Question

If ${n_R}$ and ${n_V}$ denote the number of photons emitted by a red bulb and violet bulb of equal power in a given time, then:
A) ${n_R} = {n_V}$
B) ${n_R} > {n_V}$
C) ${n_R} < {n_V}$
D) ${n_R} \geqslant {n_V}$

Answer 1

In this question, we will use the concept of the Planck's energy equation that is the relation of the Planck's constant, the wavelength of the light, and the speed of the light. Then we will consider the number of photons required to achieve that much amount of energy.

Complete solution:
In this question, It is given that ${n_R}$ and ${n_V}$ denote the number of photons emitted by a red bulb and violet bulb of equal power in a given time. We need to find the relation between the number of photons emitted by a red bulb and by the violet bulb.
We will use the Planck law to get the energy as,
$\Rightarrow E = \dfrac{{hc}}{\lambda }$
Here, the planck's constant is $h$, the speed of light is $c$, and the wavelength of the light is $\lambda$.
As we know that photons are the light whose move in packets and bundles.

If the power of each photon is $P$, then energy given out in t second is equal to $pt$. Let the number of photons be $n$, then
$\Rightarrow n = \dfrac{{Pt}}{E}$
Now we will substitute the values in the above relation.
$\Rightarrow n = \dfrac{{Pt}}{{\left( {hc/\lambda } \right)}}$
Now we simplify the above expression.
$\Rightarrow n = \dfrac{{Pt\lambda }}{{hc}}$
For red light,
$\Rightarrow {n_R} = \dfrac{{Pt{\lambda _R}}}{{hc}}$
For violet light,
$\Rightarrow {n_V} = \dfrac{{Pt{\lambda _V}}}{{hc}}$
Since, from above relation we get,
$\Rightarrow \dfrac{{{n_R}}}{{{n_V}}} = \dfrac{{{\lambda _R}}}{{{\lambda _V}}}$
And we know that,
$\Rightarrow {\lambda _R} > {\lambda _V}$
Then ${n_R}$is less than ${n_V}$
$\therefore {n_R} > {n_V}$

Hence the correct option is B.

Note: As we know that the different color of light has different wavelengths. In this question we use the concept that the wavelength of the red color light from the bulb is greater than the wavelength of the violet color light due to which the number of photons from the red color bulb will be greater than the number of photons of the violet color bulb.