# Question 1249f

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

$\textcolor{b l u e}{\underline{\textcolor{b l a c k}{E = h \cdot \nu}}}$

Here

• $E$ is the energy of the photon
• $h$ is Planck's constant, equal to $6.626 \cdot {10}^{- 34} \text{J s}$
• $\nu$ is the frequency of the photon

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

$\textcolor{b l u e}{\underline{\textcolor{b l a c k}{\nu \cdot l a m \mathrm{da} = c}}}$

Here

• $l a m \mathrm{da}$ is the wavelength of the wave
• $c$ is the speed of light in a vacuum, usually given as $3 \cdot {10}^{8} {\text{m s}}^{- 1}$

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

$\nu \cdot l a m \mathrm{da} = c \implies \nu = \frac{c}{l a m \mathrm{da}}$

Plug this into the Planck - Einstein relation to find

$E = h \cdot \frac{c}{l a m \mathrm{da}}$

$\textcolor{\mathrm{da} r k g r e e n}{\underline{\textcolor{b l a c k}{E = h \cdot c \cdot \frac{1}{l a m \mathrm{da}}}}}$

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

$E = \text{constant} \times \frac{1}{l a m \mathrm{da}}$

which is equivalent to

$E \propto \frac{1}{l a m \mathrm{da}}$

The energy of a photon, $E$, is inversely proportional to its wavelength, $l a m \mathrm{da}$, which is equivalent to saying that the energy of a photon is directly proportional to the inverse of its wavelength, $\frac{1}{l a m \mathrm{da}}$

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))))#

$h \cdot c = 1.9878 \cdot {10}^{- 25} \text{J m}$

This means that you have

$\textcolor{\mathrm{da} r k g r e e n}{\underline{\textcolor{b l a c k}{E = 1.9878 \cdot {10}^{- 25} \text{J m} \cdot \frac{1}{l a m \mathrm{da}}}}}$