Question

A point source of light of power P and wavelength \[\lambda \] is emitting light in all directions. The number of photons present in a spherical region of radius r to radius \[r + x\] with centre at the source is:
A. \[\dfrac{{P\lambda }}{{4\pi {r^2}hc}}\]
B. \[\dfrac{{P\lambda x}}{{h{c^2}}}\]
C. \[\dfrac{{P\lambda x}}{{4\pi {r^2}hc}}\]
D. None of these

Answer 1

Hint:The above problem can be resolved using the concepts and the fundamentals of the energy waves and the variables associated with these variables like the wavelength, frequency and velocity of light and many more. Moreover, these concepts are extracted from the fundamentals of the photoelectric effect. The mathematical relation for the energy stored within the region is used along with an expression for energy density. And these relations are further used to find the number of photons. And by substituting the values, one can obtain the required result.

Complete step by step answer:
Let E be the energy stored in the spherical region between \[r\] and \[r + x\] is,
Then the expression for the magnitude of this energy is,
\[E = C \times \left( {4\pi {r^2}x} \right)\] ……… (1)
Here, C is the energy density and its value is given as,
\[C = \dfrac{P}{{4\pi {r^2}c}}\]………… (2)
On substituting the value of equation 2 in 1 as,
\[\begin{array}{l}
E = C \times \left( {4\pi {r^2}x} \right)\\
E = \left( {\dfrac{P}{{4\pi {r^2}c}}} \right) \times \left( {4\pi {r^2}x} \right)\\
E = \dfrac{{Px}}{c}
\end{array}\]
The number of photons is given as,
\[N = \dfrac{E}{{h\nu }}\]
Here, h is the Planck’s constant, \[\nu \]is the frequency of energy and its value is, \[\nu = c/\lambda \]. Where c is the speed of light and \[\lambda \] is the wavelength of energy.
Solve by substituting the values as,
 \[\begin{array}{l}
N = \dfrac{1}{{h\nu }}\left( {\dfrac{{Px}}{c}} \right)\\
N = \dfrac{1}{{h\left( {\dfrac{c}{\lambda }} \right)}}\left( {\dfrac{{Px}}{c}} \right)\\
N = \dfrac{{P\lambda x}}{{h{c^2}}}
\end{array}\]
Therefore, the number of photons is \[\dfrac{{P\lambda x}}{{h{c^2}}}\] and option (B) is correct.

Note:Try to understand the fundamentals and the concept behind the photoelectric effect and the basic mathematical relations involved in this concept. Moreover, one should try to remember these formulas and apply accordingly to resolve the conditions given in such problems.