Question

If a source of power $4kW$ produces ${{10}^{20}}$ photons/second, the radiation belongs to a part of the spectrum called
A) X-rays
B) ultraviolet rays
C) microwaves
D) $\gamma$-rays

Answer 1

At first, an expression for wavelength of emitted photons is determined from its energy equation. We know that power of a source producing photons is equal to the product of the number of photons emitted per second and the energy of emitted photons. Combining both these equations, we can easily determine the wavelength of the emitted radiation, and hence, determine the part of the radiation spectrum to which this wavelength falls.

Formula used:
$1)E=hv=\dfrac{hc}{\lambda }$
$2)P=n\dfrac{hc}{\lambda }$

Complete answer:
We are given that:
- Power of the source, $P=4kW$
- Photons produced per second, $n={{10}^{20}}$
Also do we know that:

& h=6.62\times {{10}^{-34}} \\\ & c=3\times {{10}^{8}} \\\ \end{aligned}$$ where $h$ is the Planck’s constant $c$ is the speed of light Now, energy of photon is given by $$E=hv=\dfrac{hc}{\lambda }$$ where $$h$$ is the Planck’s constant $$\lambda $$ is the wavelength of photon $$v$$ is the frequency of photon Let this be equation 1. Also, power of a photon source is related to the energy of photon by $$P=nE=n\dfrac{hc}{\lambda }$$ where $P$ is the power of photon source $E$ is the energy of emitted photon $$\lambda $$ is the wavelength of photon $$n$$ is the number of photons emitted per second Let this be equation 2. Substituting the given values from the question in equation 2, we have $$P=n\dfrac{hc}{\lambda }\Rightarrow 4\times {{10}^{3}}=\dfrac{{{10}^{20}}\times 6.62\times {{10}^{-34}}\times 3\times {{10}^{8}}}{\lambda }\Rightarrow \lambda =\dfrac{19.8\times {{10}^{-26}}\times {{10}^{20}}}{4\times {{10}^{3}}}=4.96\times {{10}^{-9}}m$$ Let this be equation 3. Clearly, from equation 3, wavelength of the emitted photons lies between $${{10}^{-8}}m$$ and $${{10}^{-12}}m$$, which falls in the range of X-ray spectrum. **Therefore, the correct answer is option A.** **Note:** Here, it is important to know the ranges of wavelengths for the other options too. They are given as follows: \- Wavelengths of X – rays range in between $${{10}^{-8}}m$$ and $${{10}^{-12}}m$$, as already mentioned. \- Wavelengths of ultraviolet rays range in between $100nm$ and $400nm$. \- Wavelengths of microwaves range in between $1mm$ and $30cm$. \- Range of wavelengths of $$\gamma $$-rays is less than $100$ picometers or $$4\times {{10}^{9}}$$ inches.