© All Rights reserved @ LearnWithDash
Step-by-Step Solution
Step 1: Identify the Relevant Physical Quantities and Formula
We know that the power p delivered by a monochromatic light source is related to the energy of each photon and the number of photons emitted per second. The energy of each photon of wavelength $ \lambda $ is given by:
$ E_{\text{photon}} = \frac{h c}{\lambda} $
If n is the number of photons emitted per second, then the total power p is:
$ p = n \times \frac{h c}{\lambda} $
Consequently, the number of photons emitted per second can be found using:
$ n = \frac{p \, \lambda}{h \, c} $
Step 2: Substitute the Given Values
The given quantities are:
Power, $ p = 3.3 \times 10^{-3} \, \text{W}$
Wavelength, $ \lambda = 600 \, \text{nm} = 600 \times 10^{-9} \, \text{m}$
Planck's constant, $ h = 6.6 \times 10^{-34} \, \text{J} \cdot \text{s}$
Speed of light, $ c = 3.0 \times 10^{8} \, \text{m/s}$
Plug them into the expression:
$ n = \frac{(3.3 \times 10^{-3}) \times (600 \times 10^{-9})}{(6.6 \times 10^{-34}) \times (3 \times 10^{8})} $
Step 3: Simplify the Expression
First compute the numerator:
$ (3.3 \times 10^{-3}) \times (600 \times 10^{-9})
= 3.3 \times 600 \times 10^{-3 - 9}
= 1980 \times 10^{-12}
= 1.98 \times 10^{3} \times 10^{-12}
= 1.98 \times 10^{-9} $
Next, the denominator:
$ (6.6 \times 10^{-34}) \times (3 \times 10^{8})
= 6.6 \times 3 \times 10^{-34 + 8}
= 19.8 \times 10^{-26}
= 1.98 \times 10 \times 10^{-26}
= 1.98 \times 10^{-25} $
Therefore:
$ n = \frac{1.98 \times 10^{-9}}{1.98 \times 10^{-25}}
= 10^{16} $
Step 4: State the Final Answer
The number of photons emitted per second on average is:
$ 10^{16} $
