Magnetic Fields

Overview

Magnetic fields are produced by moving charges (currents) and permanent magnets. They exert forces on other moving charges and currents.

Magnetic Field

Symbol and Unit

• Symbol: $\vec{B}$
• SI unit: Tesla (T) = Wb/m² = kg/(A·s²)
• Also: Gauss (G), 1 T = 10⁴ G

Typical Values

Source	Field Strength
Earth's surface	25-65 μT
Bar magnet (surface)	$10^{-2}$ T
MRI machine	1-3 T
Strong laboratory	10 T
Neutron star	$10^8$ T

Magnetic Field Lines

Properties

• Form closed loops (no magnetic monopoles)
• Exit from north pole, enter south pole
• Never cross
• Density indicates field strength
• Tangent gives field direction

Sources of Magnetic Fields

Long Straight Wire (Biot-Savart Law)

$$B = \frac{\mu_0 I}{2\pi r}$$

Direction: right-hand rule (thumb = current, fingers = field)

Current Loop (at center)

$$B = \frac{\mu_0 I}{2R}$$

Solenoid (inside)

$$B = \mu_0 n I = \frac{\mu_0 N I}{L}$$

Where:

• $n = N/L$ = turns per unit length
• $N$ = total number of turns
• $L$ = length of solenoid

Toroid

$$B = \frac{\mu_0 N I}{2\pi r}$$

Magnetic Constant

$$\mu_0 = 4\pi \times 10^{-7} \text{ T·m/A} = 1.257 \times 10^{-6} \text{ T·m/A}$$

Permeability of free space

Biot-Savart Law

General formula for field from a current element:

$$d\vec{B} = \frac{\mu_0}{4\pi} \frac{I \, d\vec{l} \times \hat{r}}{r^2}$$

Magnitude:

$$dB = \frac{\mu_0 I \, dl \sin\theta}{4\pi r^2}$$

Ampère's Law

$$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enclosed}}$$

Useful for calculating $B$ with high symmetry:

• Long straight wire
• Solenoid
• Toroid

Magnetic Dipole Moment

For a current loop:

$$\vec{\mu} = IA\hat{n}$$

Where $A$ = area of loop, $\hat{n}$ = normal to loop (right-hand rule)

Units: A·m²

Earth's Magnetic Field

• Approximately dipole field
• Magnetic north ≠ geographic north
• Declination: angle between magnetic and true north
• Inclination: angle with horizontal

Examples

Example 1: Field from Wire

A long wire carries 10 A. Find $B$ at 5 cm from the wire.

$$B = \frac{\mu_0 I}{2\pi r} = \frac{4\pi \times 10^{-7} \times 10}{2\pi \times 0.05}$$ $$B = 4 \times 10^{-5} \text{ T} = 40 \text{ μT}$$

Example 2: Field from Current Loop

A circular loop (radius 10 cm) carries 5 A. Find $B$ at center.

$$B = \frac{\mu_0 I}{2R} = \frac{4\pi \times 10^{-7} \times 5}{2 \times 0.1}$$ $$B = 3.14 \times 10^{-5} \text{ T} = 31.4 \text{ μT}$$

Example 3: Solenoid

A solenoid (length 20 cm, 400 turns) carries 3 A. Find $B$ inside.

$$n = \frac{N}{L} = \frac{400}{0.2} = 2000 \text{ turns/m}$$ $$B = \mu_0 n I = 4\pi \times 10^{-7} \times 2000 \times 3$$ $$B = 7.54 \times 10^{-3} \text{ T} = 7.54 \text{ mT}$$

Example 4: Two Parallel Wires

Two long parallel wires 10 cm apart carry 5 A in opposite directions. Find $B$ at midpoint.

Fields from each wire add (same direction at midpoint):

$$B = 2 \times \frac{\mu_0 I}{2\pi r} = \frac{\mu_0 I}{\pi r}$$ $$B = \frac{4\pi \times 10^{-7} \times 5}{\pi \times 0.05} = 4 \times 10^{-5} \text{ T} = 40 \text{ μT}$$

Example 5: Using Ampère's Law

A coaxial cable has inner conductor (radius $a = 1$ mm) carrying 10 A and outer conductor (radius $b = 3$ mm) carrying 10 A in opposite direction. Find $B$ at $r = 2$ mm.

At $r = 2$ mm (between conductors):

$$\oint B \cdot dl = \mu_0 I$$ $$B(2\pi r) = \mu_0 I$$ $$B = \frac{\mu_0 I}{2\pi r} = \frac{4\pi \times 10^{-7} \times 10}{2\pi \times 0.002}$$ $$B = 10^{-3} \text{ T} = 1 \text{ mT}$$

At $r > 3$ mm: $I_{\text{enclosed}} = 0$, so $B = 0$ (fields cancel)

Magnetic Materials

Diamagnetic

• Slightly repelled by magnetic field
• $\mu < \mu_0$
• Examples: copper, silver, water

Paramagnetic

• Slightly attracted to magnetic field
• $\mu > \mu_0$
• Examples: aluminum, platinum

Ferromagnetic

• Strongly attracted, can be magnetized
• $\mu \gg \mu_0$
• Examples: iron, nickel, cobalt

Relative Permeability

$$B = \mu H = \mu_0 \mu_r H$$

Where $\mu_r$ = relative permeability