Visualizing an Electromagnetic Wave (Worksheet)
- Page ID
- 67396
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An electromagnetic wave can be described mathematically by a trigonometric function, the sine function. The mathematical description is necessary to account quantitatively for the interaction of radiation with matter. This activity will help you visualize the nature of an electromagnetic wave in terms of oscillating and moving electric and magnetic fields. By completing the activity, you will further develop your ability to examine a mathematical function, identify what the function is saying, and see how changing the parameters and variables in the function affects the graph of the function.
Resources
The following files are Mathcad documents that you can use for this activity, or you can make your own either independently or by using this one as a guide.
[../../Supplements/Ch1supp/WaveVisualizationMC7.mcd Mathcad 7 file: WaveVisualizationMC7.mcd]
[../../Supplements/Ch1supp/WaveVisualizationMC6.mcd Mathcad 6 file: WaveVisualizationMC6.mcd]
[../../Supplements/Ch1supp/WaveVisualizationMC7.pdf WaveVisualization.pdf]
Vocabulary
amplitude, wavelength, frequency, cycle, propagation, wave vector, parameter, variable, periodicity
Information
Many of the activities in this book have the same structure. You are given something to do (e.g. a model to examine or tasks to complete) and then key questions to answer and exercises to complete. The key questions guide your exploration of the model or work on the tasks, and the exercises provide the opportunity to use the knowledge you have acquired in straightforward applications. The exercises strengthen your understanding and build confidence in applying the knowledge in new contexts.
The Model: A Mathematical Description of an Electromagnetic Wave
\(E_x\) = electric field in the x-direction
\(B_y\) = magnetic field in the y-direction
x, y, z = three orthogonal directions in space
t = time
λ = wavelength
ν = frequency
k = k = 2π / λ wave vector
ω = 2πν angular frequency
\(E_x(z, t) = E_0 sin(kz − ωt)\)
\(B_y(z, t) = B_0 sin(kz − ωt)\)
Key Questions
- Which parameter in the model specifies the amplitude of the magnetic field?
- Which parameter in the sine function specifies the wavelength?
- Along which axis (x, y, or z) is the wave moving?
- What is the angle between the electric field and the magnetic field
- What is the angle between the electric field and the direction of propagation?
- If the wave has a velocity c, how far will a peak of the wave move in time τ?
- If after time τ the wave looks the same as it did initially: E(z,t) = E(z,\(t+\tau\)), what is the relationship of the distance the wave moved to the wavelength?
- If τ is the period corresponding to the time for the wave to repeat its motion, what is the frequency (cycles/sec) of the wave in terms of τ?
- Why is the 2π factor needed in the definition of k for the argument of the sine function to be in units of radians and for the function to have a periodicity in space corresponding to the wavelength?
Exercises
- Derive the equation relating the wavelength, frequency, and velocity of the wave to each other from your answers to Key Questions f and g.
- Explain why the 2π factor is needed in the definition of ω.
- Use Mathcad to prepare a graph demonstrating or illustrating which parameter specifies the amplitude of the electric field.
- Use Mathcad to prepare a graph demonstrating or illustrating which parameter specifies the wavelength of the electric field.
Problems
- Explain why using a + or - sign in sin(kz + ωt) and sin(kz − ωt) produces waves moving in opposite directions.
- Use Mathcad to prepare a graph showing the direction of propagation of the electromagnetic wave specified in the model.
- Use the “animate” feature of Mathcad to illustrate a propagating wave.