30.3 - Proportionality of electric and magnetic fields
Although the electric and magnetic field vectors of an electromagnetic wave point in
perpendicular directions, their magnitudes are strictly proportional to each other at all
positions and at all times. We graphically display the magnitudes at a particular instant
on the same coordinate system in Equation 1. Their proportionality is expressed in the
equation to the right, using a constant c that depends on two other fundamental
physical constants.
This proportionality turned out to have important implications in the study of
electromagnetic radiation. Why? Because when calculated, the value of c was very
close to the measured speed of light. This crucial discovery accelerated the
understanding of the relationship between electromagnetic radiation such as light or
radio waves, and electric and magnetic fields.Proportionality of electric and
magnetic field strengths
In an electromagnetic wave,
E = electric field strength
B = magnetic field strength
30.4 - Calculating the speed of light from fundamental constants
One of the major discoveries of 19th century physics
was that light is a form of electromagnetic radiation.
By applying and extending their knowledge of electric
and magnetic fields, physicists were able both to
create electromagnetic radiation (initially radio waves)
and to predict the speed of the waves.
James Maxwell published his four laws governing
electromagnetic phenomena in 1864, and at the same
time he predicted the existence of self-propagating
electromagnetic waves. His work enabled him to
calculate what the speed of these waves would have
to be.
The angular frequency of any wave is Ȧ = 2ʌf,
where f is its frequency in cycles per second. The angular wave number is k = 2ʌ/Ȝ,
where Ȝ is the wavelength. The speed of any wave is v = Ȝf. This means that the speed
in terms of the angular frequency and wave number is v = (2ʌ/k)(Ȧ/2ʌ), or Ȧ/k.
Maxwell had already shown that the wave speed Ȧ/k is a constant for electromagnetic
waves, a constant he had designated as c and expressed in terms of μ 0 and İ 0. We
state the relationship of c to these several variables and constants in Equation 1.
The fundamental constants μ 0 and İ 0 are used elsewhere in physics. For example, the
permeability constant is used in equations that describe the magnetic field created by
various electric current configurations. The permittivity constant is used to express one
form of Coulomb’s law. In other words, these constants determine the strengths of the
electric and magnetic forces in the physical universe.
The example problem on the right asks you to repeat Maxwell’s calculation of the value
ofc. The result is 2.998×10^8 m/s. In the year 1864, the speed of visible light in a
vacuum had been known with fair accuracy for well over a century. The English
physicist James Bradley estimated it in 1728 to be 3.1×10^8 m/s, based on his study of
“stellar aberration,” or the apparent change in the positions of stars as the Earth moves
around the Sun. Because the calculated and observed speeds were so close, Maxwell’s
results provided the first evidence that light is a kind of electromagnetic wave.Understanding that light is an electromagnetic wave, and knowing the general relationship between frequency and wavelength, sparked the
discovery of additional types of electromagnetic radiation. In 1888 Heinrich Hertz created what we would now call radio receivers and
transmitters, one of which you see in the illustration above. He proved the existence, and wavelike nature, of radiation having frequencies
around 100 MHz.Heinrich Hertz transmitter, 1888. Transformer voltage causes a spark to
jump between the postionable brass spheres, generating a radio pulse.Speed of light
Theoretical speed of radiation
= the measured speed of light
Conclusion: Light is electromagnetic
radiation!(^560) Copyright 2000-2007 Kinetic Books Co. Chapter 30