PHY2054 Review: 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 magnetic materials. They are characterized by their direction and magnitude, and are denoted by the symbol B.

• Magnetic field lines indicate the direction of the field; they emerge from the north pole and enter the south pole of a magnet.

• SI unit: Tesla (T).

Force on a Charged Particle in a Magnetic Field

A charged particle moving in a magnetic field experiences a force perpendicular to both its velocity and the magnetic field.

• Formula: $F = |q|vB\sin\phi$

• Direction: Determined by the right hand rule (RHR).

• Velocity selector: $v = \frac{E}{B}$

• Motion: Circular motion 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.

Magnetic Force on Current-Carrying Conductors

Current-carrying conductors in a magnetic field experience a force.

• Formula: $F = I l B_{\perp} = I l B \sin \phi$

• Direction: Right hand rule (RHR).

Magnetic Fields Generated by Conductors

Current in conductors generates magnetic fields.

• Long, straight conductor: $B = \frac{\mu_0 I}{2\pi r}$

• Direction: Right hand rule (RHR).

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

Magnetic Field of Loops and Solenoids

• Loops: $B = \frac{\mu_0 N I}{2R}$ (center of the loops)

• Solenoids: $B = \mu_0 n I$ (long solenoid)

• Toroidal solenoid: $B = \frac{\mu_0 N I}{2 r}$

• Direction: Right hand rule (RHR).

Magnetic Field Calculation Using Ampere’s Law

Ampere’s law relates the magnetic field around a closed loop to the current passing through the loop.

• Formula: $\sum B \Delta s = \mu_0 I_{\text{enclosed}}$

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: Changing magnetic flux through a loop.

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

Faraday's Law and Induced EMF

Faraday’s law quantifies the induced emf in a circuit due to changing magnetic flux.

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

• Slide-wire generator: $\varepsilon = |\frac{\Delta \Phi_B}{\Delta t}| = B L v$ (motional emf)

Direction of Induced EMF: Lenz's Law

Lenz’s law states that the direction of the induced emf opposes the change in magnetic flux that caused it.

• Principle: Conservation of energy.

Mutual and Self-Inductance

• Mutual inductance: $M = \frac{N_2 \Phi_{B1}}{i_1} = \frac{N_1 \Phi_{B2}}{i_2}$

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

Transformers

Transformers use electromagnetic induction to change voltage and current levels.

• 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

• 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 oscillations of electric and magnetic fields that propagate through space at the speed of light.

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

• Electromagnetic spectrum: Includes radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays.

Relationship Between Electric and Magnetic Fields

• Formula: $E = c B$

Wave Properties: Wavelength, Frequency, and Speed

• General wave: $v = \lambda f$

• Electromagnetic wave in vacuum: $c = \lambda f$

Energy Density in Electric and Magnetic Fields

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

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: $\sin \theta_{\text{crit}} = \frac{n_b}{n_a}$

Polarization and Malus’s Law

• Intensity through polarizer: $I_1 = \frac{I_0}{2}$

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