Question

A blue lamp mainly emits light of wavelength.$4500\mathop {\,A}\limits^o$ The lamp is rated at $150\,{\text{W}}$ and ${\text{8}}\%$ of the energy is emitted as visible light. The number of photons emitted by the lamp per second is:
A. $3 \times {10^{19}}$
B. $3 \times {10^{24}}$
C. $3 \times {10^{20}}$
D. $3 \times {10^{18}}$

Answer 1

Calculate the energy of the total photons in the emitted visible light. Derive the relation between the power, energy, wavelength and number of photons.

Formulae used:
The energy $E$ of a single photon is given by
$E = \dfrac{{hc}}{\lambda }$ …… (1)
Here, $h$ is the Planck’s constant, $c$ is the speed of light and $\lambda$ is the wavelength of the photon.
The relation between the power $P$ and energy $E$ is
$P = \dfrac{E}{t}$ …… (2)
Here, $t$ is the time.

Complete step by step answer:
Calculate the energy ${E_n}$ of $n$ number of photons in the emitted light.
${E_n} = nE$
Substitute $\dfrac{{hc}}{\lambda }$ for $E$ in the above equation.
${E_n} = n\dfrac{{hc}}{\lambda }$ …… (3)
Calculate the total energy of the lamp.
Rewrite equation (2) for the total energy ${E_T}$ of the lamp emitted in one second.
${E_T} = {P_T}t$
Here, ${P_T}$ is the total power of the lamp.
Substitute $150\,{\text{W}}$ for ${P_T}$ and $1\,{\text{s}}$ for $t$ in the above equation.
${E_T} = \left( {150\,{\text{W}}} \right)\left( {1\,{\text{s}}} \right)$
$\Rightarrow {E_T} = 150\,{\text{J}}$
Only ${\text{8}}\%$ of the total energy of the lamp is emitted as the visible light.
Hence, the total energy ${E_L}$ of the photons in the emitted light is ${\text{8}}\%$ of the total energy ${E_T}$ of the lamp.
${E_L} = \dfrac{8}{{100}}{E_T}$ …… (4)
The energy ${E_n}$ of the $n$ number of photons in the emitted light is equal to the total energy ${E_L}$ of the photons in the emitted light.
${E_n} = {E_L}$
Substitute $n\dfrac{{hc}}{\lambda }$ for ${E_n}$ and $\dfrac{8}{{100}}{E_T}$ for ${E_L}$ in the above equation.
$n\dfrac{{hc}}{\lambda } = \dfrac{8}{{100}}{E_T}$
Rearrange the above equation for $n$.
$n = \dfrac{8}{{100}}\dfrac{{{E_T}\lambda }}{{hc}}$
Substitute $150\,{\text{J}}$ for ${E_T}$, $4500\mathop {\,A}\limits^o$ for $\lambda$, $6.63 \times {10^{ - 34}}\,{\text{J}} \cdot {\text{s}}$ for $h$ and $3 \times {10^8}\,{\text{m/s}}$ for $c$ in the above equation.
$n = \dfrac{8}{{100}}\dfrac{{\left( {150\,{\text{J}}} \right)\left( {4500\mathop {\,A}\limits^o } \right)}}{{\left( {6.63 \times {{10}^{ - 34}}\,{\text{J}} \cdot {\text{s}}} \right)\left( {3 \times {{10}^8}\,{\text{m/s}}} \right)}}$
$\Rightarrow n = \dfrac{8}{{100}}\dfrac{{\left( {150\,{\text{J}}} \right)\left( {4500 \times {{10}^{ - 10}}\,{\text{m}}} \right)}}{{\left( {6.63 \times {{10}^{ - 34}}\,{\text{J}} \cdot {\text{s}}} \right)\left( {3 \times {{10}^8}\,{\text{m/s}}} \right)}}$
$\Rightarrow n = 2.71 \times {10^{19}}$
$\Rightarrow n \approx 3 \times {10^{19}}$
Therefore, the number of the photons emitted by the lamp per second is $3 \times {10^{19}}$.
Hence, the correct option is A.

Note: While doing solution please make sure that you have calculate the total number of photons emitted by the lamp using the energy ${\text{8}}\%$of the light emitted by the lamp and not the total energy of the lamp because this the place where mistake could be happened.