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

Hint: 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.