1. Failure of Classical Wave Theory

In the 19th century, James Clerk Maxwell’s electromagnetic wave theory successfully explained optical phenomena like interference, diffraction, and polarization. However, classical theory failed fundamentally when applied to Blackbody Radiation Distribution at atomic scales:

• Wien's Distribution Law: Derived assuming thermal equilibrium of radiation. It successfully explained blackbody radiation at short wavelengths (high frequencies), but failed significantly at longer wavelengths. $$\text{Wien's Formula: } E_\lambda d\lambda = \frac{c_1}{\lambda^5} e^{-c_2 / (\lambda T)} d\lambda$$
• Rayleigh-Jeans Law: Derived using classical equipartition of energy ($k_B T$ per mode). It matched experimental data at long wavelengths, but predicted that energy density approaches infinity as wavelength approaches zero ($\lambda \to 0$). This catastrophic breakdown at ultraviolet frequencies is famously called the Ultraviolet Catastrophe. $$\text{Rayleigh-Jeans Formula: } E_\lambda d\lambda = \frac{8 \pi k_B T}{\lambda^4} d\lambda$$

2. Max Planck's Quantum Hypothesis (1900)

To resolve the breakdown of classical mechanics, German physicist Max Planck proposed a revolutionary hypothesis:

The energy $E$ contained within a single quantum of electromagnetic radiation of frequency $\nu$ is directly proportional to its frequency:

$$E = h \nu = \frac{h c}{\lambda}$$

Where total energy emitted or absorbed by an oscillator is quantized as integral multiples:

$$E_n = n h \nu = n \frac{h c}{\lambda} \quad (n = 1, 2, 3, \dots)$$

In 1905, Albert Einstein extended Planck's quantum concept to state that light itself propagates through space as discrete localized energy packets called Photons. The key fundamental properties of photons required for NEB examinations are detailed below:

3. Photon Intensity and Photon Flux Derivation

Consider a monochromatic light source of power $P$ emitting photons of frequency $\nu$ uniformly.

1. Number of Photons Emitted per Second ($N$): $$P = \frac{\text{Total Energy}}{\text{time}} = \frac{N \cdot E}{1\text{ s}} = N h \nu \implies N = \frac{P}{h\nu} = \frac{P \lambda}{hc}$$
2. Radiation Intensity ($I$): Defined as light power per unit cross-sectional area $A$: $$I = \frac{P}{A} = \left(\frac{N}{A}\right) h \nu = n_s \cdot h \nu$$ Where $n_s = \frac{N}{A}$ is the photon flux density (number of photons falling per unit area per second).

1. Key Terminology Definitions

A. Photoelectric Effect:

The phenomenon of ejection of electrons from a clean metal surface when electromagnetic radiation (light) of suitable frequency strikes the surface. Emitted electrons are called photoelectrons and the resulting current is called photocurrent.

B. Work Function ($\phi$ or $W_0$):

The minimum energy required to liberate a bound electron from the surface of a given metal against atomic attractive forces with zero remaining kinetic energy. $$\phi = h \nu_0 = \frac{h c}{\lambda_0}$$

C. Threshold Frequency ($\nu_0$):

The minimum characteristic frequency of incident radiation below which no photoelectric emission occurs, no matter how intense the incident beam is.

D. Threshold Wavelength ($\lambda_0$):

The maximum wavelength of incident light capable of ejecting photoelectrons from a given metal surface ($\lambda_0 = c / \nu_0$).

E. Stopping Potential / Cut-off Voltage ($V_0$):

The minimum retarding (negative) potential applied to the collector anode plate relative to the cathode emitter plate that is just sufficient to stop the fastest-moving photoelectrons from reaching the collector, reducing the photocurrent strictly to zero ($I = 0$). $$e V_0 = K_{\max} = \frac{1}{2} m v_{\max}^2$$

2. Experimental Apparatus Setup

The standard laboratory apparatus for studying photoelectric phenomena consists of an evacuated glass/quartz tube containing two metallic electrodes:

3. Laws of Photoelectric Emission (Experimental Observations)

By systematically varying parameters in the setup above, Lenard and Millikan established four fundamental laws:

1. Theoretical Derivation

In 1905, Albert Einstein successfully explained the photoelectric effect by applying Planck's quantum hypothesis. He postulated that photoelectric interaction is a one-to-one collision between a single incident light photon of energy $h\nu$ and a bound electron in the metal.

When a photon strikes a surface electron, its energy $h\nu$ is completely transferred to the electron. This energy is expended in two parts:

1. Work Function ($\phi$): The minimum energy used to pull the electron out of the metal surface against atomic binding forces.
2. Maximum Kinetic Energy ($K_{\max}$): The remaining energy imparted to the emitted electron as kinetic energy: $$K_{\max} = \frac{1}{2} m v_{\max}^2$$

By Conservation of Energy:

Since $\phi = h \nu_0$ (where $\nu_0$ is threshold frequency):

$$h \nu = h \nu_0 + \frac{1}{2} m v_{\max}^2$$ $$\implies K_{\max} = \frac{1}{2} m v_{\max}^2 = h(\nu - \nu_0) = h c \left( \frac{1}{\lambda} - \frac{1}{\lambda_0} \right)$$

Relating $K_{\max}$ to the stopping potential $V_0$:

2. Explanation of Photoelectric Laws using Einstein's Equation

1. Experimental Principle (1916)

American physicist Robert A. Millikan performed precise experiments using clean alkali metals (Lithium, Sodium, Potassium) inside a ultra-high vacuum cylinder. His objective was to test Einstein's photoelectric equation and measure Planck's constant $h$.

2. Mathematical Analysis & Graphical Method

Rearranging Einstein's equation in terms of stopping potential $V_0$:

$$e V_0 = h \nu - \phi \implies V_0 = \left( \frac{h}{e} \right) \nu - \frac{\phi}{e}$$

This matches the equation of a straight line $y = m x + c$:

• Variable $y$: Stopping potential $V_0$ (Volts)
• Variable $x$: Incident light frequency $\nu$ ($\text{Hz}$)
• Slope ($m$): $m = \frac{h}{e}$ (Universal constant for all target metals!)
• x-intercept ($\nu$-axis): Threshold frequency $\nu_0$ (where $V_0 = 0$)
• y-intercept ($V_0$-axis): Retarding potential intercept $-\frac{\phi}{e}$

3. Extraction of Planck's Constant ($h$)

By calculating the experimental slope $m$ from the $V_0$ vs $\nu$ straight line and multiplying by the known charge of an electron $e = 1.602 \times 10^{-19}\text{ C}$:

$$h = e \times \text{Slope} = e \times \left( \frac{\Delta V_0}{\Delta \nu} \right)$$

Millikan obtained $h = 6.57 \times 10^{-34} \text{ J s}$, which agreed with Planck's radiation constant within 0.5% error, conclusively proving Einstein's photon hypothesis.