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 that the north pole of a compass needle points when placed 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), and the thumb points in the direction of the force for a positive charge (reverse for negative charge).

• Velocity Selector: Selects particles of a specific velocity using perpendicular electric and magnetic fields: $v = \frac{E}{B}$

• Circular Motion: In a uniform magnetic field, a charged particle moves in a circle with radius $R = \frac{mv}{|q|B}$ and 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

Current-carrying wires in a magnetic field experience a force:

• Magnitude: $F = I l B \sin \phi$

• Direction: Right-hand rule applies (current direction = fingers, field = curl, force = thumb).

Magnetic Fields Generated by Currents

• Long, Straight Conductor: $B = \frac{\mu_0 I}{2\pi r}$ (direction by RHR)

• Force Between Parallel Conductors: $\frac{F}{l} = \frac{\mu_0 I I'}{2\pi r}$

• Field at Center of Loops: $B = \frac{\mu_0 N I}{2R}$

• Field Inside a Long Solenoid: $B = \mu_0 n I$

• 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:

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

Topics Not Covered

• DC motors

• Magnetic moments

• Magnetic materials

Chapter 21: Electromagnetic Induction

Electromagnetic Induction and Magnetic Flux

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

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

Faraday’s Law and Induced emf

• Faraday’s Law: $\varepsilon = N \frac{\Delta \Phi_B}{\Delta t}$

• Slide-Wire Generator (Motional emf): $\varepsilon = |\frac{\Delta \Phi_B}{\Delta t}| = B L v$

• Lenz’s Law: The direction of induced emf 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): $\varepsilon = L \left| \frac{\Delta i}{\Delta t} \right|$

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 Storage

• Energy in Inductor: $U = \frac{1}{2} L I^2$

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

Topics Not Covered

• Generators

• Eddy currents

• R-L and L-C circuits

Chapter 22: Alternating Current

Note: The entire chapter is excluded from the exam.

Chapter 23: Electromagnetic Waves

Nature and Properties of Electromagnetic Waves

Electromagnetic waves are oscillating electric and magnetic fields that propagate through space at the speed of light. They do not require a medium and can travel through a vacuum.

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

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

• Field Strength Relationship: $E = cB$

• Wave Equation: For any periodic wave, $v = \lambda f$. For EM waves in vacuum, $c = \lambda f$

• Energy Density: $u = \varepsilon_0 E^2$

Wave Fronts and Index of Refraction

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

Reflection and Refraction

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

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

• Total Internal Reflection: $\sin \theta_{crit} = \frac{n_b}{n_a}$

Polarization of Light

• Unpolarized Light through Polarizer: $I_1 = \frac{I_0}{2}$

• Malus’s Law: $I = I_{max} \cos^2 \phi$

Topics Not Covered

• Intensity of a sinusoidal wave

• Radiation pressure

• Polarization by reflection

• Huygens’s principle