JEE Physics Dual Nature Of Radiation and Matter Notes Set 02

Photoelectric Effect

1. Electrons are ejected from a metal surface when light of frequency above the threshold frequency \( \nu_0 \) falls on it. Below \( \nu_0 \) there is no emission however intense the light is.
2. Einstein's equation: \( h\nu=\phi+K_{max} \), where \( \phi=h\nu_0 \) is the work function. The maximum kinetic energy is \( K_{max}=h\nu-\phi \) and it equals \( eV_0 \), where \( V_0 \) is the stopping potential.
3. Intensity decides the number of photons per second, hence the number of photoelectrons and the saturation current. It does not change \( K_{max} \). Doubling the intensity doubles the current.
4. Frequency decides \( K_{max} \) and the stopping potential. Emission is almost instantaneous (within about \( 10^{-9} \) s).
5. Photoelectrons come out with all kinetic energies from zero up to \( K_{max} \), because some lose energy inside the metal.
6. Graph of \( V_0 \) against \( \nu \): a straight line with slope \( h/e \) and intercept \( \nu_0 \) on the frequency axis. Graph of photocurrent against collector potential: a higher frequency needs a larger retarding potential to stop the current, while the saturation current depends on intensity.
7. Photon energy is \( E=\dfrac{hc}{\lambda}=\dfrac{12400}{\lambda(\text{Å})}\ \text{eV} \). Number of photons in energy E is \( E\lambda/hc \).

Photon Momentum and Radiation Pressure

A photon carries momentum \( p=\dfrac{h}{\lambda}=\dfrac{E}{c} \). A beam of energy E that is fully absorbed gives momentum E/c to a surface; if it is fully reflected the surface gets 2E/c.

de Broglie Waves

1. \( \lambda=\dfrac{h}{p}=\dfrac{h}{mv} \). In terms of kinetic energy K: \( \lambda=\dfrac{h}{\sqrt{2mK}} \). A graph of p against \( \lambda \) is a hyperbola.
2. For a charge q accelerated through a potential difference V: \( \lambda=\dfrac{h}{\sqrt{2mqV}} \). For an electron \( \lambda=\dfrac{12.27}{\sqrt{V}}\ \text{Å} \).
3. For particles with the same \( \lambda \) and the same potential, compare the products mq. A proton and an alpha particle have the same wavelength when \( V_\alpha=V_p/8 \).
4. The wave nature of electrons was shown by electron diffraction (Davisson and Germer experiment).