Question

Give all formulas for dual nature

Answer 1

Constants:

• $h$: Planck's constant ($6.626 \times 10^{-34} \text{ J s}$)
• $c$: Speed of light in vacuum ($3 \times 10^8 \text{ m/s}$)
• $e$: Elementary charge ($1.602 \times 10^{-19} \text{ C}$)
• $k_B$: Boltzmann constant ($1.381 \times 10^{-23} \text{ J/K}$)

I. Dual Nature of Radiation (Light as Particles - Photons)

• Photon Energy: $E = hf$ $E = \frac{hc}{\lambda}$

• Photon Momentum: $p = \frac{h}{\lambda}$ $p = \frac{E}{c}$

II. Photoelectric Effect

• Einstein's Photoelectric Equation: $K_{max} = hf - \phi$

• Work Function ($\phi$): $\phi = hf_0$ $\phi = \frac{hc}{\lambda_0}$

• Maximum Kinetic Energy in terms of Stopping Potential ($V_s$): $K_{max} = eV_s$

• Combined Equation: $eV_s = hf - \phi$ $eV_s = hf - hf_0$ $eV_s = h(f - f_0)$ $eV_s = hc\left(\frac{1}{\lambda} - \frac{1}{\lambda_0}\right)$

III. Dual Nature of Matter (de Broglie Hypothesis)

• De Broglie Wavelength ($\lambda$): $\lambda = \frac{h}{p}$

• For a particle of mass $m$ and velocity $v$: $\lambda = \frac{h}{mv}$

• For a particle with kinetic energy $K$: $\lambda = \frac{h}{\sqrt{2mK}}$

• For an electron accelerated by a potential difference $V$: $\lambda = \frac{h}{\sqrt{2meV}}$ $\lambda \approx \frac{1.227}{\sqrt{V}} \text{ nm}$

• For a gas molecule at temperature $T$: $\lambda = \frac{h}{\sqrt{3mk_BT}}$

Answer 2

The dual nature of radiation and matter is described by a set of fundamental formulas that connect properties like energy, frequency, wavelength, momentum, and kinetic energy for both photons and particles. These formulas are crucial for understanding phenomena such as the photoelectric effect and the de Broglie hypothesis.