Chapter 2
Every motor, transformer, generator, and radio transmitter Tesla ever built is an application of what this chapter covers. The physics is not optional background — it is the mechanism.
In Chapter 1 you built an electromagnet — copper wire wound around a nail, connected to a battery — and watched it pick up paper clips. You know from Chapter 1 that current creates a magnetic field (Oersted's Rule) and that a changing magnetic field creates current (Faraday's Rule). Now we need to go one level deeper: what exactly is that field, and why do some materials make it so much stronger than others?
This section introduces three quantities — B, H, and Φ — that describe magnetic fields. They are related, but they measure different things. Most textbooks introduce all three in the same paragraph and leave students confused for years. Let's go one at a time.
The B field is the magnetic field at a specific point in space. It is what you would measure if you put a sensor there. It is what pushes on a compass needle, deflects electrons in a TV tube, or makes your car's speedometer cable work. B is the result — the actual force-per-unit-charge-per-unit-speed that exists at a location.
The unit of B is the Tesla (T), named for the man this book is about. One Tesla is a strong field. Here is a scale to anchor your intuition:
| Source | B field (Tesla) | Feel |
|---|---|---|
| Earth's magnetic field | 0.00005 T (50 µT) | Too weak to feel; only a compass detects it |
| Refrigerator magnet | ~0.005 T | Barely holds a paper clip against gravity |
| Strong neodymium magnet | 0.5 – 1.5 T | Can crush fingers if two snap together |
| MRI machine | 1.5 – 7 T | Will rip metal objects across the room |
| Research superconducting magnet | 20 – 45 T | Lethal to anyone carrying metal |
The unit Tesla (T) is named for Nikola Tesla — it was adopted by the International System of Units in 1960. Every MRI scan report states the field strength in Tesla. A standard hospital MRI runs at 1.5T or 3T. When your doctor orders a "3 Tesla MRI," they are using a unit named for the man who built the motor that powers the MRI room's air handling, while the MRI itself uses a superconducting magnet measured in his honor.
Also: the Hall effect sensor in your phone's compass chip measures B directly. Every time your phone knows which way is north, it has measured the Earth's 50 µT B field.
For a wire carrying current I, with length L, sitting in a field B, the force on the wire is:
F = B × I × L (when wire is perpendicular to field)This is the equation that makes every electric motor turn. The force is proportional to B, to the current, and to the length of wire in the field. Tesla's induction motor is this equation, applied cleverly with a rotating field.
Here is where most explanations lose people. They introduce H immediately after B, give the formula B = μH, and leave the reader wondering why there are two fields instead of one.
The answer is this: B depends on what material is present. H does not.
When you run current through a coil of wire, you are creating H — the magnetic driving force. H depends only on the current and the geometry of the coil. It does not care whether there is air, iron, or ferrite inside the coil. H is determined entirely by what you do — the current you apply and how many turns of wire you use.
B is what you get as a result, after the material inside the coil has responded to your H. In air, B and H are nearly identical (with a small constant between them). In iron, B can be 5,000 to 20,000 times larger than in air for the same H. The material amplifies your effort.
The relationship, written as an equation:
B = μ0 × μr × HWhere μ0 = 4π × 10−7 is a constant (the permeability of free space — how "easy" it is for a magnetic field to exist in a vacuum), and μr is the relative permeability of the material — a number that tells you how much better the material is than vacuum at carrying a magnetic field.
| Material | μr (approx.) | What this means |
|---|---|---|
| Air, vacuum | 1 | The baseline. No amplification. |
| Aluminum, copper, wood | ~1 | Essentially the same as air. Not magnetic. |
| Ferrite (power grade) | 1,500 – 2,500 | 2,000× better than air. Used in phone chargers, switching power supplies. |
| Silicon steel (transformer core) | 1,500 – 10,000 | Up to 10,000× better than air. Every transformer core is this. |
| Soft iron (annealed) | 5,000 – 20,000 | The best common material. Electromagnet cores. |
H is the effort you put in. B is what you get out. The material in between (permeability μ) determines how efficiently effort becomes result.
An iron core with μr = 5,000 turns the same coil current into a B field 5,000 times stronger than you would get with an air core. That is why a transformer weighing 10 lbs can do what would otherwise require a coil the size of a room. The iron is doing most of the work.
If those two sentences make sense, the experiment below will make the numbers real. If they don't, re-read the water pressure analogy above.
You already built an electromagnet in Chapter 1. This experiment uses the same setup to demonstrate permeability directly. Same current (same H) — different materials (different μ) — very different results (very different B).
| Core material | Expected μr | Expected pull (relative) |
|---|---|---|
| Air (no core) | 1 | Baseline — maybe 1-2 grams |
| Wood, plastic | ~1 | Same as air — no difference |
| Aluminum | ~1 | Same as air — aluminum is not magnetic |
| Iron nail | ~200 (rusty/work-hardened) | 10-50× stronger than air |
| Soft steel bolt | ~1,000+ | Noticeably stronger than nail |
What you are measuring: The pulling force on the washer is proportional to B². Same H (same current, same coil), very different B because μr is different. The iron nail is not contributing energy — it is simply a better "pipe" for the magnetic field. The same current that barely lifts a gram through air can lift 50 grams with the right core.
This is why transformer cores are iron, not air. An air-core transformer would require a coil hundreds of times larger to move the same amount of flux. Iron is not magic — it just has μr ≫ 1.
One more quantity before Faraday's Law: magnetic flux Φ (the Greek letter phi). Flux is the total amount of B field passing through an area. If B is the intensity of the field at a point, Φ is the total field through a whole cross-section.
Φ = B × A (for uniform B perpendicular to area A)The unit of flux is the Weber (Wb). One Weber per second of change in flux produces one volt of induced EMF. That relationship is Faraday's Law, which comes next.
If you are wondering why physics needs both B and H instead of just one field, the short answer is: they represent different things that happen to be equal (times a constant) in vacuum, but diverge inside materials. H tells you about the currents and their geometry — the cause. B tells you about the actual force on a charge — the effect. In a vacuum they are proportional and you rarely need both. Inside a transformer core, motor stator, or permanent magnet, the material's response (μr) makes B and H very different, and you need both to design the system correctly.
Maxwell's equations, covered later in this chapter, are written in terms of both B and H for exactly this reason.
In 1831, Michael Faraday discovered the single most consequential fact in the history of electrical engineering: a changing magnetic flux induces an electromotive force (EMF). Every generator, transformer, induction motor, and Tesla coil in existence is an application of this one law.
When a straight conductor of length L moves with velocity v perpendicular to a uniform magnetic field B, free electrons experience a force F = qv × B, creating a charge separation — a voltage:
ε = Blvwhere B is in Tesla, L in meters, v in m/s, all three vectors mutually perpendicular. This is motional EMF — the operating principle of every electrical generator ever built. In a practical generator, conductors are wound on a rotating armature. As the armature turns, conductors sweep through the field, generating EMF continuously.
When a conductor is stationary but the magnetic field through it is changing, an EMF is still induced — because the changing field creates a circulating electric field everywhere in space (Section 2.7). For a sinusoidally varying flux Φ = Φmax · sin(2πft), the RMS induced voltage in an N-turn coil is:
εrms = 4.44 · f · N · Bmax · AThe factor 4.44 = 2π/√2, accounting for RMS conversion of a sine wave. This is the EMF equation of the transformer. Every power transformer design begins here.
The negative sign in Faraday's Law is not a mathematical convention. It is a physical law with a name: Lenz's Law (Heinrich Lenz, 1834). Stated plainly: the induced EMF drives a current in the direction that opposes the change in flux that caused it.
Eddy current braking: When a permanent magnet falls through a conductive but non-magnetic tube (copper or aluminum pipe), the changing flux induces circulating eddy currents in the pipe wall. By Lenz's Law, these currents create a field opposing the magnet's motion in both directions — retarding the fall. The retarding force is proportional to velocity, so the magnet reaches a terminal velocity. There is no friction. The effect is purely electromagnetic. See Experiment 2.1.
Back-EMF in motors: A spinning motor armature is a conductor moving through a magnetic field — a generator. It generates a voltage opposing the supply voltage (Chapter 3 covers this fully). The faster the motor spins, the larger the back-EMF, and the less net voltage drives armature current. Lenz's Law regulates motor speed automatically: the motor draws less current as it approaches its no-load speed.
Transformer loading: When a load is connected to a transformer secondary, secondary current creates MMF opposing the primary flux — Lenz's Law. This opposition allows more current to flow from the supply into the primary winding. The transformer does not "know" a load has been connected; the physics causes the primary to respond. Loading the secondary automatically draws more power from the primary source.
Engineers who understand Ohm's Law and Kirchhoff's Laws already understand magnetic circuits. The analogy is mathematically exact within the linear regime, and it is used daily in transformer and motor core design.
| Electric Circuit | Magnetic Circuit |
|---|---|
| EMF — Voltage V (volts) | MMF — Magnetomotive Force ℱ = N·I (ampere-turns) |
| Current I (amperes) | Flux Φ (webers) |
| Resistance R = ρL/A (Ω) | Reluctance ℛ = L/(μA) (A-turns/Wb) |
| Ohm's Law: V = IR | Hopkinson's Law: ℱ = Φ · ℛ |
| Conductivity σ | Permeability μ = μ0μr |
| Series resistors: R = ΣRi | Series paths: ℛ = Σℛi |
| KVL: ΣV = 0 | ΣMMF = ΣΦℛ around loop |
| KCL: ΣI = 0 at node | ΣΦ = 0 at junction |
Reluctance of a magnetic path:
ℛ = L / (μ0 μr A)where L is path length (m), A is cross-sectional area (m²).
The most important design lesson in magnetic circuits is the disproportionate influence of even a tiny air gap.
This is why transformer cores are made from tightly fitting laminations with no deliberate gap, and why motor air gaps are machined to tolerances measured in hundredths of a millimeter. The mechanical precision of the air gap in a motor stator/rotor assembly is not cosmetic — it is a primary determinant of motor performance.
Inductance L (henries) is the ratio of flux linkage to current. For a coil of N turns on a core of reluctance ℛ:
L = N² / ℛ = N² · μ0μrA / lIncreasing turns increases inductance as N². Introducing a deliberate gap decreases inductance by increasing ℛ. This is why power supply chokes have deliberate gaps — the gap linearizes the L vs. I relationship, preventing saturation under DC bias.
The relationship B = μH is not constant. When every magnetic domain in the iron has aligned with the applied field, increasing H further cannot align any more domains. The incremental permeability drops toward μ0. The material is saturated.
The B-H curve plots flux density B on the vertical axis against field strength H on the horizontal axis. Its characteristic shape has three regions:
| Material | Saturation B (approx.) |
|---|---|
| Silicon steel (grain-oriented) | 1.7–2.05 T |
| Soft iron (pure) | ~2.1 T |
| Nickel-iron alloys (permalloy) | 0.8–1.1 T |
| Cobalt-iron alloys (Permendur) | ~2.4 T |
| Ferrite (MnZn power grade) | 0.3–0.5 T |
| Amorphous metal (Metglas 2605) | ~1.56 T |
Power transformer designers work at Bmax = 1.2–1.5 T for silicon steel at 50/60 Hz — comfortably below saturation. High-frequency designs use ferrite and operate at Bmax = 0.1–0.3 T, where other loss mechanisms dominate before saturation becomes the constraint.
A time-varying magnetic field passing through a conductor induces voltages in that conductor — Faraday's Law has no exceptions. In a bulk piece of iron, those induced voltages drive circulating currents throughout the material. These are eddy currents. They dissipate power as I²R heat, heating the core without doing useful work.
The power dissipated by eddy currents per unit volume:
Pe ∝ f² × B²max × d²where f is frequency, Bmax is peak flux density, and d is the lamination thickness. The quadratic dependence on both frequency and thickness is the dominant design constraint in all power magnetics.
Solution: Laminations. Thin sheets of iron (typically 0.27–0.65 mm for power frequency), electrically insulated from each other by an oxide coating or varnish, break the eddy current paths. Currents cannot flow across insulation gaps. Each lamination acts independently. Since d is now the lamination thickness, losses are reduced by (dlam/dsolid)2.
Skin depth is the related phenomenon in conductors: at high frequency, eddy currents cause current to crowd toward the outer surface. The characteristic depth δ = √(2ρ/ωμ). For copper at 60 Hz, δ ≈ 8.5 mm. At 1 MHz, δ ≈ 66 µm. This is why Tesla coil conductors are made of wide copper strap or Litz wire — to minimize resistance at high frequency.
| Application | Frequency | Core Material | Reason |
|---|---|---|---|
| Power transformer | 50–60 Hz | Grain-oriented silicon steel | High Bsat, low hysteresis, laminated |
| Audio transformer | 20 Hz – 20 kHz | Si steel, thin laminations | Same material, tighter lamination spacing |
| SMPS transformer | 20 kHz – 1 MHz | Ferrite (MnZn, NiZn) | Insulating matrix — no bulk eddy currents |
| Tesla coil / resonant | 50 kHz – 500 kHz | Air core or ferrite | Iron losses prohibitive at these frequencies |
| Current transformer | DC – 10 kHz | Amorphous metal | Very low hysteresis, high μ |
Even in a perfectly laminated core, energy is lost driving the magnetic domains through their alignment cycle each AC cycle. The B-H curve for any real magnetic material traces out a hysteresis loop: as H is cycled, B follows a path that does not retrace itself. The area enclosed by the B-H loop is the energy dissipated per unit volume per cycle (J/m3 per cycle).
The Steinmetz formula for hysteresis power loss per unit volume:
Ph = kh × f × Bnmaxwhere kh is a material constant and n is the Steinmetz exponent — empirically approximately 1.6–2.0 for silicon steel. Steinmetz derived this experimentally in 1892. Total core loss:
Pcore = kh f Bnmax + ke f² B²maxAt low frequencies, hysteresis loss dominates (scales as f). At higher frequencies, eddy current loss dominates (scales as f2). The crossover depends on material and lamination thickness.
Electric charges are the sources and sinks of electric field lines.
Field lines begin on positive charges and end on negative charges. The net electric field flux through any closed surface equals the enclosed charge divided by ε0:
Net electric flux = Qenclosed / ε0A positive charge is a source (field lines radiate outward). A negative charge is a sink (field lines converge inward). No charge means no net flux — field lines that enter the surface must exit it. Engineering implication: charge accumulates on conductor surfaces, not in the bulk. Any net interior charge would require field lines beginning and ending mid-material — impossible in a conductor at equilibrium.
There are no magnetic monopoles. Magnetic field lines always form closed loops.
The net magnetic flux through any closed surface is always exactly zero. Unlike electric field lines, magnetic field lines have no beginning and no end. They form continuous loops — always. Cut a bar magnet in half: you get two smaller magnets, each with a north and south pole. Engineering implication: magnetic flux is conserved around a closed path. Magnetic circuits work because flux, like current at a node, must sum to zero at any branch point.
A changing magnetic field creates a circulating electric field.
Wherever B is changing with time, there exists a circulating E field around that region — even in empty space, with no conductor present. This is what drives current in a transformer secondary: not the primary current acting directly on secondary electrons (the coils are not electrically connected), but the changing magnetic field in the core creating a circulating electric field that pervades the space where the secondary winding sits. The same field is responsible for induction heating — no physical contact required.
Two things create a circulating magnetic field: electric current, and a changing electric field.
The first part (Ampere, 1826): a wire carrying current is surrounded by a circulating magnetic field. Curl the right hand's fingers around the wire with the thumb pointing in current direction — the fingers curl in the direction of B.
The second part (Maxwell's 1865 addition): even without any current-carrying wire, a region where the electric field is changing with time also has a circulating magnetic field. This is the "displacement current." It is not a current of actual moving charges — it is the rate of change of ε0E.
Without Maxwell's displacement current term, the four equations are mathematically inconsistent (Griffiths, Section 7.3.1, shows the inconsistency explicitly). With it, they become the complete description of classical electromagnetism — and they predict radio waves 23 years before Hertz demonstrated them experimentally.
What this demonstrates: Faraday induction and Lenz's Law in a bulk conductor. No circuits, no meters, no calibration required. The effect is dramatic, unambiguous, and directly tied to the physics.
In a 12-inch (30 cm) copper pipe with a 3/4-inch N42 magnet: fall time approximately 0.25–0.30 s in PVC vs. 1.0–1.8 s in copper — a factor of 4–7 slowdown. The magnet arrives gently at the bottom. There is no friction. The pipe remains at room temperature throughout the experiment (though it warms measurably after many repeated passes — the eddy current losses are real).
The falling magnet's field moves with it. The copper wall below sees increasing flux; the wall above sees decreasing flux. Faraday's Law dictates induced EMFs in both regions. These EMFs drive eddy currents — circulating loops in the pipe wall. By Lenz's Law, the currents below create a field opposing the magnet's approach (upward retarding force); the currents above create a field opposing the magnet's recession (also retarding). The retarding force is proportional to velocity (characteristic of eddy current damping), so the magnet reaches a terminal velocity where electromagnetic braking equals gravitational pull.
Donoso, J.M., Ladera, C.L., and Martín, P. (2011). "Damped fall of magnets inside a conducting pipe." American Journal of Physics, 79(2), 193–200. Quantitative analysis with full theoretical model.
What this demonstrates: the B field of a current-carrying coil; Faraday induction with visible evidence; the proportionality of induced EMF to rate of flux change.
For a 100-turn coil of 4 cm diameter (area ≈ 12.6 cm2), a bar magnet moved at 0.5 m/s changing flux density from 0 to 0.1 T generates approximately 0.25 V peak. Lighting an LED requires >1.8 V forward voltage — use a strong N52 magnet with vigorous insertion, or use a galvanometer (far more sensitive). With a good galvanometer and gentle motion, the fundamental effect is evident even at very low EMFs.
Faraday, M. (1832). "Experimental Researches in Electricity, Series I." Philosophical Transactions of the Royal Society. This is the original experiment; Faraday's apparatus was more primitive, the physics identical.