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Light and electromagnetic spectrum basics (Part 2)

17 Oct 2013  | Louis Desmarais

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For simplicity, let's consider one of the waves describing either the electric or magnetic field displayed in figure 1 in part 1. This wave can be considered as a component of the more complex electromagnetic wave. Figure 2a of this instalment shows this wave separately using x and z coordinates.

We will now use this to illustrate some of the mathematical parameters of a wave. The x and z displacement of this wave can be described by the trigonometric sine function. A general equation describing this wave can be written as shown below:

Figure 2a: A sine wave of amplitude A and wavelength λ.

In this equation, the amplitude or height, A, defines the maximum vertical or x displacement. The wavelength, 'k, is determined by the distance between adjacent crests of the wave.

Now we come to the part of the equation that may be particularly confusing, the quantity kz. Instead of using degree measure, we will use radian measure to define the argument of a trigonometric function. When considering degree measure, the range of 0° to 360° completes one cycle or complete circle. In the case of radian measure, the range of 0 to 2n radians completes the same cycle. As you may remember, a radian is a unit of angular measure equivalent to the angle subtended at the centre of an arc equal in length to the radius. In degrees, this angle equals about 57.3°. The complete circle or 360° equates to 2n radians. Thus, radian measure is a ratio of two lengths. For the sine wave in figure 2a, we know that a maximum occurs at 90°. This corresponds to a radian measure of Jt/2 radians.

The first complete cycle to the right makes 2n radians or 360°. At this point on the curve, kz = 2n. This brings us to our next parameter to consider, k. The parameter k or angular wave number defines how fast the wave oscillates with respect to z. Thus k has the units of radians per meter in the MRS system. As the wave number increases, the wavelength decreases. The simple equation below describes this mathematical relationship:

In our study of electromagnetic waves, we will also be interested in the time variation of the wave. Going back to our sine function involving displacement, we can easily change this expression to yield the time variation of the wave with amplitude A. The amplitude A now varies with time according to the following expression:

Since the function varies with respect to time, the argument of the sine function must also be changed accordingly. We now use the term cot in place of kz for the argument. The constant co, angular frequency, gives an indication of how fast the wave amplitude oscillates as t (time) increases. Since cot is measured in radians, co has the units of radians per second. One complete oscillation or period, T, equates to 2;i/co:

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