Question

Calculate the number of photons falling per second on each square meter of the earth’s surface directly below the sun.

Answer 1

Hint:In order to solve this question you have to assume that the light is monochromatic in nature and have an average wavelength of $500nm$. Also, assume that no light is being absorbed in between the sun and the earth’s surface.

Formula used:
The number of photons falling per second on each square meter of earth’s surface is given by:

$n = \dfrac{{I \times \lambda }}{{hc}}$

Where, $I$ is the intensity of light
$\lambda $ is the wavelength of light
$h$ is the Planck’s constant
$c$ is the speed of light

Complete step by step solution:
We know that the intensity of sunlight received by the surface of earth is given by

$I = 1.4 \times {10^3}W/{m^2}$

Also assume that the light is monochromatic in nature and have the average wavelength given by

$\lambda = 500nm$

On converting it into standard unit that is in meters, we get

$ \Rightarrow \lambda = 500 \times {10^{ - 9}}m$

We know that the intensity is given by,

\[I = \dfrac{P}{A}\] …….(i)

Where \[P\] is the power and \[A\] is the area
Also power is given by the energy emitted per second, that is

$P = \dfrac{{nhc}}{\lambda }$ …….(ii)

Where $n$ is the number of photons emitted per second
$\lambda $ is the wavelength of light
$h$ is the Planck’s constant
$c$ is the speed of light
Now on putting the equation (ii) in equation (i),
\[I = \dfrac{{nhc}}{{\lambda \times A}}\]

Now, for finding number of photons falling per second on each square metre of earth’s surface is given by,

\[\therefore n = \dfrac{{I \times \lambda \times 1}}{{hc}}\]

On putting all the values we have

\[ \Rightarrow n = \dfrac{{1.4 \times {{10}^3} \times 500 \times {{10}^{ - 9}}}}{{6.63 \times {{10}^{ - 34}} \times 3 \times {{10}^8}}}\]

Here, the Planck’s constant $h = 6.63 \times {10^{ - 34}}$ ,
On further solving, we get

$ \Rightarrow n = 3.5 \times {10^{21}}$

Therefore, the number of the photons falling per second on each square meter of the earth’s surface is $3.5 \times {10^{21}}$

Note:Always remember that while solving numerical, convert all the given values in their respective standard units. Also remember that the monochromatic light is the light made up of one single pure frequency. It looks to the eye as a pure colour and can never be white. Basically, a light of purely a single colour is known as monochromatic light.