Source: Lecture 12 handwritten notes | University Physics: Electricity & Magnetism
Tags: magnetic force, Lorentz force, cross product, Biot-Savart law, cyclotron radius, bar magnets
Difficulty: Introductory-Intermediate | Prerequisites: electric fields, vectors, Newton's second law
This lecture introduces the second half of classical electromagnetism: magnetism. Up to now the course has dealt with electric charges at rest (electrostatics). Here the focus shifts to what happens when charges move, because moving charges both create magnetic fields and feel forces from them. Understanding this lecture is the entry point for everything that follows, from Ampère's law to electromagnetic induction. You should be comfortable with electric fields, vector notation, and the dot product before starting.
Magnetic fields are produced by moving charges (electric currents). A charge moving through a magnetic field experiences a force perpendicular to both its velocity and the field, given by F = qv × B. Because that force is always perpendicular to the velocity, it changes direction but not speed, so a charged particle in a uniform magnetic field follows a circular path whose radius depends on mass, speed, charge, and field strength.
Magnetic field (B)
A vector field, measured in tesla (T), that exerts a force on moving electric charges. It is produced by electric currents and by permanent magnets.
Think of it as the invisible influence around a magnet or a current-carrying wire that pushes on other moving charges.
Lorentz force (magnetic component)
The force on a charged particle moving through a magnetic field: F = qv × B. Its magnitude is F = qvB sinθ, where θ is the angle between v and B.
In simple terms, this is the "sideways push" a magnetic field gives to a moving charge. The faster the charge moves, and the stronger the field, the harder the push.
Biot-Savart law
Gives the infinitesimal magnetic field dB produced by a small current element: dB = (μ0I / 4π)(ds × r̂) / r², where μ0 is the permeability of free space.
This is the magnetic version of Coulomb's law. It tells you how to calculate the magnetic field created by a tiny piece of wire carrying current.
Cross product (vector product)
An operation on two vectors A and B that produces a third vector perpendicular to both. Magnitude: |A × B| = AB sinθ. Direction: given by the right-hand rule (RHR).
Where the dot product measures how much two vectors point the same way, the cross product measures how much they point in different directions, and it gives you a new vector sticking out of the plane they share.
Cyclotron radius (Larmor radius)
The radius of the circular orbit a charged particle follows in a uniform magnetic field: R = mv / (qB).
A heavier or faster particle traces a wider circle; a stronger field or a larger charge bends the path tighter.
Every bar magnet has a north (N) and south (S) pole.
Opposite poles attract; like poles repel, similar in principle to electric charges, though magnetic monopoles have never been observed.
Field lines emerge from the north pole, curve outward, and re-enter at the south pole, forming closed loops.
The fundamental source of all magnetic fields is moving electric charge (electric current). Permanent magnets work because of microscopic current loops from electron motion inside atoms.
When a particle with charge q moves with velocity v through a magnetic field B, it experiences the force:
Magnetic (Lorentz) forceF = qv × B
Magnitude: F = qvB sinθ
The force is perpendicular to both v and B, so it never does work on the particle. Speed stays constant; only the direction of motion changes.
If v is parallel to B (θ = 0° or 180°), sinθ = 0 and the force is zero. A charge moving along a field line feels no magnetic push.
Direction is found with the right-hand rule: point fingers along v, curl them toward B; your thumb points in the direction of F for a positive charge. Reverse it for a negative charge.
A × B is a vector (unlike the dot product, which is a scalar).
The dot product, A · B, is proportional to the component of B parallel to A.
The cross product, A × B, is proportional to the component of B perpendicular to A.
Magnitude: |A × B| = AB sinθ.
Direction: perpendicular to the plane defined by A and B, with sense determined by the right-hand rule.
Anti-commutative: A × B = −B × A.
Because the magnetic force is always perpendicular to the velocity, a charged particle entering a uniform field (with v perpendicular to B) follows a circle. Setting the magnetic force equal to the centripetal force:
Circular motion conditionqvB = mv² / R
Cyclotron (Larmor) radiusR = mv / (qB)
Larger mass or higher speed means a larger radius (wider arc).
Stronger field or larger charge means a smaller radius (tighter arc).
The period of the orbit, T = 2πm / (qB), depends only on mass, charge, and field, not on speed. This is the principle behind the cyclotron particle accelerator.
The magnetic force does no work (it is always perpendicular to displacement), so kinetic energy KE = ½mv² stays constant in a purely magnetic field.
If an electric field is also present, the electric force can do work and change the particle's speed. The potential energy in an electric field is U = qEd (for a uniform field over distance d).
Biot-Savart lawdB = (μ0I / 4π) · (ds × r̂) / r²
μ0 = 4π × 10−7 T·m/A is the permeability of free space.
ds is a tiny length element of wire in the direction of current I.
r̂ is the unit vector from the wire element to the point where B is being calculated, and r is the distance.
To find the total field from a full wire, integrate dB over the entire current-carrying path.
The cyclotron radius relationship (R = mv/qB) is the operating principle of mass spectrometers, which separate ions by mass-to-charge ratio. It is also the basis for cyclotrons and synchrotrons used in medical imaging and particle physics.
The Lorentz force on electrons in the Earth's magnetic field traps charged particles from the solar wind into the Van Allen radiation belts and steers them toward the poles, producing the aurora.
Students often think the magnetic force can speed up a charged particle. It cannot. Because F is always perpendicular to v, the force changes the particle's direction only, never its speed or kinetic energy.
Students sometimes apply the right-hand rule and forget to flip the result for a negative charge. For electrons (q < 0), the force direction is opposite to what the right-hand rule gives directly.
It is common to confuse the dot product and the cross product in magnetic force problems. The magnetic force uses the cross product (v × B); the dot product gives zero when the vectors are perpendicular, which is exactly backwards from how the magnetic force behaves.
Expect problems asking you to find the direction of the magnetic force using the right-hand rule, especially with charges moving at various angles to B.
Deriving R = mv/(qB) by equating the magnetic force to the centripetal force is a classic exam derivation. Know it cold.
Questions comparing the magnetic force to the electric force are common. The key distinction: magnetic force does no work; electric force does.
The Biot-Savart law sets up the next several lectures. You probably will not be tested on integrating it yet, but understanding its structure (current creates B, the cross product gives direction) is expected.
1. True or false: A magnetic field can increase the kinetic energy of a charged particle.
2. Fill in the blank: The magnitude of the cross product A × B is AB ___.
3. True or false: If a charged particle moves parallel to a magnetic field, the magnetic force on it is at its maximum.
4. Fill in the blank: The cyclotron radius of a charged particle is R = ___ / (qB).
5. True or false: The cross product is commutative (A × B = B × A).
Tap or hover on each answer to reveal it.
Q: A proton moves east at 5 × 106 m/s through a uniform magnetic field of 0.3 T directed north. What is the magnitude and direction of the force on the proton?
A: F = qvB sinθ = (1.6 × 10−19)(5 × 106)(0.3)(sin 90°) = 2.4 × 10−13 N. By the right-hand rule (fingers east, curl toward north), the force points upward (out of the ground).
Q: An electron (m = 9.11 × 10−31 kg) moves at 2 × 107 m/s perpendicular to a 0.5 T field. What is the radius of its circular orbit?
A: R = mv/(qB) = (9.11 × 10−31)(2 × 107) / (1.6 × 10−19 × 0.5) = 2.28 × 10−4 m, or about 0.23 mm.
Q: Explain why the magnetic force cannot change the speed of a charged particle, even though it can change the particle's direction of travel.
A: Work equals force times displacement in the direction of force (W = F · ds). Because the magnetic force is always perpendicular to the velocity (and therefore to the displacement), the dot product F · ds = 0 at every instant. No work means no change in kinetic energy, which means speed remains constant.
Q: Two particles, one with mass m and one with mass 2m, enter the same uniform magnetic field at the same speed and charge. How do their cyclotron radii compare?
A: Since R = mv/(qB), and speed, charge, and field are the same, the heavier particle has twice the radius: R2 = 2R1.
Q: In the Biot-Savart law, what is the role of the cross product ds × r̂?
A: It determines the direction of the magnetic field produced by a current element. The field is perpendicular to both the current direction (ds) and the line from the element to the observation point (r̂). Its magnitude (sinθ) means a point directly along the wire axis gets no field contribution from that element, while a point broadside gets the maximum.
This lecture connects directly to Ampère's law and Faraday's law, which you will meet in the next few weeks. Ampère's law gives a shortcut for calculating B in situations with high symmetry, much as Gauss's law did for electric fields. Faraday's law later shows that a changing magnetic field creates an electric field, closing the loop between electricity and magnetism and leading toward Maxwell's equations.
The cross product reviewed here will return in force (pun intended) when you study torque on current loops and the magnetic dipole moment. If the cross product still feels unfamiliar, getting comfortable with it now will pay off repeatedly.
magnetic forceLorentz forcecross productvector productright-hand ruleRHRbar magnetmagnetic polesBiot-Savart lawcyclotron radiusLarmor radiuscircular motion magnetic fieldcharged particle in B fieldqvBmv/qBpermeability of free spaceμ0PHYS 212electricity and magnetismmass spectrometercentripetal force