Question #1249f

1 Answer
Feb 16, 2017

Here;s how you can do that.

Explanation:

The idea here is that you must prove that the energy of a photon and its wavelength have an inverse relationship, i.e. when the wavelength increases, the energy decreases and vice versa.

To do that, you can start from the Planck - Einstein relation, which shows that the energy of a photon is directly proportional to its frequency

color(blue)(ul(color(black)(E = h * nu)))

Here

  • E is the energy of the photon
  • h is Planck's constant, equal to 6.626 * 10^(-34)"J s"
  • nu is the frequency of the photon

Now, you should also know that frequency and wavelength have an inverse relationship as given by the equation

color(blue)(ul(color(black)(nu * lamda = c)))

Here

  • lamda is the wavelength of the wave
  • c is the speed of light in a vacuum, usually given as 3 * 10^8"m s"^(-1)

Rearrange this equation to find an expression for the frequency of the wave in terms of its wavelength

nu * lamda = c implies nu = c/(lamda)

Plug this into the Planck - Einstein relation to find

E = h * c/(lamda)

color(darkgreen)(ul(color(black)(E = h * c * 1/(lamda))))

Since the product between h and c is always constant, you can say that

E = "constant" xx 1/(lamda)

which is equivalent to

E prop 1/(lamda)

The energy of a photon, E, is inversely proportional to its wavelength, lamda, which is equivalent to saying that the energy of a photon is directly proportional to the inverse of its wavelength, 1/(lamda)

You can even find a numeral value for the product between h and c

h * c = 6.626 * 10^(-34)"J" color(red)(cancel(color(black)("s"))) * 3 * 10^(8)"m" color(red)(cancel(color(black)("s"^(-1))))

h * c = 1.9878 * 10^(-25)"J m"

This means that you have

color(darkgreen)(ul(color(black)(E = 1.9878 * 10^(-25)"J m" * 1/(lamda))))