Question

How does energy relate to wavelength and frequency?

Answer 1

Hint: The relationship between energy and wavelength or frequency can be well understood if you think in terms of a photon. We know that a photon is a discrete unit of quantized energy. Think about where the photon gets its energy from by employing Planck’s quantum theory. Given that photons are emitted in the range of the electromagnetic spectrum, i.e., radiation with increasing frequency and decreasing wavelength or vice versa,
determine the correlation between the two and energy.

Formula Used:
$c=\nu\lambda$
$E=h\nu$
$E=\dfrac{hc}{\lambda}$

Complete Solution:
Let us begin by looking at the question in the context of photons.

Photons are elementary particles that are essentially quantized electromagnetic radiation. This quantization implies that they hold packets of energy. The energy that they carry depends upon the kind of electromagnetic radiation that they quantized. This quantization of energy is consistent with Planck’s quantum theory which states that

Energy is absorbed or radiated by atoms in discrete packets of energy called quanta, which are now known to be photons.

Each quantum, or photon consists of a specific amount of energy that it can absorb or emit, which is directly proportional to the frequency of radiation.

We know that the electromagnetic spectrum consists of electromagnetic radiation of varying frequencies and wavelengths. Therefore, according to Planck’s quantum theory we have:
$E \propto \nu \Rightarrow E = h\nu$, where E is the photon energy,
h ($6.626 \times 10^{-34}\;J.s$) is the proportionality (Planck’s) constant and $\nu$ is the frequency of radiation.

Now, all electromagnetic radiation travels in the form of photons at the speed of light $c \approx 3 \times 10^8\;ms^{-1}$.
If $\lambda$ is the wavelength of any radiation, then,
$c=\nu \times \lambda \Rightarrow \nu = \dfrac{c}{\lambda}$

Plugging this back into our energy expression in place of $\nu$, we get:
$E = h\dfrac{c}{\lambda}$

In essence, energy is directly proportional to the frequency of radiation but is inversely proportional to the wavelength. This means that energy increases with an increase in frequency or decrease in wavelength, and energy decreases with a decrease in frequency or an increase in wavelength.

Note:
Remember that Planck’s quantum theory, contrary to Maxwell's electromagnetic wave theory, suggests that radiant energy is not absorbed or emitted continuously but discontinuously in the form of small packets of energy called photons. Planck’s quantum theory was able to explain phenomena like the blackbody spectrum and photoelectric effect where Maxwell’s electromagnetic wave theory failed to do so since Maxwell’s theory entailed a continuous energy emission/absorption distribution.