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Physics 30: Electromagnetic Radiation

eMATH Activity

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Relativistic Wavelength of an Electron

Purpose:

To extend the concept of the relativistic wavelength of an electron.

Background:

de Broglie showed that if it was possible for light to have momentum, then it was also possible for particles to have a wavelength. His formula is simple:

where p is the momentum of the electron given by the formula you learned in Chapter 9, p=mv.

Unfortunately, this isn't the whole story. Einstein showed how the speed of an object affects its mass. In fact, as an electron's speed increases, so does its mass! This effect is unnoticeable for small speeds, but as speed increases, the effect can't be ignored.

It stands to reason that as the speed of an object approaches the speed of light (c), relativistic effects can't be ignored.  In this case, the mass of the electron gets larger as its speed increases, and its wavelength is affected. 

The purpose of this eMATH is to see how the relativistic and non-relativistic wavelengths of an electron compare as the speed of the electron increases.

Instructions:

1. Download and open the accompanying Excel spreadsheet.

This spreadsheet has a table that shows the relativistic and non-relativistic wavelengths of an electron as a function of its speed.  Take some time to examine the table and the accompanying graph.  Pay attention to the wavelength values at low speeds and at speeds where the electron is moving near the speed of light.

Analysis:

1.

The table has a column that shows the electron speed as a proportion of the speed of light (c).  This is often how speed of subatomic particles is represented, especially when the particle has a speed approaching the speed of light.  Look closely at the graph.  Can you determine from the graph where the two lines representing the non-relativistic and relativistic wavelengths separate?  At approximately what speed does this happen? (State your answer in terms of c.)
2. The table does not have a calculated value for the wavelength of an electron when it is moving at the speed of light.  Use the formula given in the Physics Insight on p. 727 of the textbook to determine the wavelength of an electron when it is moving at the speed of light.  To do this, enter the speed of light (in m/s) into the formula for v.  What result do you get for the wavelength?  Can you explain why this is so?
3. The calculation in question 2 should have yielded a fraction with zero in the denominator.  What does this tell you about the possibility of any object that has a mass moving at the speed of light? (Hint: What would be the energy of an object with infinite mass moving at the speed of light?  Use the formula for kinetic energy to explore this idea.)
4.

Determine a range of speeds that would be acceptable for a person to use, where the relativistic effects don't have to be taken into account.
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