Dual Nature of Radiation and Matter

Given, temperature of sun, $T_s=6000 \mathrm{~K}$

Radius of sun, $R_s=7.2 \times 10^5 \mathrm{~km}=7.2 \times 10^8 \mathrm{~m}$

Radius of earth, $R_e=6000 \mathrm{~km}=600 \times 10^3 \mathrm{~m}$

Distance between earth and sun,

$d=15 \times 10^7 \mathrm{~km}=15 \times 10^{10} \mathrm{~m}$

Intensity of light on the earth,

$\begin{aligned} I & =\frac{\text { Total energy emitted by sun }}{4 \pi d^2} \times\left(\pi R_e{ }^2\right) \\ & =\frac{\sigma\left(T_s^4 \cdot 4 \pi R_s^2 \times \pi R_e{ }^2\right)}{4 \pi d^2}=\frac{\sigma T_s^4 R_s{ }^2 \times \pi R_e{ }^2}{d^2} \\ & \begin{aligned} & =\frac{5.67 \times 10^{-8} \times(6000)^4 \times\left(7.2 \times 10^8\right)^2 \times 3.14 \times (6000\times10^3)^2}{\left(15 \times 10^{10}\right)^2} \\ & =19.2 \times 10^{16} \end{aligned} \end{aligned}$

In radioactivity, electrons are emitted in the form of $\beta\left({ }_{-1} e^{\circ}\right)$ particles. It is also called negative $\beta$-decay, where an unstable nucleus emits an energetic electron and an anti-neutrino and a neutron in the nucleus becomes a proton that remains in the product nucleus.

In photoelectric effect, when light rays of frequency greater than or equal to threshold frequency incident on metal surface, then electrons are emitted from metal surface. This phenomenon is called photoelectric effect. For photoelectric emission it is not necessary that element is unstable but for radioactivity, it is necessary that elements are unstable. Hence, assertion is true but reason is false.

As we know that, wavelength of a particle

$\begin{aligned} & \lambda =\frac{h}{\sqrt{2 m E}}=\frac{h}{\sqrt{2 m}} \cdot \frac{1}{\sqrt{E}} \\ \Rightarrow \quad & \log \lambda =\log \left(\frac{h}{\sqrt{2 m}} \cdot \frac{1}{\sqrt{E}}\right) \\ \Rightarrow \quad & \log \lambda =\log \frac{h}{\sqrt{2 m}}+\log \frac{1}{E^{1 / 2}} \\ \Rightarrow \quad & \log \lambda =\log \left(\frac{h}{\sqrt{2 m}}\right)-\frac{1}{2} \log E \\ \Rightarrow \quad & \log \lambda =-\frac{1}{2} \log E+\log \frac{h}{\sqrt{2 m}} \end{aligned}$

So, equation of straight line is $y=m x+c$.

This is equation of line with slope $-\frac{1}{2}$.

The photoelectric effect is the emission of electrons (called photoelectrons) when light strikes a metal surface. Hence, photoelectric effect can take place only with an electron bound in the atom. Electron is a lepton whereas proton is fermion.