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CLASS 12 • PHYSICS
Dual Nature of Radiation
1. Einstein's Equation
The Master Equation
$h\nu = \phi_0 + K_{max}$
Energy = Work Function + Kinetic Energy
Work Function ($\phi_0$)
$\phi_0 = h\nu_0 = \frac{hc}{\lambda_0}$
Min energy to remove electron
Kinetic Energy ($K_{max}$)
$K_{max} = eV_0$
$V_0$ is Stopping Potential
2. De-Broglie Wavelength
Basic Formula
$\lambda = \frac{h}{p} = \frac{h}{mv}$
In Terms of Energy (K)
$\lambda = \frac{h}{\sqrt{2mK}}$
For Accelerated Electron
$\lambda = \frac{1.227}{\sqrt{V}} \text{ nm}$
V is accelerating voltage
For Gas Molecule
$\lambda = \frac{h}{\sqrt{3mkT}}$
3. Graph Trends
Stopping Potential ($V_0$) vs Freq ($\nu$)
$V_0 = (\frac{h}{e})\nu - \frac{\phi_0}{e}$
Slope = $h/e$ (Universal Constant)
Photo-current vs Intensity
Straight Line passing through origin (Directly Proportional)
Effect of Frequency
Higher Frequency $\to$ More Kinetic Energy $\to$ More Negative Stopping Potential.
4. Photon Picture
Energy of 1 Photon
$E = h\nu = \frac{hc}{\lambda}$
Calculation Trick: $hc \approx 1240 \text{ eV nm}$
Momentum
$p = \frac{h}{\lambda} = \frac{E}{c}$
Power of Source
$P = nE$
n = number of photons emitted per sec
5. Important Experiments
Davisson & Germer
Proved WAVE nature of electrons using Nickel crystal diffraction.
Lenard & Hallwachs
Observed Photoelectric effect (Particle nature of light).
6. Common Exam Traps
- ⚠️ Intensity vs Frequency: Intensity increases CURRENT (number of electrons). Frequency increases KINETIC ENERGY (speed of electrons). Don't mix them!
- ⚠️ Slope of V vs $\nu$: The slope is always $h/e$. It does NOT depend on the metal used.
- ⚠️ Unit Conversion: Work function is usually in $eV$. Convert to Joules ($\times 1.6 \times 10^{-19}$) before using in $h\nu$ equation.
7. Golden Theory Rules
- One-to-One: One photon interacts with ONLY one electron. Energy is not distributed.
- Instantaneous: Photoelectric emission is instantaneous ($10^{-9}$ sec). There is no time lag.
- Neutrality: Photons are electrically neutral and are not deflected by Electric or Magnetic fields.
