Question

The number of photons of light having wavelength $100nm$ which can provide $1J$ energy is nearly:
(A) ${10^7}$ photons
(B) $5 \times {10^{18}}$ photons
(C) $5 \times {10^{17}}$ photons
(D) $5 \times {10^7}$ photons

Answer 1

Energy of photons is equal to the product of the number of photons with planck’s constant and the frequency of the light. Planck’s constant is a constant having value equal to $6.26 \times {10^{ - 34}}$ .

Complete step by step solution:
First of all let us talk about wavelength of light, speed of light and planck’s constant.
Speed of light: The speed by which light travels in the air, is known as speed of light, The value of speed of light in air is constant and has the value $3 \times {10^8}m/\sec$ .
Wavelength of light: It is defined as the distance between the identical points in the adjacent cycle of a waveform along a wire. The unit of wavelength is centimetre, metre and millimetres. It is represented by $\lambda$ .
Frequency: It is defined as the number of occurrences of a repeating event per unit of time. It is measured in the unit of hertz. It is represented by $\nu$ .
Planck’s constant: It is a constant having value equal to $6.26 \times {10^{ - 34}}$ . It is represented by $h$ .
Now energy of photons is equal to the product of the number of photons with Planck's constant and the frequency of the light. $E = nh\nu$ , where $E$ is the energy of photons, $n$ is the number of photons, $h$ is the Planck's constant and $\nu$ is the frequency of light.
Frequency is equal to the ratio of speed of light to the wavelength of light. $\nu = \dfrac{c}{\lambda }$ .
$E = nh\dfrac{c}{\lambda }$ and we have to find the number of photons and we are given the energy of photons, Planck's constant, speed of light and wavelength of light.
$E = 1J,\lambda = 100nm,c = 3 \times {10^8}$ and $h = 6.26 \times {10^{34}}$ . Putting these values in the formula we will get the value of the number of photons as $n = \dfrac{{E\lambda }}{{hc}} = \dfrac{{1 \times 100 \times {{10}^{ - 9}}}}{{6.62 \times {{10}^{34}} \times 3 \times {{10}^8}}} = 5 \times {10^{17}}$
Hence, the number of photons of light having wavelength $100nm$ which can provide $1J$ energy is nearly $5 \times {10^{17}}$ photons.
So option C is correct.

Note:
The other units of measuring wavelength are nanometre which is equal to $1nm = {10^{ - 9}}m$ and millimetre which is equal to $1mm = {10^{ - 3}}m$ .
In general energy is defined as the rate of doing work. It is of many types: heat, chemical, physical, etc.