Magnetic Fields

Part of Electrical Theory

How magnetic fields form around currents, how they interact with conductors, and how this interaction drives every motor and generator ever built.

Why This Matters

Every motor, generator, transformer, relay, solenoid, speaker, and microphone runs on magnetic field interaction. Without understanding magnetic fields, these devices are black boxes that either work or don’t. With this understanding, you can design and build them — winding the right coils, choosing appropriate core materials, and troubleshooting when they fail.

In rebuilding scenarios, you’ll need to build electric generators from wound wire coils and magnets, construct motors for pumps and machine tools, and wind transformers for voltage conversion. All of this requires understanding how magnetic fields form, how they’re shaped by geometry and materials, and how they interact with moving conductors.

What a Magnetic Field Is

A magnetic field is a region of space where magnetic forces act. It’s described by field lines that run from the north pole to the south pole of a magnet (by convention — the lines continue inside the magnet from south to north, forming closed loops).

Field strength is measured in:

• Tesla (T): SI unit; 1 Tesla = 1 Weber per square meter = very strong
• Gauss: Older unit; 1 Tesla = 10,000 Gauss
• Common references: Earth’s field ≈ 0.0001–0.00006 T (0.5–1 Gauss); neodymium magnet ≈ 1–1.4 T; electromagnet coil ≈ 0.5–2 T

Flux (Φ, phi) is the total magnetic “flow” through an area: Φ = B × A × cos(θ), where B is field strength, A is area, θ is angle between field and area normal.

Magnetic Field Around a Current-Carrying Conductor

When current flows through a wire, a circular magnetic field forms around it. The direction is given by the right-hand rule:

• Point your right thumb in the direction of conventional current (+ to −)
• Your curled fingers indicate the direction the magnetic field circles around the wire

For electron flow (actual direction, − to +), use the left-hand rule instead.

Field strength around a wire: B = (μ₀ × I) / (2π × r)

Where:

• μ₀ = 4π × 10⁻⁷ T·m/A (permeability of free space)
• I = current in amps
• r = distance from wire in meters

A wire carrying 10A at 1cm distance: B = (4π×10⁻⁷ × 10) / (2π × 0.01) = 200 × 10⁻⁶ T = 200 μT

This is weak — about twice Earth’s field. To get useful magnetic fields, you need coils.

Coil Fields: Adding Up Contributions

A coil multiplies the magnetic field. Each turn contributes its own field, and they all add in the same direction along the coil axis.

Solenoid field strength: B = μ₀ × μᵣ × N/L × I

Where:

• μᵣ = relative permeability of core (1 for air, 1000–10000 for iron)
• N = number of turns
• L = length of coil in meters
• I = current in amps

Example: Air-core solenoid, 500 turns, 10cm long, 2A: B = (4π×10⁻⁷) × 1 × (500/0.1) × 2 = 12.6 × 10⁻³ T = 12.6 mT

Same solenoid with iron core (μᵣ = 1000): B = 12.6 mT × 1000 = 12.6 T

This is why iron cores are used in electromagnets — they multiply field strength by their permeability.

Force on a Current-Carrying Conductor

When a current-carrying conductor sits in an external magnetic field, it experiences a force. This is the operating principle of all electric motors.

Force on a wire: F = B × I × L × sin(θ)

Where:

• F = force in newtons
• B = magnetic flux density in tesla
• I = current in amps
• L = length of wire in field
• θ = angle between wire and field (maximum when perpendicular)

Direction of force — the motor rule (Fleming’s left-hand rule for motors):

• Left hand (motors: current flows in, motion comes out)
• Point index finger in direction of magnetic field (N to S)
• Point middle finger in direction of conventional current
• Thumb points in direction of force (motion)

Example: A wire 10cm long carrying 5A in a 0.5T field, perpendicular: F = 0.5 × 5 × 0.1 × 1 = 0.25 N

In a motor, this force acts on many turns of wire in a strong field, producing torque.

Electromagnetic Induction: Moving Fields Create Current

The reverse of the motor effect — move a conductor through a magnetic field (or change the field around a stationary conductor), and current is induced. This is electromagnetic induction, discovered by Faraday in 1831.

Faraday’s law: EMF = -N × dΦ/dt

The induced voltage (EMF) equals the number of turns times the rate of change of magnetic flux through the coil. The negative sign means the induced current opposes the change (Lenz’s law).

Key implications:

• Faster movement = larger induced EMF
• More turns = larger induced EMF
• Stronger field = larger induced EMF
• No change = no EMF (DC current in a stationary coil produces no induction)

Direction of induced current — the generator rule (Fleming’s right-hand rule):

• Right hand (generators: motion goes in, current comes out)
• Index finger in direction of field (N to S)
• Thumb in direction of conductor motion
• Middle finger indicates direction of induced current

Hysteresis and Core Losses

When an iron core is repeatedly magnetized in alternating directions (as in AC transformers and motors), the magnetic domains in the iron must repeatedly flip. This requires energy and generates heat. This is hysteresis loss.

Minimizing hysteresis:

• Use soft magnetic materials (low coercivity): soft iron, silicon steel
• Avoid hard magnetic materials (high coercivity): tool steel, permanent magnet materials
• The area of the BH hysteresis loop represents energy lost per cycle

Eddy current losses: Alternating fields in a solid conductor induce circulating currents within the conductor itself, dissipating energy as heat. Solution: laminate the core — use thin sheets of steel insulated from each other. Current cannot circulate through the insulation. The thinner the laminations, the lower the eddy current loss.

Combined core loss ∝ f × B² (approximately)

Doubling frequency doubles core loss. Doubling flux density quadruples it. This is why power transformers designed for 50 Hz fail (overheat) if accidentally connected to 400 Hz.

Building and Testing Electromagnets

Simple electromagnet construction:

1. Cut a core from soft iron rod or stack iron laminations
2. Wind magnet wire (enamel-insulated copper) uniformly in one direction
3. More turns = stronger magnet; more current = stronger magnet
4. Connect to DC battery — test by picking up iron nails, washers, or chips

Testing field direction: Sprinkle iron filings on paper placed over the magnet — they align with field lines, showing the field pattern.

Measuring field strength approximately:

• Count how many steel washers the magnet can lift in a stack
• Compare to known magnets for relative calibration
• Proper measurement requires a gaussmeter (Hall effect probe)

Practical electromagnet current limits:

• Wire heats up when carrying current
• Power dissipated: P = I² × R
• For continuous operation, limit wire temperature below insulation rating
• Larger wire diameter allows more current; more turns but smaller wire may overheat

Understanding magnetic fields bridges the gap between electrical theory and electromagnetic machinery — the motors, generators, and transformers that make an electrical civilization possible.