Dual Nature of Matter and Radiation

• The de Broglie wavelength of a non-relativistic particle of mass $m$ and charge $q$ accelerated through potential $V$ is:
  $$\lambda = \frac{h}{\sqrt{2mqV}}$$
• For a proton ($m_p = m$, $q_p = e$):
  $$\lambda_p = \frac{h}{\sqrt{2 m e V}}$$
• For an $\alpha$-particle ($m_\alpha = 4m$, $q_\alpha = 2e$) accelerated through potential $V_\alpha$:
  $$\lambda_\alpha = \frac{h}{\sqrt{2 (4m)(2e) V_\alpha}} = \frac{h}{\sqrt{16 m e V_\alpha}}$$
• Setting $\lambda_\alpha = \lambda_p$:
  $$\sqrt{16 m e V_\alpha} = \sqrt{2 m e V} \implies 16 V_\alpha = 2 V \implies V_\alpha = \frac{2V}{16} = \frac{V}{8}$$

According to Einstein's Photoelectric Equation:

Given that incident frequency is $\nu = 2\nu_0$:

Since some emitted electrons lose energy through internal collisions within the metal lattice before escaping, their actual kinetic energies vary continuously in the range $0 \le K \le K_{\max}$. Thus, the maximum kinetic energy of emitted photoelectrons can be $h\nu_0$.

• For a point source of light of power $P$, the intensity of radiation at distance $d$ follows the inverse square law:
  $$I = \frac{P}{4\pi d^2} \propto \frac{1}{d^2}$$
• Photoelectric current $i$ is directly proportional to the intensity of incident radiation (number of photons incident per unit area per second):
  $$i \propto I \implies i \propto \frac{1}{d^2}$$