Magnetic Field & Magnetic Forces

Earth's magnetic field is essentially a giant bar magnet buried inside the planet. But here's the confusion: the geographic North pole of Earth is actually a magnetic South pole. The compass needle's N-pointing end is attracted to it — opposite poles attract. So technically, when you say "north" on a compass, you're pointing at a magnetic south pole.

Earth's magnetism isn't a buried permanent magnet. It's generated by the geodynamo effect: circulating molten iron currents in Earth's outer core produce electric currents, which generate magnetic fields. The field isn't perfectly aligned with the rotation axis (the angle difference is "magnetic declination"), and it wanders over geological time. Every ~200,000 years or so, it reverses completely — N and S flip. The last reversal was ~780,000 years ago, and we may be overdue for another.

Magnetic monopoles would revolutionize physics — and despite decades of searching, none have ever been found. A monopole would be a single "N" or "S" that could exist alone. Some grand unified theories predict they should exist, but experimental searches from particle colliders to polar ice cores have come up empty. Their existence would explain why electric charge is quantized (Dirac's argument), but until one shows up, magnetism stays "dipolar only."

These symbols are universal in physics diagrams, but students often get them backwards. Here's the mnemonic: imagine an arrow (like from a bow) flying toward you or away from you.

• ⊙ (circle with a dot) — the arrow is flying toward you. You see the pointy tip head-on, which looks like a dot. So ⊙ means "out of the page."
• ⊗ (circle with a cross) — the arrow is flying away from you. You see the fletching (feathers) at the back, which form an X. So ⊗ means "into the page."

This convention applies to any vector, not just B. You'll see it for velocity, force, or current direction too. A current ⊙ means "current flowing out of the page at that point."

This is one of the most important and counterintuitive facts in all of electromagnetism: the magnetic force on a moving charge does ZERO work. It can bend the particle's path, but it cannot change its speed or kinetic energy — ever.

Why? Work is W = F · d · cos(angle between F and d). The magnetic force F is always perpendicular to v (because F = qv × B is a cross product, and cross products are always perpendicular to both inputs). Displacement d is in the direction of v. So F and d are always at 90°, cos 90° = 0, and W = 0 always.

What the force does do: it changes direction but never magnitude of velocity. That's exactly the definition of circular motion at constant speed. So charged particles in uniform B fields always trace circles (or helices, if they have a parallel component of velocity).

The energy has to come from somewhere. If you want to accelerate a charged particle, you can't use a B field alone — you need an electric field to do the actual work. This is why particle accelerators alternate between E-field "kicks" (which do the work) and B-field "steering" (which bends the path into a circle to return it to the next kick). Cyclotrons, synchrotrons, even the LHC all follow this pattern.

Proof the magnetic force can still matter: even though it does no work, it can still redirect a particle into a wall, push a charged metal rod sideways, or trap particles in Earth's magnetosphere. "No work" doesn't mean "no effect" — just no energy transfer.

The formula F = qv × B has a q out front — the signed charge. If q is negative (electron: q = −e), then the sign of F flips. Mechanically: compute the cross product v × B using the right hand as if the charge were positive, then mentally flip the direction if the charge is actually negative.

Don't use a "left-hand rule" for negative charges. This is a common recipe taught by some textbooks and tutorials, but it causes confusion when you also have to deal with RHR #1 (field around a wire), where the rule is the same regardless of charge sign. Stick with the right hand always — just flip the final answer for negative charges.

Why this matters in practice: in a cathode-ray tube (old CRT TV), electrons are steered by magnetic fields. Positive ions in the same fields would move the opposite way. Mass spectrometers (next slide) exploit this to separate positive ions from negative ones on opposite sides of the detector.

In 1897, J.J. Thomson used a velocity selector to measure the charge-to-mass ratio of the electron — the experiment that discovered the electron itself. His setup:

1. Accelerate electrons from a hot cathode with a known voltage V. Energy: ½mv² = eV, so v = √(2eV/m).
2. Pass them through a velocity selector with tunable E and B. Find the combination that gives straight-through motion: v = E/B.
3. Combine: (E/B)² = 2eV/m → e/m = E²/(2VB²).

Thomson got e/m ≈ 1.76 × 10¹¹ C/kg, which is 1836 times larger than the e/m of a hydrogen ion. This showed that the mysterious "cathode rays" weren't atoms — they were something 1836× lighter, which Thomson called "corpuscles" and we now call electrons. This was the first evidence that atoms had substructure.

Modern applications: every mass spectrometer starts with a velocity selector to ensure all particles have the same speed before they enter the mass-measurement stage. Without it, lighter and heavier ions with different speeds would mix together.

The magnetron in a microwave oven works on exactly this principle. An electron is accelerated by an electric field and then enters a magnetic field, where it loops in a circle. As it passes slits in a resonant cavity, it excites electromagnetic oscillations at the cyclotron frequency — and those oscillations are the 2.45 GHz microwaves that cook your food.

Sanity check: for a 2.45 GHz magnetron, with m_e = 9.11 × 10⁻³¹ kg and e = 1.6 × 10⁻¹⁹ C, we need B = 2π m f / e ≈ 0.088 T ≈ 880 gauss. That's the actual field strength of magnets inside your microwave oven.

What about particles that have a velocity component along B? They trace a helix — circling in the plane perpendicular to B while drifting steadily along B. This is how Earth's magnetic field traps cosmic rays in the Van Allen belts: they spiral along field lines from pole to pole. Auroras are the glow when these trapped particles finally hit the upper atmosphere.

R = mv/(qB) scaling: double the mass → double the radius. Double the speed → double the radius. Double the charge → half the radius. Double the field → half the radius. This is exactly what mass spectrometers exploit on the next slide.

Before mass spectrometry, chemists thought each element had a single, well-defined atomic weight. Then in the 1910s, J.J. Thomson (same man who discovered the electron) built an early mass spectrometer and made a surprising discovery: neon gas wasn't a single mass — it split into two beams, at masses 20 and 22 atomic mass units.

These weren't two different elements (both were chemically identical neon). They were isotopes — same number of protons and electrons but different numbers of neutrons in the nucleus. This was the first direct experimental evidence that atoms have substructure within their nuclei.

Every element has isotopes. Carbon has ¹²C (most common, stable), ¹³C (rare, stable, used in NMR), and ¹⁴C (radioactive, basis of carbon dating). Uranium has ²³⁵U (fissile, used in bombs and reactors) and ²³⁸U (not fissile, most of natural uranium). Mass spectrometry is how we distinguish them — exactly the physics of this slide.

Modern applications: drug testing, forensic chemistry, dating fossils, exploring planetary atmospheres (the rovers on Mars use mini mass spectrometers to identify gases), detecting doping in athletes, even figuring out where beef came from. All based on R = mv/(qB).

The wire formula comes directly from the single-charge formula. Here's the elegant one-paragraph derivation:

Say the wire has length L, cross-section A, and contains n charge carriers per unit volume, each with charge q and drift velocity v. The total number of charges in the wire is n·A·L. The total force on all of them:

F = (nAL) · qvB sin φ

But the current is I = nqvA, so nqvA·L = I·L. Substituting:

F = I L B sin φ

The formula is the same as for a single charge, just rewritten in terms of the wire's macroscopic current instead of microscopic particle drift.

Why this matters in motors: every electric motor exploits this force. A current flowing through a coil inside a magnetic field feels a force, which becomes a torque, which turns a shaft. From tiny drone motors to 3-ton locomotive motors, all of them are F = IL × B at scale.

Why it matters in loudspeakers: a loudspeaker voice coil is a wire sitting in a permanent magnet's field. Audio-frequency AC current pushed through the coil creates a force that oscillates the speaker cone back and forth — which is the sound you hear. Your phone's earpiece, your car's subwoofer, concert arena mains — all F = ILB.

This is the #1 source of torque errors. The angle θ in τ = NIAB sin θ is measured from the normal vector of the loop (perpendicular to the loop's face), not from the loop's face itself.

Example: "B is parallel to the loop's face" means θ = 90° from the normal → maximum torque. "B is perpendicular to the loop's face" (passing through the loop) means θ = 0° from the normal → zero torque.

Why zero torque when θ = 0°? When the loop's normal aligns with B, the forces on opposite sides still exist but act along the same line — they cancel out as both a force and a torque. This is the stable equilibrium position that every current loop wants to find, just like a compass needle in Earth's field.

Example 20.5 — circular coil: 50 loops, R = 30 cm, I = 8 A, B = 5 T, θ = 90° → τ_max = 50 × 8 × π(0.3)² × 5 = 565.5 N·m. This is the kind of torque that runs industrial motors.

There are TWO right-hand rules in magnetism, and students conflate them:

• RHR #1 (field around a wire): thumb along I, curl fingers — they show B. Used for Slide 11's formula.
• RHR #2 (force on a charge/wire in a field): fingers along v (or I), curl toward B, thumb = F. Used for Slides 4–9.

When to use which: if you're computing the field produced by a current, use RHR #1. If you're computing the force felt by a current or charge in an external field, use RHR #2. Getting these mixed up leads to sign errors.

Why the field is circular around a wire: by symmetry. A long straight wire looks the same if you rotate it around its axis or slide it along its length. The magnetic field must respect those symmetries, which forces it to be circular (rotational symmetry) and the same at every point along the length (translational symmetry). These two constraints leave only one possible answer.

With static charges, like charges repel. With currents, like currents attract. This flipped rule comes directly from the RHR: wire 1 creates a B field that circles around it. At wire 2's location, this field points in a specific direction. Apply RHR #2 to wire 2's current in that field, and the resulting force points toward wire 1 when currents are parallel.

Walk through the geometry: wires both carry current "up." By RHR #1, wire 1's B field at wire 2's location points into the page. Wire 2's current "up" in a field "into page" gives F = IL × B pointing toward wire 1 (by RHR #2). Symmetric reasoning gives the same attraction for wire 1.

The definition of the ampere (pre-2019): one ampere was defined as the current that, when flowing in two infinitely long parallel wires one meter apart, produces a force of exactly 2 × 10⁻⁷ N per meter of length between them. This was an experimental definition based on this exact formula. In 2019, the SI redefinition fixed the elementary charge e to be exactly 1.602176634 × 10⁻¹⁹ C, so the ampere is now defined from charge directly.

The word "solenoid" is used two ways in engineering:

• Pure solenoid (the physics version): just a wire coil producing a uniform B field. Used in MRI machines, particle accelerators, and scientific experiments.
• Solenoid actuator (the industrial version): a coil with a movable iron core inside. When current flows, B inside the coil magnetizes the core and pulls it deeper into the coil. This converts electrical energy into linear motion — used in car starter motors, door locks, industrial valves, pinball flippers, washing machine water valves. When you hear "click" on a device, it's usually a solenoid actuator firing.

Example 20.8 — a solenoid for an electron beam experiment: N = 800 turns, L = 15 cm, I = 5 A. Then n = 800/0.15 = 5333 turns/m. B = (4π × 10⁻⁷)(5333)(5) ≈ 0.033 T. That's about 660× Earth's field — enough to visibly deflect a compass from meters away.

Three ways to make the field stronger: more current I, more turns per meter n, or add an iron core (which multiplies B by the iron's permeability, typically ×1000 or more). Industrial electromagnets combine all three to reach ~1 T.

The formula B = μ₀I/(2πr) only works for a single long straight wire. Ampère's Law works for ANY current distribution:

• Solenoid: pick a rectangular Amperian loop with one side inside and one side outside. Only the inside segment contributes → B·L = μ₀·nLI → B = μ₀nI (Slide 13's formula!).
• Toroid (donut-shaped coil): circular Amperian loop inside the toroid → B·(2πr) = μ₀·N·I → B = μ₀NI/(2πr).
• Coaxial cable: different Amperian loops at different radii give different enclosed currents, yielding the field in every region.
• Current sheets, slabs, planes: all solvable by choosing the right Amperian path.

Ampère's Law is one of Maxwell's equations. In its final form (∇×B = μ₀J + μ₀ε₀ ∂E/∂t), it's the third of the four fundamental equations of electromagnetism. Without it, we wouldn't have radio, microwaves, or any wireless technology — the ∂E/∂t term is exactly what makes electromagnetic waves possible.

In the 1800s, people knew current flowed in metals but didn't know whether the actual moving charges were positive or negative. Both explanations fit macroscopic observations.

Edwin Hall's 1879 experiment settled it: pass a current through a metal strip in a perpendicular magnetic field. If positive charges are moving (conventional current), Lorentz force pushes them to one side, making that side positive. If negative electrons are moving the other way, Lorentz force pushes them to the opposite side of the strip. By measuring which side of the strip becomes positive (the "Hall voltage"), you can directly identify the sign of the carriers.

Hall showed that in most metals, the Hall voltage has the sign expected for negative carriers — electrons. This was 18 years before the electron was formally discovered by Thomson.

Modern Hall sensors are in every modern device: they detect brushless motor position (drones, car alternators), measure current in electric vehicles, sense wheel rotation in ABS brakes, detect lid closure on laptops, and form the basis of the "magnetic card reader" in hotel key systems.

The Quantum Hall Effect (Klaus von Klitzing, 1980) showed that the Hall conductance is quantized in integer multiples of a fundamental constant e²/h. This won him the 1985 Nobel Prize and is now used as the international standard for electrical resistance.

All materials have electrons with "spin" magnetic moments. In most substances, these spins point in random directions and cancel out macroscopically. Iron is different: its atoms have unpaired inner electrons, and quantum mechanical effects (exchange interaction) cause neighboring atoms to spontaneously align their spins parallel to each other in groups called "domains."

In unmagnetized iron, different domains point in random directions and cancel. But when you apply an external field, the domains rotate and grow, aligning with B. This amplifies the external field by a factor of hundreds to thousands. If the domains remain aligned after B is removed, you've created a permanent magnet.

The Curie temperature is the temperature at which thermal motion overcomes the alignment — above it, iron loses its ferromagnetic properties and becomes paramagnetic. For iron, T_Curie = 770 °C. Heat a bar magnet red-hot and it demagnetizes instantly.

Only four elements are ferromagnetic at room temperature: iron, nickel, cobalt, and gadolinium. Everything else you see labeled "magnetic" is an alloy or compound containing these elements.

Diamagnetic materials are actually repelled by magnetic fields — including you. You're weakly diamagnetic (mostly from the water in your body). With a strong enough field (~16 T), you can actually levitate a frog or even a small human. This is the "floating frog experiment" from Andre Geim's 2000 research. He later won the Nobel Prize for graphene.

If you're stuck on a magnetism problem, ask: is the charge moving, or is there a current?

• Single charge moving: F = qvB sin φ. Circular orbit? R = mv/(qB).
• Current-carrying wire: F = ILB sin φ.
• Current loop in B: τ = NIAB sin θ.
• Current producing a B field: use Ampère's Law. Long wire → B = μ₀I/(2πr). Solenoid → B = μ₀nI.
• Two currents interacting: F/L = μ₀I₁I₂/(2πr). Same direction attract.

90% of ch. 20 problems are one of these five categories. Identify which one you're in, pick the formula, and plug in.

• Forgetting sin φ: if the angle isn't given, it's often 90° (sin = 1) — but double-check. Never assume.
• Confusing φ with its supplement: sin φ = sin(180°−φ), so "30° from the wire" and "150° from the wire" give the same force. Either angle is fine.
• Using cos instead of sin: F = qvB sin φ always. If you're using cos, you're off by a complementary angle.
• Forgetting to flip for electrons: if q is negative, the direction reverses from what RHR gives.
• Mistaking RHR #1 for RHR #2: thumb/curl rules look similar. Thumb = I, curl = B is RHR #1 (field from current). Fingers = v, curl toward B, thumb = F is RHR #2 (force on charge/current).
• Not converting to SI: 5 cm is 0.05 m. 200 mA is 0.2 A. 50 gauss is 0.005 T. Always.