What is the wavelength of an electron moving at 3.00 percent of the speed of light?
If we call the speed of light 𝑐, then we’re told that the electron in this example is moving at 3.00 percent of that speed, a speed we’ll call 𝑣. We want to know the electron’s wavelength, which we’ll call 𝜆. We can use the de Broglie relationship to figure out the wavelength of this electron given its speed.
That relationship says that an object’s wavelength is equal to Planck’s constant over the object’s mass times its velocity. We’ve been told the object’s speed but what about ℎ, Planck’s constant, and its mass, 𝑚? The mass of an electron we’ll treat as 9.1 times 10 to the negative 31st kilograms. Planck’s constant we’ll treat as exactly 6.626 times 10 to the negative 34th joule seconds. And 𝑐, the speed of light in terms of which our velocity is given, we’ll treat as exactly 3.00 times 10 to the eighth meters per second.
When we plug these values into our equation and enter them on our calculator, we find that 𝜆 has a value of 80.9 times 10 to the negative 12 meters or 80.9 picometres. That’s the wavelength of this electron.