An ultraviolet lamp emits light of wavelength 400 nm at the rate of 400 W. An infrared lamp emits light of wavelength 700 nm, also at the rate of 400 W. (a) Which lamp emits photons at the greater rate and (b) what is that greater rate?

The expression of emission rate is given by,

$R=\frac{P}{E_p}$

Here, P is power.

The energyof a photon of wavelength $\lambda$ is given by,

$E_p=\frac{hc}{\lambda }$

Here, h is Planck’s constant and c is the speed of light.

(a)

The wavelength of the photon and energy of the photon has an indirect relationship, so; the wavelength of the photon will be larger when the energy is smaller. Here, the wavelength of infrared light is more than ultraviolet light.

Therefore, the energy of the infrared photon is less than the ultraviolet photon. But the power emitted by both lamps is the same.

$\mathrm{Number}\mathrm{of}\mathrm{photons}\mathrm{emitted}\mathrm{per}\mathrm{second}=\frac{\mathrm{Energy}\mathrm{emitted}\mathrm{per}\mathrm{second}}{\mathrm{Energy}\mathrm{of}\mathrm{each}\mathrm{photon}}$

Here, the energy emitted per second is constant and equal to 400 W.

$\mathrm{Number}\mathrm{of}\mathrm{photons}\mathrm{emitted}\mathrm{per}\mathrm{second}\propto \frac{1}{\mathrm{Energy}\mathrm{of}\mathrm{each}\mathrm{photon}}$

From the above relation, it can be observed that if the energy of the infrared photon is less than the ultraviolet photon, then the number of photons emitted by the infrared light will be more.

Therefore, the infrared lamp emits photons at a greater rate.

The expression to calculate the emission rate is given by,

$R=\frac{P\lambda }{hc}$…… (1)

Substitute the below values in eq 1.

$$\begin{gathered} P=400w \\ \lambda =700nm \\ h=6.626\times {10}^{-34}\mathrm{J}·\mathrm{s} \\ \mathrm{c}=3\times {10}^8\mathrm{m}/\mathrm{s} \end{gathered}$$

$$\begin{gathered} \mathrm{R}=\frac{\left(400\mathrm{W}\right)\left(700\mathrm{nm}\right)}{\left(6.626\times {10}^{-34}\mathrm{J}·\mathrm{s}\right)\left(3\times {10}^8\mathrm{m}/\mathrm{s}\right)} \\ =\frac{\left(400\mathrm{W}\right)\left(700\mathrm{nm}\right)\left({\displaystyle \frac{1\mathrm{m}}{{10}^9\mathrm{nm}}}\right)}{\left(6.626\times {10}^{-34}\mathrm{J}·\mathrm{s}\right)\left(3\times {10}^8\mathrm{m}/\mathrm{s}\right)} \\ =1.4\times {10}^{21}\mathrm{photons}/\mathrm{s}. \end{gathered}$$

Therefore, the rate of emitted protons is$1.4\times {10}^{21}\mathrm{photons}/\mathrm{s}.$