Question

How many photons are emitted per second by a 5 mW laser operating at 632.8 nm?
A. \[1.6 \times {10^{16}}\]
B. \[1.6 \times {10^{13}}\]
C. \[1.6 \times {10^{10}}\]
D. \[1.6 \times {10^3}\]

Answer 1

Hint: To find the required number of photons emitted, we are using the formula
\[E = \dfrac{{nhc}}{\lambda }\] Where, E is the energy of laser beam of wavelength \[\lambda \], h is the Planck’s constant, c is the speed of light and n is the number of photon emitted.

Complete step-by-step answer:
To calculate the required number of photons emitted.
Calculating the energy of the beam from the given data in the question.
Given:-
\[P = 5{\text{ mW}}\]
\[ \Rightarrow P = 5{\text{ mW}}\]
\[t = 1\sec \]

Using the formula: \[E = P \times t\], …………………..(i) for energy calculation .We get:
Substituting the given values of P and t in the equation (i)
We have,
\[E = 5 \times {10^{ - 3}}\]J………………….(ii)
Now we take the formula \[E = \dfrac{{nhc}}{\lambda }\], we can also modify it as:
\[n = \dfrac{{E\lambda }}{{hc}}\]……………………………………………..(iv)
Substituting the given values of E, h, \[\lambda \] and c in eqn (iv)[ we take \[h = 6.626 \times {10^{ - 34}}{\text{J - s}}\].
We get the required number of photons:-
\[
  n = \dfrac{{5 \times {{10}^{ - 3}} \times 632.8 \times {{10}^{ - 19}}}}{{6.626 \times {{10}^{ - 34}} \times 3 \times {{10}^8}}} \\
   \Rightarrow \dfrac{{3196}}{{19.876}} \times {10^{14}} \\
   \Rightarrow 1.591 \times {10^{16}} \\
   \approx 1.6 \times {10^{16}} \\
\]
Hence Option (A) is the correct answer.

Note: To solve such types of questions one should have the ability to implement the short and quick fact based formula. We use the for \[E = \dfrac{{nhc}}{\lambda }\] not \[\dfrac{{hc}}{\lambda }\] because it is the energy of single electron while on multiplying the n with \[\dfrac{{hc}}{\lambda }\] we get the energy of the laser beam. One should also carefully convert the given values into the SI units.