Electromagnetic Spectrum and Wave Properties: Interference, Diffraction, and the Photoelectric Effect

Electromagnetic Spectrum

Overview of the Electromagnetic Spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation, which differ in wavelength and frequency. Visible light (400 to 700 nm) is only a small portion of this spectrum.

• Shorter wavelength (higher frequency) light has higher energy.

• Radio waves have the lowest energy; gamma rays have the highest energy.

• High-energy electromagnetic radiation (e.g., UV, X-ray, gamma) can damage biological molecules and is called ionizing radiation.

Order of electromagnetic radiation by increasing wavelength:

• X-ray < visible < infrared

Order by increasing frequency or energy:

• Infrared < visible < X-ray

Order by increasing wavelength (for visible light):

• Blue < Green < Red

Order by increasing frequency or energy (for visible light):

• Red < Green < Blue

Wave Behavior Properties

Interference

Interference is the interaction between waves, such as electromagnetic or ocean waves. There are two main types:

• Constructive interference: Occurs when waves are in phase and their amplitudes add, resulting in a larger wave.

• Destructive interference: Occurs when waves are out of phase and their amplitudes cancel each other, resulting in a smaller or zero amplitude.

Example: When two sound waves of the same frequency and phase meet, the sound is louder (constructive). If they are out of phase, the sound can be reduced or silenced (destructive).

Diffraction

Diffraction occurs when traveling waves encounter an obstacle or opening that is about the same size as their wavelength, causing the waves to bend around the obstacle.

• Traveling particles do not diffract; this is a property unique to waves.

• When light passes through slits separated by a distance comparable to its wavelength, an interference pattern is observed, which is characteristic of all light waves.

Example: The pattern of light and dark bands seen when light passes through two closely spaced slits (Young's double-slit experiment).

Two-Slit Interference

When light passes through two slits, the difference in path length determines whether the resulting interference is constructive or destructive.

• Constructive interference occurs when the path difference is a whole number multiple of the wavelength.

• Destructive interference occurs when the path difference is a half-integer multiple of the wavelength.

Example: The bright and dark fringes observed on a screen in the double-slit experiment are due to constructive and destructive interference, respectively.

The Photoelectric Effect

Introduction to the Photoelectric Effect

The photoelectric effect is the phenomenon where electrons are emitted from a metal surface when light shines on it. The emitted electrons are called photoelectrons.

• Classical wave theory predicted that increasing light intensity or decreasing wavelength would always eject electrons, possibly after a lag time.

• However, experiments showed that electrons are only emitted if the light has a frequency above a certain threshold frequency, regardless of intensity.

Einstein's Quantum Explanation

Einstein proposed that light energy is delivered in discrete packets called quanta or photons. The energy of a photon is proportional to its frequency:

• Energy of a photon:

$E = h\nu$

• Where $E$ is energy, $h$ is Planck's constant ($6.626 \times 10^{-34}\ \text{J}\cdot\text{s}$), and $\nu$ is frequency.

• Since $c = \lambda \nu$, energy can also be written as:

$E = \frac{hc}{\lambda}$

• Where $c$ is the speed of light ($3.00 \times 10^8\ \text{m/s}$), and $\lambda$ is wavelength.

Key Points:

• One photon at the threshold frequency gives an electron just enough energy to escape the atom (binding energy $\phi$).

• If the photon has more energy than the binding energy, the excess becomes the kinetic energy of the ejected electron:

$KE = h\nu - \phi$

Problem Solving: Photon Energy Calculations

To find the number of photons in a light pulse:

1. Calculate the energy of a single photon using its wavelength:

$E_{\text{photon}} = \frac{hc}{\lambda}$

2. Divide the total energy of the pulse by the energy per photon:

$\text{Number of photons} = \frac{E_{\text{pulse}}}{E_{\text{photon}}}$

Example: A laser pulse with a wavelength of 337 nm and total energy of 3.83 mJ contains:

• Convert 337 nm to meters: $337 \times 10^{-9}$ m

• Calculate $E_{\text{photon}}$:

$E_{\text{photon}} = \frac{6.626 \times 10^{-34}\ \text{J}\cdot\text{s} \times 3.00 \times 10^8\ \text{m/s}}{337 \times 10^{-9}\ \text{m}} = 5.90 \times 10^{-19}\ \text{J}$

• Convert 3.83 mJ to J: $3.83 \times 10^{-3}$ J

• Calculate number of photons:

$\text{Number of photons} = \frac{3.83 \times 10^{-3}\ \text{J}}{5.90 \times 10^{-19}\ \text{J}} = 6.49 \times 10^{15}$ photons

Practice Problems

• How many photons per second are emitted by a 1-watt light bulb (all light at 525 nm)?

• What wavelength (in nm) has sufficient energy per photon to dislodge an electron from sodium (binding energy 275 kJ/mol)?

Summary Table: Electromagnetic Radiation Properties

Type	Wavelength (nm)	Frequency (Hz)	Energy
Radio	>106	<109	Lowest
Microwave	106 - 103	109 - 1012	Low
Infrared	103 - 700	1012 - 1014	Moderate
Visible	700 - 400	4.3×1014 - 7.5×1014	Moderate
Ultraviolet	400 - 10	7.5×1014 - 3×1016	High
X-ray	10 - 0.01	3×1016 - 3×1019	Very High
Gamma ray	<0.01	>3×1019	Highest

Key Equations

• Relationship between wavelength, frequency, and speed of light:

$c = \lambda \nu$

• Energy of a photon:

$E = h\nu = \frac{hc}{\lambda}$

• Kinetic energy of ejected electron (photoelectric effect):

$KE = h\nu - \phi$

Summary

• The electromagnetic spectrum includes all forms of electromagnetic radiation, with visible light being only a small part.

• Wave properties such as interference and diffraction are fundamental to understanding light behavior.

• The photoelectric effect demonstrates the particle nature of light and led to the development of quantum theory.