendicular to displacement does zero work. Consequently:
The speed of the particle never changes.
The kinetic energy never changes.
The magnetic field can change the direction of motion but not the magnitude.
This is a short-answer exam question. The key sentence is: "The magnetic force is always perpendicular to the velocity of the particle, so F · v = 0 at every instant, and the work done is zero."
For an ideal (infinitely long) solenoid with n turns per unit length carrying current I:
Inside the solenoid: B = μ₀nI, uniform and directed along the axis.
Outside the solenoid: B ≈ 0.
This is derived using Ampere's law with a rectangular loop: one side inside, one side outside, with the enclosed current being nI per unit length.
Common distractors:
B = μ₀I / (2πr) is the field around a long straight wire, not a solenoid.
B = μ₀n²I has the wrong dependence on n.
B = 0 inside is wrong; the field is zero outside.
The force on a straight wire segment of length L carrying current I in a field B is:
F = IL × B (vector form)
|F| = ILB sin θ (magnitude)
The factors that determine the magnitude are: the current I, the length L, the field strength B, and the angle θ between the wire and the field. All four matter.
For an EM wave propagating in the +x direction:
E, B, and the propagation direction are mutually perpendicular.
If E oscillates along the y-axis, then B oscillates along the z-axis (and vice versa).
The direction follows E × B = direction of propagation. If E is along +y and propagation is along +x, then B must be along +z.
Magnetic force on a charge: F = qv × B, |F| = qvB sin θ
Cyclotron radius: r = mv / (qB)
Cyclotron frequency: ω = qB / m, period T = 2πm / (qB)
Solenoid field: B = μ₀nI
Force on a wire: F = ILB sin θ
EM wave relationship: E × B points in the propagation direction; E/B = c
Mass spectrometers exploit r = mv / (qB) to separate ions by mass: heavier ions curve less and land farther from the entrance slit. MRI machines use a large solenoid to produce the uniform field needed to align hydrogen nuclei in your body. Electric motors rely on F = ILB to convert current into rotational motion. Electromagnetic waves carry radio signals, light, and Wi-Fi.
Students sometimes write r = mv² / (qB) for the cyclotron radius. This is incorrect; the v in the centripetal acceleration formula cancels one power of v in the force, leaving r = mv / (qB).
Thinking the magnetic force changes the speed of a charged particle. It cannot. Speed is constant; only direction changes.
Believing that reversing the sign of the charge changes the radius. It does not (assuming the same magnitude of charge). Only the direction of circulation changes.
Confusing the solenoid field formula B = μ₀nI with the straight-wire formula B = μ₀I / (2πr). These describe completely different geometries.
⚠️ Question 3A asks for the cyclotron radius. The correct answer is r = mv / (qB). Watch for the mv² / (qB) distractor.
⚠️ Question 3B asks how the path changes for an electron. The answer is that the direction of rotation reverses. The radius, speed, and kinetic energy are unchanged.
⚠️ Question 7 tests the solenoid field. The answer is B = μ₀nI. Do not pick the straight-wire formula.
⚠️ Question 9 tests EM wave polarisation. If E is along y and propagation is along +x, B is along z.
⚠️ Question 14 (multi-select) asks which factors determine the force on a wire. All four (B, L, I, θ) are correct.
⚠️ Question 17 (short answer) asks why the magnetic force does zero work. Write about perpendicularity of F and v.
Fill in the blank: The radius of a charged particle's circular path in a uniform B field is r = ______.
True or False: The magnetic force can increase the kinetic energy of a moving charged particle.
Fill in the blank: Inside a long solenoid with n turns per unit length and current I, the magnetic field is B = ______.
True or False: If the charge of a particle is reversed, the radius of its circular path in a B field changes.
Fill in the blank: In an EM wave propagating in the +x direction with E along y, B oscillates along the ______ axis.
Answers: 1. mv / (qB). 2. False. 3. μ₀nI. 4. False (only the direction of rotation changes). 5. z.
Q: What is the radius of the circular path of a particle with mass m and charge q moving at speed v perpendicular to a uniform field B?
A: r = mv / (qB). Set the magnetic force qvB equal to the centripetal force mv²/r and solve for r.
Q: If the particle were an electron instead of a positive charge, how would the path change?
A: The direction of rotation would reverse. The radius and speed remain the same (assuming the same magnitude of charge and velocity).
Q: Explain why the magnetic force does zero work on a moving charged particle.
A: The magnetic force F = qv × B is always perpendicular to the velocity v. Work is W = ∫F · ds = ∫F · v dt. Since F ⊥ v, the dot product F · v = 0 at every instant, so the total work is zero. The magnetic force changes direction but not speed.
Q: What is the magnetic field inside a long solenoid with n turns per unit length carrying current I?
A: B = μ₀nI, directed along the axis of the solenoid. This follows from Ampere's law applied to a rectangular loop straddling the solenoid wall.
Q: Which factors determine the magnitude of the magnetic force on a straight current-carrying wire?
A: The field strength B, the current I, the wire length L, and the angle θ between the wire and the field. The force magnitude is F = ILB sin θ.
The cyclotron radius connects to the Hall effect (where charge carriers in a conductor curve in a B field, creating a transverse voltage) and to the design of particle accelerators. The solenoid field formula is the starting point for inductance (L = μ₀n²Al for a solenoid of length l and cross-sectional area A), which feeds directly into RL and RLC circuit analysis. Electromagnetic waves tie together everything in the course: Maxwell's equations unify electricity and magnetism and predict that light is an EM wave.
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