Question

How much energy is in a wave that has a length of 150 nm?

Answer 1

Hint: The formula that defines Einstein’s photon theory or Planck relation or Plank-Einstein relation should be used to solve this problem. The formula relates the energy of a photon and the wavelength through Plank’s constant and the speed of light.
Formula used:
\[\begin{align}
  & E=hf \\
 & E=h\dfrac{c}{\lambda } \\
\end{align}\]

Complete step by step answer:
Consider the formula that defines Einstein's photon theory or Planck relation or Plank-Einstein relation.
Planck's equation is given as follows.
\[E=hf\]
Here E represents the energy of a photon, h is Planck's constant and f is the frequency of the photon.
This frequency can be represented in the form of the speed of light and the wavelength of the photon.
\[E=h\dfrac{c}{\lambda }\]
 Here E represents the energy of a photon, h is Planck's constant, c is the speed of light and \[\lambda \]is the wavelength of the photon.
From the given information we have the data as follows.
The wavelength of a wave is 150 nm.
\[\lambda =150\times {{10}^{-9}}\,m\]
Consider the formula to compute the value of the energy.
\[E=h\dfrac{c}{\lambda }\]
Substitute the values of Planck's constant, the speed of light and the wavelength of the wave in the above equation.
\[E=6.6223\times {{10}^{-34}}\times \dfrac{3\times {{10}^{8}}}{150\times {{10}^{-9}}}\]
Continue further computation.
\[\begin{align}
  & E=\dfrac{6.6223\times 3}{150}\times {{10}^{-17}} \\
 & E=0.1325\times {{10}^{-18}} \\
\end{align}\]
Thus, the value of the energy of a wave in terms of Joule is,
\[E=1.325\times {{10}^{-18}}J\]

\[\therefore \] The energy in a wave that had a length of 150 nm is \[1.325\times {{10}^{-18}}J\]

Note: As the wavelength of the wave is given, so, we have converted the formula from frequency into the wavelength. If in case, they give the value of frequency, then, we can use the direct formula. In the question they haven’t asked for any particular unit of energy, so, we will represent the answer in terms of Joule.