Question

\(11 \times 10^{11}\) Photons are incident on a surface in \(10 \mathrm{~s}\). These photons correspond to a wavelength of \(10 \AA\). If the surface area of the given surface is \(0.01 \mathrm{~m}^{2}\), the intensity of given radiations is \(\ldots \ldots\) $\left\\{\mathrm{h}=6.625 \times 10^{-34} \mathrm{~J} . \mathrm{s}, \mathrm{c}=3 \times 10^{8}(\mathrm{~m} / \mathrm{s})\right\\}$ (A) \(21.86 \times 10^{-3}\left(\mathrm{~W} / \mathrm{m}^{2}\right)\) (B) \(2.186 \times 10^{-3}\left(\mathrm{~W} / \mathrm{m}^{2}\right)\) (C) \(218.6 \times 10^{-3}\left(\mathrm{~W} / \mathrm{m}^{2}\right)\) (D) \(2186 \times 10^{-3}\left(\mathrm{~W} / \mathrm{m}^{2}\right)\)

Short answer

The intensity of the given radiations is \(2.186 \times 10^{-3} ~\mathrm{W.m^{-2}}\).

Step-by-step solution

Calculate the energy of a single photon

According to the Planck's equation, the energy (E) of a photon can be determined using the formula: \(E = \dfrac{hc}{\lambda}\) where h = Planck's constant = \(6.625 \times 10^{-34}~J.s\) c = speed of light = \(3 \times 10^8~m.s^{−1}\) λ = wavelength of photon = \(10 \mathrm{~\mathring{A}} = 10^{-10}~m\) Now, substitute these values in the formula and calculate the energy of a single photon: \(E = \dfrac{(6.625 \times 10^{-34}) (3 \times 10^{8})}{10^{-10}}\)

Calculate the total energy of all the photons

The total energy can be found by multiplying the energy of a single photon by the total number of photons: Total photonic energy \(=\) Energy of a single photon \(×\) number of photons \(= E \times 11 \times 10^{11}\) Now substitute the expression for E we found in step 1: Total photonic energy \(= \dfrac{(6.625 \times 10^{-34}) (3 \times 10^{8})(11 \times 10^{11})}{10^{-10}}\)

Calculate the intensity

Intensity (I) is defined as the total energy incident on an area per unit time. In this case, we are given the area of the surface and the time duration. We can calculate the intensity using the formula: \(I = \dfrac{\text{Total photonic energy}}{\text{Area} \times \text{Time}}\) Substitute the values given in the problem and the expression for total photonic energy derived in step 2: \(I = \dfrac{\dfrac{(6.625 \times 10^{-34}) (3 \times 10^{8})(11 \times 10^{11})}{10^{-10}}}{(0.01)(10)}\)

Solve the expression to find the intensity

Now, compute the intensity using the expression above: \(I = \dfrac{(6.625 \times 10^{-34}) (3 \times 10^{8})(11 \times 10^{11})}{(10^{-10})(0.01)(10)} = 2.186 \times 10^{-3} ~\mathrm{W.m^{-2}}\) Upon solving the expression, we find that the intensity of the given radiations is \(2.186 \times 10^{-3} ~\mathrm{W.m^{-2}}\), which corresponds to option (B).