Question

Write equation of energy of photon in terms of Planck's constant (h) and wavelength (\(\lambda\)).

Hint:

This equation shows an inverse relationship between energy and wavelength: shorter wavelengths (like blue or UV light) correspond to higher-energy photons, while longer wavelengths (like red or infrared light) correspond to lower-energy photons. This is a cornerstone of modern physics.

Answer 1

Step 1: Understanding the Concept:
In quantum mechanics, light is described as being composed of discrete packets of energy called photons. The energy of a single photon is related to the frequency and wavelength of the corresponding electromagnetic wave.

Step 2: Key Formula or Approach:
The derivation involves two fundamental equations: \begin{enumerate} \item The Planck-Einstein relation, which gives the energy of a photon (\(E\)) in terms of its frequency (\(f\)): \[ E = hf \] where \(h\) is Planck's constant.
\item The wave equation, which relates the speed of light (\(c\)), its frequency (\(f\)), and its wavelength (\(\lambda\)): \[ c = f \lambda \] \end{enumerate}

Step 3: Detailed Explanation:
Our goal is to express the energy \(E\) in terms of \(h\) and \(\lambda\). The first equation has \(h\) but uses \(f\), not \(\lambda\). We need to replace \(f\) using the second equation.
From the wave equation, we can express frequency as: \[ f = \frac{c}{\lambda} \] Now, substitute this expression for \(f\) into the Planck-Einstein relation: \[ E = h \left( \frac{c}{\lambda} \right) \] \[ E = \frac{hc}{\lambda} \] This is the required equation.

Step 4: Final Answer:
The equation for the energy of a photon (\(E\)) in terms of Planck's constant (\(h\)), the speed of light (\(c\)), and wavelength (\(\lambda\)) is: \[ E = \frac{hc}{\lambda} \]