Dual Nature of Radiation and Matter | Class 12 Physics (NCERT Ch 11)

CBSE Class 12 › Unit VII: Dual Nature

Dual Nature of Radiation and Matter

NCERT Chapter 11 • Photoelectric Effect & de Broglie Wavelength

NCERT 2025–26 Unit VII • ~4 Marks JEE Main • 1 Question

1. Electron Emission

Free electrons in metals require minimum energy to escape the surface. This energy can be supplied via:

Types of Electron Emission:

Thermionic Emission: By heating the metal (e.g., filament in vacuum tubes).
Field Emission: By applying a strong electric field (~10⁸ V/m).
Photoelectric Emission: By irradiating with light of suitable frequency.

2. Photoelectric Effect

Discovered by Hertz (1887) and studied in detail by Hallwachs & Lenard.

Experimental setup for studying the photoelectric effect. — Light incident on the photosensitive plate causes electron emission.

Key Definitions:

Work Function ( $\phi_0$ ): Minimum energy to eject an electron (unit: eV).
Threshold Frequency ( $\nu_0$ ): Minimum frequency for photo-emission: $\nu_0 = \phi_0 / h$ .
Photoelectrons: Electrons emitted due to light incidence.

3. Experimental Study

Key observations from Lenard’s experiments:

Photocurrent ∝ Intensity (for $\nu > \nu_0$ ).
Stopping potential $V_0$ independent of intensity.
$V_0$ increases linearly with frequency $\nu$ .
No emission if $\nu < \nu_0$ (even for high intensity).
Emission is instantaneous (< 10⁻⁹ s).

A. Effect of Intensity

Graph showing linear relation between photocurrent and light intensity. — Photocurrent $\propto$ Intensity (if $\nu > \nu_0$ ).

B. Effect of Potential & Frequency

Graph showing variation of photocurrent with potential for different frequencies. — Higher frequency radiation requires a larger (more negative) stopping potential ( $V_0$ ) to stop the electrons.

4. Why Wave Theory Failed

Classical wave theory predicted:

KE should increase with intensity (but it depends only on $\nu$ ).
No threshold frequency should exist (but $\nu_0$ is observed).
Time lag for emission at low intensity (but emission is instantaneous).

These contradictions led to the photon model.

5. Einstein’s Photoelectric Equation

Albert Einstein (1905) explained the effect using light quanta (photons). Awarded Nobel Prize in 1921.

Diagram explaining the energy balance in Einstein's photoelectric equation.

Einstein’s Photoelectric Equation 3 Marks

Photon Energy: Each photon has energy $E = h\nu$ .
Energy Conservation:
Photon energy = Work function + Max KE of electron.
$h\nu = \phi_0 + K_{max}$

Mathematical Form:
$K_{max} = h\nu - \phi_0$
Since $K_{max} = eV_0$ :
$eV_0 = h\nu - \phi_0$

Threshold Frequency:
For $K_{max} = 0$ : $h\nu_0 = \phi_0$ → $\nu_0 = \phi_0 / h$

Slope of $V_0$ vs $\nu$ Graph:

From $V_0 = \left(\dfrac{h}{e}\right)\nu - \dfrac{\phi_0}{e}$ , the slope is $\dfrac{h}{e}$ . Millikan verified this experimentally.

Graphs of variation of photocurrent with different intensities, keeping the same frequency. Different metals and stopping potential

Note:

The graph of $V_0$ vs $\nu$ is plotted for a single photosensitive material. Different metals give different parallel lines (same slope $h/e$ , different threshold frequencies $\nu_0$ ). The x-intercept gives $\nu_0 = \phi_0 / h$ , which is unique to each metal.

Stopping Potential vs Frequency: Linear Equation
From Einstein’s equation: $eV_0 = h\nu - \phi_0$
⇒ $V_0 = \left(\dfrac{h}{e}\right)\nu - \dfrac{\phi_0}{e}$

Graph Interpretation:

Slope = $\dfrac{h}{e}$ → used to experimentally determine Planck’s constant.
X-intercept = threshold frequency $\nu_0 = \dfrac{\phi_0}{h}$ .
Y-intercept = $-\dfrac{\phi_0}{e}$ → depends on work function of the material.
All metals give parallel lines (same slope, different intercepts).

6. Particle Nature of Light: Photons

Photon Properties:

Energy: $E = h\nu = \dfrac{hc}{\lambda}$
Momentum: $p = \dfrac{h\nu}{c} = \dfrac{h}{\lambda}$
Speed: $c$ (in vacuum)
Electrically neutral (not deflected by E/B fields)
In collisions: Energy & momentum conserved, but photon number may change (absorption/emission).

7. Wave Nature of Matter

Louis de Broglie (1924) proposed that moving particles exhibit wave-like behavior.

de Broglie Wavelength Derivation 2 Marks

For a photon: $\lambda = \dfrac{h}{p}$ (from $E = pc$ and $E = h\nu$ ).
de Broglie hypothesized this applies to all matter.

For a particle of mass $m$ and velocity $v$ :
Momentum $p = mv$
$\lambda = \dfrac{h}{mv} = \dfrac{h}{p}$

For electron accelerated by voltage $V$ :
$K = eV = \dfrac{p^2}{2m}$ → $p = \sqrt{2meV}$
$\lambda = \dfrac{h}{\sqrt{2meV}} = \dfrac{1.227}{\sqrt{V}} \text{ nm}$

$\lambda = \dfrac{h}{p}$

8. Davisson-Germer Experiment

First experimental proof of matter waves (1927).

Schematic of Davisson-Germer experiment proving wave nature of electrons. — Electron diffraction pattern was observed, confirming de Broglie’s hypothesis.

Key Steps:

Electron beam accelerated by voltage $V$ directed at nickel crystal.
Scattered electrons detected at various angles.
Peak intensity observed at specific angle (e.g., 50° for 54V).
Bragg’s law applied: $n\lambda = 2d\sin\theta$ .
Calculated $\lambda$ matched de Broglie’s formula.

Practice Time!

Solve numericals on Work Function and de Broglie Wavelength: Chapter 11 Important Questions →

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