Unit 3: Magnetic Fields

Every magnet has north and south poles. Like poles repel, opposite attract. Magnetic field lines form closed loops — exit N and enter S outside the magnet, then return from S to N inside. There are no magnetic monopoles (unlike electric charges where single + or − can exist).

The magnetic field \(\vec{B}\) is measured in teslas (T = N/(A·m)). 1 T is very strong; Earth's surface field is ~50 μT. Key properties:

• \(\vec{B}\) is tangent to field lines; line density ∝ field strength
• Field lines never start or end (∇·B = 0, Gauss's law for magnetism)
• The magnetic force does no work (\(\vec{F} \perp \vec{v}\) always)

The magnetic force on a moving charged particle:

Right-hand rule (for positive charge): fingers along \(\vec{v}\), curl toward \(\vec{B}\), thumb = \(\vec{F}\) direction. For negative charges, reverse the force direction.

Circular motion in uniform B: When \(\vec{v} \perp \vec{B}\), the magnetic force is the centripetal force:

Cyclotron frequency (independent of speed and radius!):

If \(\vec{v}\) has a component parallel to \(\vec{B}\), the motion is helical: circular in the perpendicular plane plus constant drift along the field.

Hall Effect: When a current-carrying conductor sits in a perpendicular B-field, charge carriers are deflected to one side, creating a measurable transverse voltage:

where w is the conductor width, n is charge carrier density, and \(v_d\) is drift velocity. The Hall voltage's sign reveals whether charge carriers are positive or negative — historically crucial for discovering that current in metals is carried by electrons.

The magnetic force on a straight wire segment:

Same RHR: fingers along current (\(\vec{\ell}\)), curl toward \(\vec{B}\), thumb = force.

Torque on a current loop in B-field:

where \(\vec{\mu}\) is the magnetic dipole moment: \(N\) turns, current \(I\), area vector \(\vec{A}\) (RHR: curl fingers with current, thumb = direction of \(\vec{A}\)).

The torque tends to align \(\vec{\mu}\) with \(\vec{B}\). Potential energy: \(U = -\vec{\mu}\cdot\vec{B}\). This is the working principle behind electric motors.

Every current produces a magnetic field. The simplest and most important result:

Long straight wire:

Direction: RHR — thumb along current, fingers curl in direction of \(\vec{B}\) (concentric circles around the wire).

Force between two parallel wires:

Parallel currents attract; anti-parallel repel. This is how the ampere is defined.

Circular loop (center): \(B = \mu_0 I/(2R)\). Solenoid (inside): \(B = \mu_0 n I\) where \(n = N/L\) turns per meter. Field is uniform inside, ~zero outside.

The Biot-Savart law gives the \(d\vec{B}\) contribution from a current element \(Id\vec{\ell}\):

This is the magnetic analog of \(d\vec{E} = k\frac{dq}{r^2}\hat{r}\) — both inverse-square laws.

Key results via integration:

• Finite straight wire: \(B = \frac{\mu_0 I}{4\pi r}(\cos\theta_1 - \cos\theta_2)\)
• Circular loop (on axis): \(B = \frac{\mu_0 I R^2}{2(R^2+x^2)^{3/2}}\)
• Loop center (x=0): \(B = \mu_0 I/(2R)\)

olen integrate: \(\vec{B} = \int d\vec{B}\). Exploit symmetry to cancel components — the direction comes from \(d\vec{\ell} \times \hat{r}\) at each segment.

Ampere's Law relates the line integral of \(\vec{B}\) around a closed loop to the enclosed current:

Strategy (mirrors Gauss's Law): (1) Choose Amperian loop exploiting symmetry. (2) B is constant and ∥ or ⟂ on each segment. (3) Find \(I_{\text{enc}}\). (4) Solve for B.

Standard results (must memorize):

• Long straight wire: \(B = \mu_0 I/(2\pi r)\)
• Inside wire (uniform J): \(B = \mu_0 I r/(2\pi R^2)\)
• Ideal solenoid: \(B = \mu_0 n I\) inside, B ≈ 0 outside
• Toroid: \(B = \mu_0 N I/(2\pi r)\) inside the winding, B = 0 outside

Note: This is the static form. Maxwell will later add the displacement current term \(\mu_0\varepsilon_0 d\Phi_E/dt\) to handle time-varying E-fields.