Magnetism

Big Picture

Magnetic fields are produced by moving charges (currents) and by intrinsic magnetic moments of elementary particles. A magnetic field exerts a force on a moving charge — crucially, the force is perpendicular to both the velocity and the field. This leads to circular and helical motion, which is exploited in mass spectrometers, particle accelerators, and plasma confinement devices.

ELI-10: Explain It Like I'm 10

A magnet has a north pole and a south pole. Like poles repel; opposite poles attract. Break a magnet in half and you get two smaller magnets, each with its own north and south — there is no such thing as an isolated magnetic pole. Magnetic fields come from moving electric charges: electrons spinning around atoms create tiny magnets. When many of these tiny magnets align, you get a permanent magnet like a refrigerator magnet or a compass needle.

Assumptions

For the worked examples and derivations in this topic, we assume:

1. Non-relativistic velocities: The charged particle's speed v is much less than the speed of light c, so relativistic mass increase is negligible.
2. Uniform magnetic field: Unless otherwise stated, B is constant in magnitude and direction throughout the region of interest.
3. No other forces: Gravity, electric fields, and drag are negligible or absent. When we say a particle moves in a circle, we mean the magnetic force is the only force acting.
4. Steady currents: Currents are constant in time (DC), which keeps the magnetic field static. Time-varying fields belong to the next topic (electromagnetic induction).
5. Thin wires: The current-carrying wire's thickness is negligible compared to the distances involved, so we can treat it as a one-dimensional line.
6. Vacuum permeability: We use μ₀, the permeability of free space, unless a material (like iron) is explicitly introduced.

12.1 Magnetic Fields and Forces

Core Ideas

Magnetic field B. SI unit: tesla (T). 1 T = 1 N/(A·m). For reference, Earth's field ≈ 50 μT, a refrigerator magnet ≈ 5 mT, an MRI magnet ≈ 1.5–3 T.

Magnetic force on a moving charge:

F = qv × B

Magnitude: F = qvBsinθ, where θ is the angle between v and B.

Right-hand rule (cross product for positive charge): Point the fingers of your right hand in the direction of v, then curl them toward B (the shorter arc). Your outstretched thumb points in the direction of F for a positive charge. For a negative charge (like an electron), the force direction is opposite to your thumb.

Alternative right-hand rule — "flat hand" method (for F = qv × B):

1. Open your right hand flat, fingers together and thumb perpendicular (like making an "L" or "finger-gun" shape).
2. Align your fingers with v (velocity direction of the positive charge).
3. Orient your palm so that B points out from your palm. Think: your palm "feels" the field pushing into it.
4. Your thumb now points in the direction of the magnetic force F on a positive charge. For a negative charge, flip the result.

Text-based right-hand rule illustration for a specific case:

Imagine a proton moving to the east (v → ), in a magnetic field pointing north (B ↑):

        (thumb: FORCE = upward, out of page toward you)

           |
           |  F (up)
           |
       ____|____
      /    |     \
     |  o  |      |  <-- hand with fingers east, palm facing north
      \____|_____/
           |
           |
           v (fingers = v, east)

• Fingers point east (v).
• Palm faces north (B enters palm).
• Thumb points up (out of the page toward you) → that's the force on the positive charge.

Key Properties

• The force is zero if the charge is stationary (v = 0) or moving parallel to the field (sinθ= 0).
• The force is always perpendicular to both v and B → does NO work on the particle (changes direction, not speed).
• In a uniform B with v ⊥ B: circular motion with radius r = mvqB.

Why Magnetic Forces Do No Work

Work is defined as W = ∫F · dr, the force dotted with the displacement. The magnetic force is F = qv × B. The instantaneous displacement dr = v  dt, so:

dW = F · dr = (qv × B) · v  dt = q  dt  (v × B) · v

The cross product v × B is always perpendicular to v. The dot product of a vector with a perpendicular vector is zero. Therefore dW = 0 at every instant — the magnetic force does zero work.

Consequences: A static magnetic field can steer a charged particle (change its direction) but can never accelerate it (change its kinetic energy). The speed remains constant; only the velocity vector rotates. This is why cyclotrons and synchrotrons use electric fields to add energy to particles and magnetic fields to bend them into circular paths.

Subtlety — the magnetic force CAN transfer energy in a larger system: When you hold two magnets near each other and feel them repel, it feels like work is being done. The resolution: the magnetic field converts the kinetic energy of aligned electron spins into bulk motion; it's the total electromagnetic energy that changes, not the kinetic energy of individual charged particles. In a wire carrying current, the magnetic force on the electrons is perpendicular to their drift velocity — but the force is transmitted to the lattice (the wire itself), and that can do macroscopic work. For a single isolated charge in a pure B field, however, the magnetic force does precisely zero work.

Circular Motion in a Uniform Magnetic Field

When v ⊥ B, the magnetic force provides the centripetal force:

qvB = mv²r   ⇒  r = mvqB

The angular frequency (cyclotron frequency) is:

ω= vr = qBm,  f = ω2π = qB2πm

Notice ω is independent of speed — all particles of the same q/m ratio complete one revolution in the same time. This is the key insight behind the cyclotron.

Worked Example 1: Cyclotron Motion

Problem: A proton (m = 1.67 × 10^-27 kg, q = +e = 1.60 × 10^-19 C) enters a uniform magnetic field of B = 0.500 T with speed v = 2.00 × 10⁶ m/s, perpendicular to the field. Find: (a) the radius of its circular path, (b) the cyclotron frequency, (c) the period of one revolution.

Solution:

(a) Radius:

r = mvqB = (1.67 × 10^-27)(2.00 × 10⁶)(1.60 × 10^-19)(0.500)

r = 3.34 × 10^-218.00 × 10^-20 = 0.0418 m = 4.18 cm

(b) Cyclotron frequency:

f = qB2πm = (1.60 × 10^-19)(0.500)2π(1.67 × 10^-27)

f = 8.00 × 10^-201.049 × 10^-26 = 7.63 × 10⁶ Hz = 7.63 MHz

(c) Period:

T = 1f = 17.63 × 10⁶ = 1.31 × 10^-7 s = 131 ns

Check: Do the units work? kg · m/sC · T = kg · m/sC · N/(A·m) = kg·m/sC·N/(A·m). Since 1 N = 1 kg·m/s² and 1 A = 1 C/s, this simplifies to meters. ✓

Conceptual Example: The Mass Spectrometer

A mass spectrometer separates ions by their mass-to-charge ratio. Here is the step-by-step logic:

1. Ionization: A sample is vaporized and ionized — atoms gain or lose electrons, giving them a net charge +q. Doubly ionized atoms carry +2e, etc.
2. Acceleration (velocity selector stage): The ions pass through crossed electric and magnetic fields (a "velocity selector"). Only ions with speed v = E/B pass straight through. This ensures all ions entering the main chamber have the same speed v.
3. Deflection: The mono-speed ions enter a uniform magnetic field B (perpendicular to v). Each ion follows a semicircular path of radius r = mv/(qB).
4. Detection: The ions strike a photographic plate or electronic detector at different positions. Since v, q, and B are known, measuring r gives the mass:

m = qBrv

Lighter ions curve more sharply (smaller r), heavier ions curve more gently (larger r). The instrument produces a "mass spectrum" — a plot of detector signal vs. mass. This lets chemists identify unknown compounds, measure isotopic ratios, and even date archaeological samples (via radiocarbon dating with accelerator mass spectrometry).

ELI-10: Explain It Like I'm 10

A magnetic field pushes on a moving charged particle sideways — the push is always perpendicular to the particle's path. So the magnetic force never speeds it up or slows it down; it only turns it. In a uniform field, the particle goes around in circles. Heavier particles make bigger circles, which is how a mass spectrometer sorts atoms by weight — like a race track where heavier cars need a wider turn radius.

12.2 Magnetic Force on Current-Carrying Wires

Core Idea

A straight wire of length L carrying current I in a magnetic field B experiences a force:

F = IL × B

Magnitude: F = ILBsinθ.

This follows directly from F = qv × B: a current I represents charge q moving at drift velocity v_d through the wire. The total charge in a segment of length L is q = (nA L)e and I = nA e v_d, so q v_d = IL. Substituting gives F = IL × B.

Worked Example 2: Force on a Current-Carrying Wire

Problem: A straight copper wire of length L = 0.80 m carries a current I = 5.0 A from west to east. A uniform magnetic field of B = 0.30 T points north-to-south (i.e., downward at 45° into the ground — the field makes an angle of 90° with the wire). Find the magnitude and direction of the magnetic force on the wire.

Simplified version: Let the wire run along the +x direction (east), and the magnetic field point in the +y direction (north), making θ= 90°.

Solution:

The magnitude:

F = ILBsinθ= (5.0)(0.80)(0.30)sin90° = (5.0)(0.80)(0.30)(1.0) = 1.2 N

For the direction, use the right-hand rule for F = IL × B:

• Fingers point in the direction of current (east, +x).
• Curl fingers toward B (north, +y).
• Thumb points up (out of the plane, +z direction).

So the force is 1.2 N upward (perpendicular to both the wire and the field).

Real-world check: Is 1.2 N reasonable? That is about the weight of a small apple — a modest but easily measurable force. If you increase the current to 50 A in a 2 T field with a 1 m wire, you get F = (50)(1)(2) = 100 N, about the weight of a 10 kg mass. This is why high-current bus bars in power plants must be mechanically braced against magnetic forces during short circuits.

Torque on a Current Loop — The Electric Motor Principle

Consider a rectangular loop of wire (width w, height h, area A = wh, N turns) carrying current I in a uniform magnetic field B. The loop's normal vector n̂ makes an angle φ with B.

The forces on opposite sides of the loop produce a couple — equal and opposite forces that create a net torque but no net force. The torque magnitude is:

τ= NIABsinφ

Maximum torque: When φ= 90° (the plane of the loop is parallel to B), sinφ= 1, and τ_max = NIAB.

Zero torque: When φ= 0° or 180° (the plane of the loop is perpendicular to B), sinφ= 0 and τ= 0. This is an equilibrium position; in a DC motor, a commutator reverses the current each half-turn so the torque always pushes in the same rotational direction.

Worked Example 3: Torque on a Motor Coil

Problem: A simple DC motor has a rectangular coil of N = 100 turns, each of area A = 20 cm² = 2.0 × 10^-3 m². The coil carries I = 0.50 A in a magnetic field of B = 0.25 T. Find: (a) the maximum torque on the coil, (b) the torque when the coil plane makes a 30° angle with the field (φ= 60° between n̂ and B).

Solution:

(a) Maximum torque (sinφ= 1):

τ_max = NIAB = (100)(0.50)(2.0 × 10^-3)(0.25) τ_max = (100)(0.50)(5.0 × 10^-4) = 0.025 N·m

(b) At φ= 60°:

τ= τ_maxsin60° = (0.025)(0.866) = 0.0217 N·m

Interpretation: The torque varies sinusoidally with angle. A real DC motor uses a split-ring commutator to keep the torque acting in the same direction by flipping the current direction each half-rotation. Without the commutator, the coil would oscillate back and forth rather than spin continuously.

ELI-10: Explain It Like I'm 10

Run current through a wire in a magnetic field and the wire gets pushed sideways. Wrap the wire into a coil, and the coil twists — that is an electric motor. The twisting force (torque) depends on the current, the strength of the magnet, and the size of the coil. Every motor in your house — fans, blenders, power tools — works on this principle.

12.3 Sources of Magnetic Fields

Core Ideas

Moving charges create magnetic fields. The Biot-Savart law gives the field from a small current element I  ds:

dB = μ₀4π I  ds × r̂r²

For situations with high symmetry, Ampère's law is much simpler:

∮B · ds = μ₀ I_enc

Where μ₀ = 4π × 10^-7 T·m/A (permeability of free space), and I_enc is the total current passing through the surface bounded by the closed path of integration (the "Amperian loop").

Using Ampère's law — the recipe:

1. Identify symmetry: The problem must have enough symmetry that B is constant along a well-chosen closed path and either parallel or perpendicular to ds.
2. Choose an Amperian loop: A closed path that matches the symmetry (circle for a straight wire or toroid, rectangle for a solenoid or infinite current sheet).
3. Evaluate the circulation: ∮B · ds = B × (length of path where B is parallel to ds).
4. Find I_enc: The total current piercing the surface bounded by the loop.
5. Solve for B.

Right-hand rule for the field around a straight wire: Point your right thumb in the direction of the current. Your curled fingers show the direction of the magnetic field lines circling the wire. The field lines form concentric circles; the field is tangent to these circles.

Worked Example 4: Solenoid Field via Ampère's Law

Problem: An ideal solenoid has n turns per unit length (n = N/L) and carries current I. Use Ampère's law to derive the magnetic field inside.

Solution:

An ideal solenoid is infinitely long (or long enough that end effects are negligible), tightly wound, and produces a uniform field inside parallel to the axis. The field outside is approximately zero (field lines spread out and weaken).

Step 1 — Symmetry analysis: By the right-hand rule, the field inside points along the solenoid axis (say, +z). It is uniform in magnitude and direction throughout the interior. Outside, B ≈ 0.

Step 2 — Choose the Amperian loop: Draw a rectangular loop of height h with one side of length ℓ inside the solenoid (parallel to the axis) and the opposite side outside, plus two short vertical segments connecting them.

   ← outside (B≈0)
   ┌──────────────────────────┐
   │                          │  ← vertical segment
   │   INSIDE                  │
   │   B →  (uniform, parallel │
   │        to axis)          │
   │   side of length ℓ       │
   │                          │  ← vertical segment
   └──────────────────────────┘

Step 3 — Evaluate the circulation ∮B · ds:

The closed integral breaks into four segments:

• Inside segment (length ℓ, B ∥ ds): B · ds = B  ds, contribution = +Bℓ.
• Outside segment (length ℓ, B ≈ 0): contribution = 0.
• Two vertical segments: B is perpendicular to ds (field lines are horizontal inside, zero outside), so B · ds = 0.

Total circulation: ∮B · ds = Bℓ+ 0 + 0 + 0 = Bℓ.

Step 4 — Find I_enc: The current piercing the rectangle equals the number of turns enclosed by the loop times the current per turn:

I_enc = (turns per unit length) × (length) × I = nℓI

Step 5 — Apply Ampère's law:

∮B · ds = μ₀ I_enc   ⇒  Bℓ= μ₀ (nℓI)

B = μ₀ n I

where n = N/L is the number of turns per unit length.

Numerical check: A solenoid with n = 2000 turns/m and I = 2.5 A produces:

B = (4π × 10^-7)(2000)(2.5) = 6.28 × 10^-3 T = 6.28 mT

This is about 125 times Earth's magnetic field — strong enough to noticeably attract iron filings but far weaker than an MRI magnet.

Real-solenoid note: A finite solenoid has a weaker and non-uniform field near the ends. The field at the exact center of a finite solenoid of length L and radius R is:

B_center = μ₀ n I L√L² + 4R²

For L ≫ R, this approaches μ₀ n I, confirming the idealization.

Key Results (Summary)

• Biot-Savart (general): dB = μ₀4π I  ds × r̂r² — the field contribution from a tiny current element.
• Long straight wire: B = μ₀ I2πr — circular field lines around wire, right-hand rule for direction.
• Solenoid (tightly wound coil, length L, N turns): B = μ₀NLI = μ₀ n I inside. Nearly uniform.
• Toroid: B = μ₀ N I2πr inside the donut; B = 0 outside. The field is completely confined.
• Force between two parallel wires: Two parallel wires carrying currents I₁ and I₂ separated by distance d attract if currents are parallel, repel if anti-parallel. Force per unit length: F/L = μ₀ I₁ I₂2πd. This is the basis for the SI definition of the ampere.

ELI-10: Explain It Like I'm 10

Every electric current creates a magnetic field circling the wire — like a tiny invisible tornado of magnetism wrapping around it. A solenoid (a coil of wire) concentrates this field into a strong, uniform magnet inside the coil — just like a bar magnet. This is how electromagnets work: turn on the current, magnetism appears; turn it off, it disappears. Scrapyard cranes, doorbells, and MRI machines all use electromagnets.

Common Misconceptions

1. "Magnetic monopoles exist"

Misconception: Cutting a bar magnet in half separates the north and south poles, giving you a north-only piece and a south-only piece.

Reality: Every time you break a magnet, you get two smaller dipoles, each with its own north and south. No experiment has ever detected an isolated magnetic monopole (a "north charge" without a corresponding south). All known magnetic fields are produced by dipoles or by moving electric charges. Paul Dirac showed theoretically that if monopoles exist, they would explain electric charge quantization — but none have been found. Modern grand unified theories predict they might exist at extremely high energies, but for all practical physics, magnetic monopoles do not exist.

2. "Magnetic forces can do work"

Misconception: A magnet can pick up a paperclip, lifting it against gravity — that is work being done. Therefore magnetic forces do work.

Reality: The magnetic force F = qv × B does zero work on an isolated moving charge because it is always perpendicular to velocity. However, in a macroscopic object, the interaction is more subtle: (a) the magnetic field induces magnetization in the paperclip, creating an attractive force that is not described by the simple Lorentz force law for a single charge; (b) in current-carrying conductors, the magnetic force on the electrons is transmitted through collisions to the atomic lattice — the lattice moves, and the mechanical force on the lattice does work. The electromagnetic field's energy does change; it is just that the pure v × B Lorentz force on a point charge contributes zero to the particle's kinetic energy. The statement "magnetic forces do no work" is always true for a single point charge in a pure magnetic field; it requires careful qualification for extended objects and time-varying fields.

3. "The magnetic field does work in a motor"

Misconception: Since a motor converts electrical energy to mechanical work, and it uses magnetic fields, the magnetic field must be doing the work.

Reality: The energy conversion in a motor comes from the electric power supply. The current in the coil carries electrical energy (from the battery or outlet). The magnetic field provides the mechanism to convert that electrical energy into torque, but the energy itself comes from the current source. The magnetic field is a "catalyst" — it enables the conversion without being consumed. More precisely, the back EMF (covered in the next topic on induction) is what governs energy transfer: the electrical power IE balances the mechanical power τω.

4. "Stronger magnet = stronger force, always"

Misconception: The magnetic force on a charged particle is always larger in a stronger magnetic field.

Reality: For a given particle speed and charge, F = qvBsinθ. If the particle is moving parallel to the field (θ= 0), the force is zero regardless of how strong B is. The orientation of velocity relative to the field matters just as much as the field strength.

Topic Summary

• Magnetic force on a moving charge: F = qv × B. Perpendicular to both → does no work on isolated charges; changes direction, not speed.
• Circular motion in uniform B: r = mv/(qB), cyclotron frequency f = qB/(2πm), independent of speed.
• Force on current-carrying wire: F = IL × B.
• Torque on a current loop (τ= NIABsinφ) — the motor principle. Maximum when loop plane is parallel to field; zero when perpendicular.
• Mass spectrometer: velocity selector + uniform B separates ions by m/q; radius r = mv/(qB).
• Ampère's law (∮B · ds = μ₀ I_enc) relates magnetic field circulation to enclosed current. Use when symmetry is high.
• Long straight wire: B = μ₀ I/(2πr). Solenoid: B = μ₀ nI (uniform inside, approx. zero outside).
• Biot-Savart law: dB = μ₀4π I  ds × r̂r² — the general law; Ampère's law is the integral form for symmetric cases.
• Magnetic monopoles have never been observed; all magnets are dipoles.
• Right-hand rules govern direction of force (v × B) and direction of field around a current-carrying wire.

Essential Equations

Concept Check

1. A charged particle moves parallel to a magnetic field. What is the magnetic force on it? Why?
2. Why does a magnetic field do no work on an isolated charged particle? Explain using the definition of work and the Lorentz force.
3. Describe the right-hand rule for finding the direction of the magnetic force F = qv × B.
4. Describe the right-hand rule for finding the direction of the magnetic field around a straight current-carrying wire.
5. A current loop in a uniform magnetic field experiences a torque. Under what orientation is the torque maximum? Minimum (zero)?
6. How does inserting an iron core into a solenoid affect its magnetic field? Why?
7. In a mass spectrometer, why do lighter ions curve more sharply than heavier ions of the same charge?
8. Two parallel wires carry current in the same direction. Do they attract or repel? What if the currents are opposite?
9. An electron and a proton enter the same uniform magnetic field with the same speed, perpendicular to the field. Which one has the larger orbit radius? Which one circles in the opposite direction?
10. What would happen to the cyclotron frequency if the particle's speed doubled? Explain.

Open Educational References

• OpenStax, College Physics, Chapter 22: Magnetism
• OpenStax, University Physics, Volume 2, Chapters 11–12: Magnetic Forces, Sources of Magnetic Fields