The Timpson Lyceum · University Physics II (2026) – Electricity & Magnetism

Magnetic Field and Magnetic Forces – Lecture 9 (University Physics II)

Overview

Everything up to this point has concerned electric fields and their effect on charges. This lecture opens the second half of the course by introducing the magnetic field $\vec{B}$ — a field that acts only on moving charges, and whose force is perpendicular to both the velocity and the field itself. Two facts are taken as experimental givens rather than derived: magnetic monopoles do not exist, and a moving charge in a magnetic field feels a force $\vec{F} = q\vec{v} \times \vec{B}$ . From those two starting points follow Gauss's law for magnetism (the second of Maxwell's equations), the circular and helical paths charged particles trace out in a uniform field, the velocity selector and the measurement of the electron's mass, and finally the force and torque on current-carrying wires and loops. The recurring difficulty is geometric: magnetism is irreducibly three-dimensional, and the right-hand rule has to be applied deliberately rather than intuitively.

Key Concepts

Magnets, poles, and monopoles

Poles. Like poles repel and opposite poles attract. A compass needle is simply a finely balanced magnet; the end that points toward the Arctic is called its north pole. Because that end is attracted there, the Earth's magnetic south pole lies in the far north — the Earth behaves as a magnet oriented opposite to the naming convention.
No monopoles. Breaking a magnet in half yields two smaller magnets, each with a north and a south end. No isolated pole has ever been found, and the theory here assumes none exists. This is the single deepest difference between magnetism and electrostatics, where isolated positive and negative charges are routine.
Mapping $\vec{B}$ . An electric field is mapped by placing a small test charge and recording the force on it. Since there is no test monopole, a magnetic field is mapped with a compass instead: the direction the compass needle points at each location is the direction of $\vec{B}$ there.

The magnetic force on a moving charge

Two experimental facts. A current-carrying wire deflects a nearby compass — so moving charges produce magnetic fields. And a charge given a velocity near a current-carrying wire feels a force perpendicular to its motion. (The underlying explanation involves special relativity and length contraction; it is not derived here.)
The force law. Observation shows the force reverses when $\vec{v}$ reverses, doubles when $q$ doubles, doubles when $v$ doubles, and always points perpendicular to $\vec{v}$ . These facts are captured by

\vec{F} = q\,\vec{v} \times \vec{B}

Magnitude. The magnitude of a cross product is the product of magnitudes times the sine of the angle between the vectors:

F = |q|\,v B \sin\theta

where $\theta$ is the angle between $\vec{v}$ and $\vec{B}$ . When $\vec{v} \parallel \vec{B}$ , $\sin\theta = 0$ and there is no force at all. The force is maximum when they are perpendicular.

Right-hand rule. Point the index finger along $\vec{v}$ , the middle finger along $\vec{B}$ ; the thumb gives the direction of $\vec{v} \times \vec{B}$ . Never bend the middle finger backward past the index finger — the fingers must sweep from the first vector to the second. The cross product is not commutative: $\vec{B} \times \vec{v} = -\,\vec{v} \times \vec{B}$ .
Sign of the charge. A negative charge feels the force in the opposite direction from a positive charge with the same velocity.
Neutral particles feel nothing. Magnetic fields act only on charged particles, and only when those charges are moving.
Units. The SI unit of $B$ is the tesla (T); $1\ \text{T} = 10^4$ gauss. The Earth's field is roughly half a gauss to a gauss, so one tesla is an extremely strong field.
Drawing convention. A field into the page is drawn as $\times$ (the fletching of a receding arrow); a field out of the page is drawn as $\odot$ (the oncoming point).

Magnetic flux and Gauss's law for magnetism

Magnetic flux is defined exactly as electric flux, with $\vec{E}$ replaced by $\vec{B}$ : for a uniform field through a flat area,

\Phi_B = B A \cos\phi

where $\phi$ is the angle between $\vec{B}$ and the area vector (the normal to the surface). Flux is maximum when the area vector is parallel to $\vec{B}$ and zero when it is perpendicular. For a non-uniform field or curved surface, integrate over patches: $\Phi_B = \int \vec{B} \cdot d\vec{A}$ . For a closed surface, the outward normal is taken as positive by convention.

Units. Flux is measured in webers (Wb), equivalently $\text{T}\cdot\text{m}^2$ .
Gauss's law for magnetism. Gauss's law for electricity says the net electric flux through a closed surface equals the enclosed charge over $\varepsilon_0$ ; an enclosed dipole gives zero net flux because every outgoing field line has a matching incoming one. Because a magnet is always a source paired with a sink and can never be split, the enclosed "pole strength" is always zero:

\oint \vec{B} \cdot d\vec{A} = 0

This is the second of Maxwell's four equations, and it is a direct statement that magnetic monopoles do not exist.

Motion of a charged particle in a uniform field

Circular paths. With $\vec{v}$ perpendicular to a uniform $\vec{B}$ , the force is always perpendicular to the velocity, so the particle turns continuously and traces a circle. Positive and negative charges circle in opposite senses — the reason cloud-chamber tracks reveal a particle's sign, and how the positron was identified as an electron-mass particle of opposite charge.
Magnetic fields do no work. Work requires a force component along the displacement, and the magnetic force is always perpendicular to the velocity. A magnetic field can therefore change a charged particle's direction but can never change its speed or kinetic energy.
Radius. Setting the magnetic force equal to the mass times the centripetal acceleration, $qvB = mv^2/R$ , gives

R = \frac{mv}{|q|B}

A stronger field or a larger charge tightens the circle; a larger mass or a larger speed widens it.

Angular speed and period. Since $\omega = v/R$ ,

\begin{aligned} \omega &= \frac{|q|B}{m} \\ T &= \frac{2\pi}{\omega} = \frac{2\pi m}{|q|B} \end{aligned}

Remarkably, neither depends on the speed or the radius. Two identical particles launched at very different speeds in the same field complete their (very different-sized) circles in exactly the same time. This is the principle behind the cyclotron, which flips the polarity of its accelerating gaps at that fixed frequency.

Helical motion. If the velocity has a component parallel to $\vec{B}$ , that component is untouched (no force acts along $\vec{B}$ ) while the perpendicular component circles. The result is a helix. The pitch — the distance advanced per revolution — is the parallel velocity times the period, $p = v_\parallel T$ .
Magnetic bottles. In a non-uniform field that converges at both ends, the force acquires a component that pushes particles back toward the center, trapping them. This is one way hot plasma is confined.

Applications: velocity selector and the mass of the electron

Velocity selector. Crossed electric and magnetic fields push a charge in opposite directions. The forces balance when $qE = qvB$ , so only particles with

v = \frac{E}{B}

travel straight through. Slower particles are deflected by the stronger electric force; faster ones by the stronger magnetic force.

Lorentz force law. Electric and magnetic fields are the only fields exerting forces on a charged particle, so the total force is their sum:

\vec{F} = q\left(\vec{E} + \vec{v} \times \vec{B}\right)

Maxwell's equations describe how the fields behave; the Lorentz force law is the separate ingredient that tells how those fields act on matter. Together with Newtonian mechanics, they form the complete framework of classical electromagnetism.

Thomson's $e/m$ experiment. Electrons accelerated from rest through a potential difference $V$ gain kinetic energy $eV$ , so $\tfrac{1}{2}mv^2 = eV$ and $v = \sqrt{2eV/m}$ . Passing them through a velocity selector fixes $v = E/B$ independently. Setting the two expressions equal and solving for the mass gives $m = 2eVB^2/E^2$ — the route by which the electron's mass was first determined.

Magnetic forces on currents

Force on a wire segment. The charges in a wire drift with velocity $v_d$ , and the total charge in a short length is $nqA\,dl$ (carriers per unit volume × volume × charge each). Since $I = nqv_dA$ , the force on that segment is

d\vec{F} = I\,d\vec{l} \times \vec{B}

For a straight wire of length $L$ in a uniform field, this becomes $\vec{F} = I\vec{L} \times \vec{B}$ , with magnitude $ILB\sin\theta$ . A wire carrying no current feels no force, and a wire parallel to $\vec{B}$ feels no force. Note that current itself is never treated as a vector — it is a global property of the wire — so the direction is carried by $d\vec{l}$ .

Force on a closed loop. In a uniform field, the net force on any current loop is zero. The forces on opposite sides cancel.
Torque on a loop. A loop in a uniform field can still experience a torque. For a rectangular loop of sides $a$ and $b$ carrying current $I$ , the forces on the two sides of length $a$ are $F = IaB$ , each acting at a lever arm $b/2$ about the rotation axis, so

\tau = 2\left(\frac{b}{2}\right)(IaB)\sin\phi = IAB\sin\phi

where $A = ab$ is the loop area and $\phi$ is the angle between the area vector and $\vec{B}$ — the same angle used for flux, which is why torque and flux tie together naturally.

Magnetic dipole moment. Define $\vec{\mu} = I\vec{A}$ , with direction set by the right-hand rule (curl the fingers along the current; the thumb gives $\vec{\mu}$ ). The torque and potential energy then mirror the electric-dipole results exactly:

\begin{aligned} \vec{\tau} &= \vec{\mu} \times \vec{B} \\ U &= -\vec{\mu} \cdot \vec{B} = -\mu B \cos\phi \end{aligned}

The potential energy follows from integrating the work done by the torque, $W = \int \tau\, d\phi$ , and requiring that it equal the change in $U$ .

Why magnets attract. In a truly uniform field a dipole feels torque but no net force. Attraction and repulsion between magnets arise entirely from the non-uniformity of the field — the curvature of the field lines gives the forces on opposite sides of the dipole components that no longer cancel. An infinitely large, perfectly uniform magnet would not attract an iron nail at all, and the faster the field varies in space, the stronger the attraction or repulsion.
DC motor. Loops of current sitting between magnet poles feel torques that all drive rotation the same way, so the assembly spins. Magnetic fields cannot do work on a lone moving charge, but they can drive current loops and thereby do useful mechanical work.

Worked Example

A proton ( $q = 1.6 \times 10^{-19}\,\text{C}$ , $m = 1.67 \times 10^{-27}\,\text{kg}$ ) moves in a uniform field $B = 0.5\ \text{T}$ pointing along the $x$ -axis. Its initial velocity is $v_x = 1.5 \times 10^5\ \text{m/s}$ and $v_y = 2.0 \times 10^5\ \text{m/s}$ , with $v_z = 0$ . Find the initial force, the initial acceleration, the radius of the helix, the period, and the pitch.

Only the component of velocity perpendicular to $\vec{B}$ contributes to the force. Here $\vec{B}$ lies along $x$ , so $v_x$ produces nothing and only $v_y$ matters; by the right-hand rule the force points along $-z$ :

\begin{aligned} F &= |q|\,v_\perp B \\ &= (1.6 \times 10^{-19}\,\text{C})(2.0 \times 10^5\,\text{m/s})(0.5\,\text{T}) = 1.6 \times 10^{-14}\,\text{N} \end{aligned}

The acceleration follows from Newton's second law:

\begin{aligned} a &= \frac{F}{m} \\ &= \frac{1.6 \times 10^{-14}\,\text{N}}{1.67 \times 10^{-27}\,\text{kg}} = 9.5 \times 10^{12}\ \text{m/s}^2 \end{aligned}

The radius uses the perpendicular velocity only, since that is the part executing circular motion:

\begin{aligned} R &= \frac{m v_\perp}{|q|B} \\ &= \frac{(1.67 \times 10^{-27}\,\text{kg})(2.0 \times 10^5\,\text{m/s})}{(1.6 \times 10^{-19}\,\text{C})(0.5\,\text{T})} = 4.2 \times 10^{-3}\ \text{m} \end{aligned}

The period depends only on mass, charge, and field — not on the speed or the radius:

\begin{aligned} T &= \frac{2\pi m}{|q|B} \\ &= \frac{2\pi (1.67 \times 10^{-27}\,\text{kg})}{(1.6 \times 10^{-19}\,\text{C})(0.5\,\text{T})} = 1.3 \times 10^{-7}\ \text{s} \end{aligned}

Finally, the pitch is how far the parallel motion advances in one full revolution:

\begin{aligned} p &= v_\parallel T \\ &= (1.5 \times 10^5\,\text{m/s})(1.3 \times 10^{-7}\,\text{s}) = 2.0 \times 10^{-2}\ \text{m} \end{aligned}

The proton therefore spirals along the field in a helix of radius about 4.2 mm, completing over seven million turns per second while advancing 2 cm per turn — and its speed never changes, since the magnetic force does no work.

Summary

Magnetic poles always come in pairs; no monopole has been found, so a magnetic field is mapped with a compass rather than with an isolated test pole.
The magnetic force on a moving charge is $\vec{F} = q\vec{v} \times \vec{B}$ , with magnitude $|q|vB\sin\theta$ — perpendicular to both $\vec{v}$ and $\vec{B}$ , zero when they are parallel, and reversed for a negative charge.
Because there are no monopoles, the net magnetic flux through any closed surface is zero, $\oint \vec{B} \cdot d\vec{A} = 0$ — the second of Maxwell's equations.
A magnetic field can never do work on a charge; it changes direction only. A perpendicular velocity gives a circle of radius $R = mv/|q|B$ , and adding a parallel component makes it a helix.
The angular speed $\omega = |q|B/m$ and period $T = 2\pi m/|q|B$ are independent of speed and radius.
Crossed $\vec{E}$ and $\vec{B}$ fields select the single velocity $v = E/B$ ; combined with an accelerating potential, this is how the electron's mass was measured. The total force on a charge is the Lorentz force $\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$ .
A current-carrying wire feels $d\vec{F} = I\,d\vec{l} \times \vec{B}$ . A closed loop in a uniform field feels zero net force but a torque $\vec{\tau} = \vec{\mu} \times \vec{B}$ , where $\vec{\mu} = I\vec{A}$ , with potential energy $U = -\vec{\mu} \cdot \vec{B}$ .
Magnets attract or repel only because their fields are non-uniform; a perfectly uniform field produces torque but no net force.