1. Light and Radiation

Electromagnetic waves (2/4)

Let us investigate further the properties of plane electromagnetic waves. We found on page 1 of this chapter that monochromatic waves are characterised in space by the wavelength λ, and in time by the period T or frequency f with f=1/T.

These features can be described with a sinusoidal function for the electric field E, oscillating around E=0 and taking on a maximum value which is called the amplitude E_o of the wave. The spatial periodicity of E(x) along the x-axis can thus be written as

E (x) = E_{o} sin 2 π \frac{x}{λ}

Periodicity of the electric field E along the distance x.

...and the temporal periodicity along the t-axis by

E (t) = E_{o} sin 2 π \frac{t}{T}

Periodicity of the electric field E in time t.

Combining the spatial and temporal features into one equation yields finally the electric field which is a function of two variables, x and t:

E (x, t) = E_{o} sin 2 π (\frac{x}{λ} - \frac{t}{T})

Polarised light ↓

Equations ↓

The sine function is periodic in multiples of 2π of its argument. Hence, with

x = 0, λ, 2 λ, 3 λ, ...

x = n λ

t = 0, T, 2 T, 3 T, ...

t = n T

where n is an integer number, the sine functions in the left column take on the same value $sin 2 π n = 0$ .

More specifically, the displacement E(x,t) of the electric field always takes on the same value (not necessarily zero) when the wave has propagated along the x-axis by multiples of λ and along the t-axis by multiples of T. The condition for a constant value of the displacement is a constant value of the argument (or: the phase) of the sine function:

\frac{x}{λ} - \frac{t}{T} = const .

We calculate the velocity of this constant displacement by differentiating x with respect to t:

x = \frac{λ t}{T} + λ \cdot const.

\Rightarrow

\frac{d x}{d t} = \frac{λ}{T}

Derivatives ↓

In mathematics, you are familiar with functions f of a variable x, i.e., f(x). Differentiating f with respect to x is then written as the derivative f’.

In physics there are many different functions of interest, e.g. the position, the velocity, the electric and the magnetic field and so on, which you might differentiate with respect to time t, position x and other variables. Then the “ ’ ” as a symbol to denote the differentiation is no more useful since it is unclear as to which variable you want to differentiate. This ambiguity can be easily solved when writing the derivative as a quotient of two differentials which include explicitely the function and the variable. In the case discussed here the differentials are dx and dt, and the derivative of x with respect to t is then:

\frac{d x}{d t}

The symbol “d” preceeding the quantity which follows denotes an infinitely small difference of that quantity, i.e., a differential.

Read more about functions of several variables and their derivatives in Supplement 2.6.

The term dx/dt is the so-called phase velocity c of the wave, and it follows:

c = \frac{λ}{T} = f λ

This result has been obtained from purely geometrical considerations. It therefore holds with all kinds of wave phenomena, i.e. waves on water, sound waves,... With electromagnetic waves this is the speed of light, and its value in vacuum is

c=2.998·10⁸ m/s

or 300 000 km/s, approximately. The speed of light in air is slightly below the vacuum value, in water it is around 225 000 km/s, and glass it is around 200 000 km/s. Read more about the speed of light in supplement 1.3.

Question: Time lapse of sunlight and moonlight reaching the earth ↓

Question: The light year and the distance to Alpha Centauri ↓

The magnetic field B of the electromagnetic wave is given by the same type of equation as the electric field:

B (x, t) = B_{o} sin 2 π (\frac{x}{λ} - \frac{t}{T})

Both E and B oscillate perpendicularly to each other, as shown in the graph on the previous page. Moreover, both oscillate perpendicularly to the direction of propagation, here: the x-axis.

Understanding Spectra from the Earth

1. Light and Radiation

Electromagnetic waves (2/4)