Light

Recall that:

Light is electromagnetic radiation.

Light is produced by oscillating charges.

Light interacts with matter by causing charges to oscillate.

Light waves are transverse waves.

Light travels with a speed, c, in a vacuum; c = 3.00 ´ 10⁸ m/s.

Light can be characterized by a frequency, n , and a wavelength, l .

c = ln

Light travels slower than c in air, water, and other transparent media.

Light produces an oscillating electric field at any point in space that it illuminates.

Light also produces an oscillating magnetic field which is perpendicular to the electric field and to the direction of propagation.

The SI name for s^-1 is "hertz," abbreviated hz. (Named for Heinrich Hertz.)

The height of a wave on water or the magnitude of the electric field in a light wave is called the "amplitude."

The Electromagnetic Spectrum

The frequency of electromagnetic radiation ranges from essentially zero to over 10²⁴ s^-1. Conversely, the wavelength ranges from millions of miles to smaller than the nucleus of atoms.

The full range of electromagnetic radiation is technically light, but our eyes are only sensitive to a very small fraction of the total range of frequencies or wavelengths. There is an instructive diagram of the electromagnetic spectrum on page 295 of your text.

A summary of the names we give to various regions of the spectrum is

³ 10³ m Long radio waves

~ 100 m AM radio band

1 m to 10 m FM radio band

~ 1 cm Microwave band

10^-6 m to 10^-3 m Infrared

390 nm to 710 nm Visible

10 nm to 390 nm Ultraviolet

~ 0.1 nm X-rays

£ 0.01 nm g -rays

The oscillating charges that produce light in the various spectral regions turn out to be vibrating quit fast. Let's look at the frequencies of light in three regions:

AM radio band: This one is easy because radio stations identify themselves by the frequency of their radiation.

790 khz is 790000 hz or 790000 vibrations per second.

Orange light at about 600 nm. 600 nm = 6 ´ 10^-7 m.

 

Moderate energy g-ray. 0.01 pm = 1 ´ 10^-14 m.

Notice that I used the term "moderate energy g -ray" above. But we haven't said anything about the energy of light yet. We will come back to this ideal later.
 
 

Three (Little) Problems in Physics

In 1900 many scientists (including physicists) felt that physics was complete. All of the major questions had been answered. All that remained was to make more and more accurate measurements of the physical constants.

The major theories were complete:

Newtons's laws covered mechanics - e.g. planetary motion,

James Clerk Maxwell's theory of electromagnetism covered, everything electrical.

Thermodynamics covered the properties of bulk matter, and

Statistical thermodynamics made the connection from the properties of individual atoms and molecules to the properties of bulk matter.

Black body radiation.

The photo-electric effect

Line spectra.

Black Body Radiation

We see most objects by their reflected light.

However, objects also emit radiation called "thermal radiation" which depends on their temperature.

To avoid the confusion of thermal radiation with reflected light people work with a "black body." An ideal black body absorbs all the light that falls on it so that any radiation that is emitted is entirely due to the characteristics of thermal radiation.

The experimental "spectrum" of ideal black body radiation fit onto a nice smooth curve.

(A spectrum is a plot of intensity of radiation against frequency or wavelength.)

The black body spectrum has a peak whose position depends on the temperature of the body.

Our sun is an approximate black body whose spectrum peaks at the visible portion of the spectrum. The thermal radiation corresponds to a temperature of approximately 4,000 K. (People radiate in the infrared at about 10 m m.)

There was a "good" theory based on the principles everyone believed which worked well at long wavelengths (low frequency) but failed miserably at short wavelengths. (The ultraviolet catastrophe.)

Max Planck found the formula for a curve which fit the data - too well. In order to derive his curve he had to assume that light energy was "quantized." That is, that the energy of a light wave depended on the frequency of the light (and not on its intensity). Further the energy of the beam was some multiple of a fundamental energy given by

.

It looked as if the light beam consisted of a number of "particles" called "photons," and that the intensity of the beam gave the number of photons in the beam, but the energy of the photons is given by Planck's equation.
 
 

The Photoelectric Effect

It was known in 1900 that when you shine light (of appropriate frequency) on the bare surface of a metal electrons are ejected.

You can measure the velocity of the ejected electrons and hence their kinetic energy. It was expected that the higher intensity of the beam the higher the kinetic energy of the electrons would be.

But, in fact, the kinetic energy of the electrons depends on the frequency of the light rather than the intensity. With a more intense beam you get more electrons, but their energy still depends on the frequency.

Albert Einstein used Planck's idea that the energy of a photon depended on the frequency of the light to derive an equation which explained the photoelectric effect.

The basic idea is that each photon kicks one electron out of the metal. Part of the hnenergy of the photon goes to allow the electron to escape from the metal and the rest is converted into kinetic energy of the electron.

(Einstein also used Planck's idea to explain the low temperature heat capacity of metals, which had been another of the "little" problems.)

Line Spectra

At the same time people were looking at black body radiation the were also looking at the light emitted from hot gases, like H, and He, etc. Instead of finding a smooth distribution of emitted frequencies they found that gases, H for example, emitted only at a small number of frequencies.

The emission from hot gasses were diffracted through a prism to give a series of lines of different colors instead of a smooth distribution of frequencies. The came to be known as line spectra.

Although there was no explanation for why this should be, people were able to find a formula which fit the wavelengths of light atomic hydrogen emission. The wavelengths fit into groups called "series."

One such group called the Balmer series fit an equation, called the Rydberg equation, of the form

,

The constant, R, is called the "Rydberg constant." It has the value, 1.0974 ´ 10⁷ m^-1 (or 91.125 nm^-1)

People suspected that some of the other series might be obtained by changing the 2² to, maybe, 1², or 3², and so on. That is, maybe there were series whose wavelengths fit the equation

,

,

When people looked for spectral lines with wavelengths given by these formulas they found them!

The series from the first formula is called the Lyman series after the physicist who found it, and the series from the second formula is called the Paschen series after its discoverer. There were even series found with the 2² replaced with 4² and 5².
 
 

The Bohr Model of the Hydrogen Atom

By the time all this information was known it had been learned that atoms consisted of a central heavy and relatively small core - called the nucleus - surrounded by the relatively light electrons.

The question is, how are the electrons arranged?

Niels Bohr assumed that the electrons move around the nucleus like th planets move around the sun.

But the physics of Newton and Maxwell predicted that such an atom would radiate away its energy and collapse.

Bohr assumed that for unknown reasons the atom wouldn't radiate away its energy and collapse, but that there were only certain allowed orbits.

He assumed that Planck's formula for the energy of a photon was correct and then converted the Rydberg equation for wavelength into an equation in energy.

Remember that n = c/l and that Planck's equations gives the energy of a photon as E = hn, so E = hc/l .

Multiply the Rydberg formula by hc and we get an equation that has units of energy.

Based on his modified Rydberg formula, he deduced that the energy of the allowed orbits had to fit a formula of the form

,