Electromagnetic Waves – Physics with Calculus Study Notes

Electromagnetic Waves

Introduction

Electromagnetic waves are a fundamental concept in physics, describing the propagation of oscillating electric and magnetic fields through space. These waves are essential for understanding light, radio waves, X-rays, and many other phenomena. This chapter explores their production, propagation, spectrum, energy, and polarization.

The Production of Electromagnetic Waves

Oscillating Charges and Wave Generation

• Electromagnetic fields are produced by oscillating electric charges.

• When an electric charge accelerates, it emits electromagnetic radiation.

• The electric field produced by an antenna connected to an AC generator propagates away from the antenna, similar to a wave on a string.

• A magnetic field is also generated, perpendicular to both the electric field and the direction of propagation.

• The directions of the electric field (E), magnetic field (B), and wave propagation can be determined using the right-hand rule.

Example: A radio antenna emits electromagnetic waves as electrons oscillate back and forth, creating changing electric and magnetic fields that propagate outward.

The Propagation of Electromagnetic Waves

Speed of Light

• All electromagnetic waves travel through a vacuum at the same speed, known as the speed of light (c).

• In materials (e.g., air, water), light slows down but typically not below half its vacuum speed.

• The speed of light is extremely large and was first measured using astronomical observations and later in laboratory experiments (e.g., Fizeau's notched mirror method).

• The value of the speed of light is given by electromagnetic theory:

$c = 3.00 \times 10^8\ \mathrm{m/s}$

• Astronomical distances are often measured in light-years, the distance light travels in one year.

• The Doppler effect applies to electromagnetic waves: the observed frequency changes if the source or observer is moving, though the speed in vacuum remains constant.

Example: Light from the Andromeda Galaxy takes about 2 million years to reach Earth.

The Electromagnetic Spectrum

Frequency and Wavelength Relationship

• All electromagnetic waves in vacuum have the same speed, so their wavelength ($\lambda$) and frequency ($f$) are related by:

$c = f \lambda$

• The electromagnetic spectrum is the full range of electromagnetic wave frequencies.

Regions of the Electromagnetic Spectrum

• Radio waves: Lowest frequencies, used for communication (radio, TV).

• Microwaves: Used for cooking and telecommunications; wavelengths from 0.3 mm to 30 cm.

• Infrared: Felt as heat; used in remote controls.

• Visible light: Narrow frequency range detectable by the human eye.

• Ultraviolet: Higher frequencies than visible light; causes tanning and can damage skin.

• X-rays: Even higher frequencies; used in medical imaging.

• Gamma rays: Highest frequencies; produced in nuclear reactions, used to destroy cancer cells and sterilize food.

Energy and Momentum in Electromagnetic Waves

Energy Density

• The energy density in an electric field is:

$u_E = \frac{1}{2} \varepsilon_0 E^2$

• In a magnetic field:

$u_B = \frac{1}{2} \frac{B^2}{\mu_0}$

• The total energy density of an electromagnetic wave is:

$u = u_E + u_B$

• It can be shown that $u_E = u_B$, so:

$u = \varepsilon_0 E^2 = \frac{B^2}{\mu_0}$

Intensity and Radiation Pressure

• Intensity (I): The energy delivered by a wave to a unit area per unit time.

• Electromagnetic waves also carry momentum:

$p = \frac{U}{c}$

• This leads to radiation pressure:

$P = \frac{I}{c}$

• Radiation pressure can affect objects, such as causing the curvature of a comet’s dust tail.

Polarization

Nature of Polarization

• Polarization refers to the direction of the electric field in an electromagnetic wave.

• Polarized light: Electric fields are aligned in the same direction.

• Unpolarized light: Electric fields are in random directions.

Polarizers and the Law of Malus

• A polarizer transmits only the component of light in its polarization direction.

• The Law of Malus gives the intensity of transmitted light:

$I = I_0 \cos^2 \theta$

• For unpolarized light passing through a polarizer, the transmitted intensity is half the initial intensity:

$I = \frac{1}{2} I_0$

• Combining a polarizer and analyzer, the final intensity depends on the angle between their axes.

Applications and Effects of Polarization

• Liquid crystal displays (LCDs) use polarization properties of liquid crystals, which can be controlled by voltage.

• Scattering by atoms or molecules can partially or completely polarize light (e.g., sky polarization).

• Reflection from smooth surfaces can also polarize light.

Example: Sunlight scattered at right angles to the Sun is polarized; polarized sunglasses reduce glare by blocking horizontally polarized light reflected from surfaces.

Summary Table: Electromagnetic Spectrum

Region	Frequency Range	Wavelength Range	Applications/Notes
Radio Waves	~104–109 Hz	~104–0.1 m	Radio, TV communication
Microwaves	~109–1012 Hz	0.3 mm–30 cm	Cooking, radar, telecom
Infrared	~1012–4×1014 Hz	0.7 μm–1 mm	Heat, remote controls
Visible Light	4×1014–7.5×1014 Hz	400–700 nm	Human vision
Ultraviolet	7.5×1014–1017 Hz	10–400 nm	Tanning, sterilization
X-rays	1017–1020 Hz	0.01–10 nm	Medical imaging
Gamma Rays	>1020 Hz	<0.01 nm	Nuclear reactions, cancer treatment

Key Equations

• Speed of light: $c = 3.00 \times 10^8\ \mathrm{m/s}$

• Wave relationship: $c = f \lambda$

• Energy density (electric): $u_E = \frac{1}{2} \varepsilon_0 E^2$

• Energy density (magnetic): $u_B = \frac{1}{2} \frac{B^2}{\mu_0}$

• Total energy density: $u = \varepsilon_0 E^2 = \frac{B^2}{\mu_0}$

• Intensity: $I = c u$

• Momentum: $p = \frac{U}{c}$

• Radiation pressure: $P = \frac{I}{c}$

• Law of Malus: $I = I_0 \cos^2 \theta$

• Transmitted intensity (unpolarized): $I = \frac{1}{2} I_0$

Additional info: The above notes expand on the lecture outline by providing definitions, equations, and examples for each major topic in the chapter. The table summarizes the electromagnetic spectrum for quick reference.