Question

A 100W sodium lamp radiates energy uniformly in all directions. The lamp is located at the centre of a large sphere that absorbs all the sodium light which is incident on it. The wavelength of the sodium light is $589nm$
A.What is the energy per photon associated with sodium light?
B.At what rate are the photons delivered of the sphere?

Answer 1

The energy of light is inversely proportional to its wavelength. Wavelength is related to energy with speed of light and the Planck’s constant. Total Power of light can be defined as a summation of energy carried by each photon.
Formula Used:
1. $E=\dfrac{hc}{\lambda }$
2. $n=\dfrac{P}{E}$
Where $E=$energy, $h=$ planck’s constant, $c$ is the speed of light, $\lambda =$ wavelength, $P=$ power

Complete step by step solution:
To solve the above question, let us first understand a few concepts.
Einstein said that light is made of particles, which are like packets of energy. These particles are called photons. When these photos are travelling, they travel like a wave.
The properties of a particle and a wave are very different and this is the difference between Classical Physics and Quantum Mechanics.
The equation we will be using for this numerical is:
$E=\dfrac{hc}{\lambda }$
Where, Where $E=$energy, $h=$ planck’s constant, $c$ is the speed of light, $\lambda =$ wavelength
Now, comes to the solution part,a. we have $\lambda =589\times {{10}^{-9}}m$
Then energy per photon is given by $E=\dfrac{hc}{\lambda }$
We know that Planck’s constant and speed of light have constant values which are
$h=6.6\times {{10}^{-34}},c=3\times {{10}^{8}}$
Substituting the above values, we get
$E=\dfrac{6.6\times {{10}^{-34}}\times 3\times {{10}^{8}}}{589\times {{10}^{-9}}}$
Solving the above equation, we get:
$E=3.37\times {{10}^{-9}}J$
When we convert this value into electron volt, we get the value as: $2.11eV$
Hence, the amount of Energy per photons associated with sodium light was found to be $2.11V$
B.
given $P=100W$
let the number of photons delivered to this sphere be equal to $n$
so, $n=\dfrac{P}{E}$
where; $P=$ power, $E=$energy.
Substituting the values we get:
$n = \dfrac{{100}}{{3.37 \times {{10}^{ - 19}}}}$
$n = 2.96 \times {10^{20}}photon{\sec^-1 }$
Hence, the rate at which photons deliver was found to be $2.96 \times {10^{20}}photon{\sec^-1 }$ .

Note:
-Photons are one of the types of elementary particles, it is the quantum of the electromagnetic field including electromagnetic radiation such as light and radio waves and the force carrier for the electromagnetic force. They are massless and move with the speed of light in vacuum .
-Photons play a key role in quantum mechanics and also feature dual nature one is wave and other one is particle form. There are many scientists working on it. The main characters are Albert Einstein, Max Planck , Bose etc .