Question

What is the energy, in Joules, of a wave with a frequency of $6.78 \times {10^7}Hz$ ? What type of wave is this?

Answer 1

To approach the question, we have to revise our concept of quantization of energy. As we know, light is an electromagnetic wave and has dual nature i.e. it exhibits both wave and particle nature. Here, we have to consider the particle nature of the light and find the relation between energy and frequency.

Formula used:
We know the relation between and energy is given by:
$E = h\nu$
Where, $E$ is the energy, $h$ is Planck's Constant and $\nu$ is the frequency.
$c = \lambda \nu$
Where, $c$ is the speed of light with the value of $3 \times {10^8}m/s$, $\lambda$ is the wavelength of the wave and $\nu$ is the frequency of the wave.

Complete step by step answer:
Max Plank formulated the concept of quantization of energy. The quantization of energy means that energy is emitted or absorbed only in integral multiples of a small unit of energy called quantum. Plank’s equation states that the energy and the frequency of a quantum of electromagnetic radiation are proportional with the proportionality constant $h$.Mathematically it is represented by:
$E = h\nu \,\,\,\,\,\,\,\,\,......(1)$
The value of Plank’s constant comes out to be $6.626 \times {10^{ - 34}}\,J - s$ .

Now, we have the value of frequency from the question which is given by the following:
$\nu = 6.78 \times {10^7}\,Hz\,\,\,\,\,\,\,......(2)$
We have to put the value of equation (2) and Plank’s constant in equation (1), we get,
$E = 6.626 \times {10^{ - 34}} \times 6.78 \times {10^7}$
$\therefore E = 4.49 \times {10^{ - 26}}\,J$

We now have to find the type of wave. For this, we have to calculate the wavelength of the wave with the given frequency. We know the relation between frequency and wavelength to be:
$c = \lambda \nu \,\,\,......(3)$
From equation (3) we get,
$\lambda = \dfrac{c}{\nu }\,\,\,......(4)$
We put the values in equation (4) and get,
$\lambda = \dfrac{{3 \times {{10}^8}}}{{6.78 \times {{10}^7}}}$
$\therefore \lambda = 4.425\,m$
The value of wavelength comes out to be $4.425\,m$ which falls in the radio waves range which is from $1\,mm$ to $100\,km$ . But as our frequency is in megahertz, the wavelength falls particularly into the range of FM radio waves with its wavelength range from $1\,m$ to $10\,m$.

So, the value of energy is $4.49 \times {10^{ - 26}}\,J$ and it falls into the range of FM radio waves.

Note: A photon is referred to as a packet of electromagnetic energy. This implies that energy is discrete and not continuous. Photons having higher frequency will have higher energy. Also, the photon is a massless particle. The energy of photons is usually expressed in electron volt (eV). The Photoelectric Effect and Compton Effect are the two phenomena that prove the particle nature of the light.