PHY2054 Study Guide: Magnetic Fields, Electromagnetic Induction, and Electromagnetic Waves

Chapter 20: Magnetic Field and Magnetic Forces

Basic Properties of Magnetic Fields

Magnetic fields are vector fields that exert forces on moving charges and current-carrying conductors. The direction of the magnetic field at any point is the direction a north pole of a compass needle points at that location.

• Magnetic field symbol: B, measured in teslas (T).

• Magnetic field lines: Point from north to south outside a magnet and indicate the field's direction and strength (denser lines = stronger field).

Force on a Charged Particle in a Magnetic Field

A charged particle moving in a magnetic field experiences a force given by:

• Magnitude: $F = |q|vB \sin \phi$

• Direction: Determined by the right-hand rule (RHR): Point fingers in the direction of velocity (v), curl toward the magnetic field (B), thumb points in the direction of force for a positive charge (opposite for negative charge).

Velocity Selector

A velocity selector uses perpendicular electric and magnetic fields to select particles of a specific velocity:

• Formula: $v = \frac{E}{B}$

Circular Motion of Charged Particles in a Magnetic Field

Charged particles move in circular paths in a uniform magnetic field due to the magnetic force acting as a centripetal force.

• Radius: $R = \frac{mv}{|q|B}$

• Angular frequency: $\omega = \frac{|q|B}{m}$

• Application: Mass spectrometers use this principle to separate ions by mass-to-charge ratio.

Magnetic Force on Current-Carrying Conductors

A current-carrying wire in a magnetic field experiences a force:

• Formula: $F = I l B \sin \phi$

• Direction: Right-hand rule (RHR) applies: fingers in direction of current, curl toward B, thumb gives force direction.

Magnetic Fields Generated by Currents

• Long, straight conductor: $B = \frac{\mu_0 I}{2\pi r}$ (direction by RHR: thumb in current direction, fingers curl in B direction)

• Force between parallel conductors: $\frac{F}{l} = \frac{\mu_0 I I'}{2\pi r}$

• Magnetic field at center of loops: $B = \frac{\mu_0 N I}{2R}$

• Magnetic field inside a long solenoid: $B = \mu_0 n I$

• Magnetic field inside a toroidal solenoid: $B = \frac{\mu_0 N I}{2\pi r}$

Ampère’s Law

Ampère’s law relates the integrated magnetic field around a closed loop to the electric current passing through the loop:

• Formula: $\sum B_{||} \Delta s = \mu_0 I_{enclosed}$

Chapter 21: Electromagnetic Induction

Electromagnetic Induction

Electromagnetic induction is the process by which a changing magnetic flux induces an electromotive force (emf) in a conductor.

• Cause: Change in magnetic flux through a loop.

Magnetic Flux

• Formula: $\Phi_B = B_{\perp} A = B A \cos \theta$

Faraday’s Law of Induction

• Induced emf: $\varepsilon = N \frac{\Delta \Phi_B}{\Delta t}$

Motional emf and Slide-Wire Generator

• Formula: $\varepsilon = |\frac{\Delta \Phi_B}{\Delta t}| = B L v$

Lenz’s Law

The direction of the induced emf (and current) is such that it opposes the change in magnetic flux that produced it.

Mutual and Self-Inductance

• Mutual inductance (M): Describes emf induced in one coil due to changing current in another.

• Self-inductance (L): $\varepsilon = L |\frac{\Delta i}{\Delta t}|$

Transformers

• Voltage ratio: $\frac{V_2}{V_1} = \frac{N_2}{N_1}$

• Power conservation: $V_1 I_1 = V_2 I_2$

Magnetic Energy Stored in an Inductor

• Total energy: $U = \frac{1}{2} L I^2$

• Energy density: $u = \frac{B^2}{2\mu_0}$

Chapter 23: Electromagnetic Waves

Nature of Electromagnetic Waves

Electromagnetic waves are oscillating electric and magnetic fields that propagate through space at the speed of light.

• Speed of light: $c = \frac{1}{\sqrt{\varepsilon_0 \mu_0}} = 3.00 \times 10^8 \text{ m/s}$

• Electromagnetic spectrum: Range of all possible frequencies of electromagnetic radiation.

Relationship Between Electric and Magnetic Fields

• Formula: $E = cB$

Wave Properties

• Wave speed: $v = \lambda f$

• For electromagnetic waves in vacuum: $c = \lambda f$

Energy Density in Fields

• Electric field energy density: $u = \varepsilon_0 E^2$

Wave Fronts and Index of Refraction

• Index of refraction: $n = \frac{c}{v}$

Law of Reflection and Snell’s Law

• Law of reflection: $\theta_r = \theta_a$

• Snell’s law: $n_a \sin \theta_a = n_b \sin \theta_b$

Refraction and Total Internal Reflection

• Critical angle for total internal reflection: $\sin \theta_{crit} = \frac{n_b}{n_a}$

Polarization of Light

• Unpolarized light through a polarizer: $I_1 = \frac{I_0}{2}$

• Malus’s law (intensity after polarizer at angle $\phi$): $I = I_{max} \cos^2 \phi$

Additional info: Topics such as DC motors, magnetic moments, magnetic materials, generators, eddy currents, R-L and L-C circuits, and certain aspects of electromagnetic waves (e.g., intensity of a sinusoidal wave, radiation pressure, polarization by reflection, Huygens’s principle) are explicitly excluded from the exam and are not covered here.