a) What is quantization of charge? Describe the theory of Millikan's oil drop experiment to determine the number of charges on an oil drop. b) What is P N junction diode? Discuss its applications as…

(a) Quantization of Charge and Millikan’s Oil-Drop Experiment

Quantization of Charge

Charge is said to be quantized because it exists in discrete packets (quanta) of a fundamental unit, rather than being infinitely divisible. The smallest such unit is the elementary charge ($e$), equal to $1.602\times 10^{-19}$ coulombs. All observed charges are integer multiples of $e$: $$q=\pm ne(n=1,2,3,\dots )$$ This was first inferred from experiments like cathode-ray tubes (J.J. Thomson, 1897) and later confirmed by Robert Millikan’s oil-drop experiment (1909–1913).

Millikan’s Oil-Drop Experiment

Objective: Measure the charge of an electron ($e$) and verify its quantization.

Theory and Setup:

1. Oil Droplets: Tiny oil drops are sprayed into a chamber where they acquire a net charge by friction (typically $-ne$ due to electron attachment).
2. Electric and Gravitational Fields:
   • A uniform electric field ($E$) is applied between two plates.
   • The droplet’s motion is observed under gravity ($mg$) and electric force ($qE$).
3. Balancing Forces: By adjusting $E$, the droplet can be held stationary. At equilibrium: $$qE=mg\Rightarrow q=\frac{mg}{E}$$
   • $m$ is determined from Stokes’ law (viscous drag on a falling sphere): $$m=\frac{1}{2}\rho d^3\text{(density ρ, diameter d)},$$ and terminal velocity $v$ under gravity: $$6\pi \eta rv=\frac{4}{3}\pi r^3\rho g\Rightarrow r=\sqrt{\frac{9\eta v}{2\rho g}}.$$
4. Charge Quantization: Millikan measured $q$ for hundreds of drops and found all values were integer multiples of $1.602\times 10^{-19}$ C.

Key Observations:

• The smallest observed charge was $e=1.602\times 10^{-19}$ C.
• All measured charges were $q=ne$, confirming charge is quantized.

Limitations:

• Assumes spherical droplets and negligible air buoyancy.
• Later refined to account for relativistic effects and quantum tunneling.

(b) P-N Junction Diode and Full-Wave Rectifier

P-N Junction Diode

A P-N junction diode is formed when a p-type (hole-rich) and n-type (electron-rich) semiconductor are joined. Its I-V characteristic shows:

• Forward Bias: Low resistance, current flows (holes/electrons recombine).
• Reverse Bias: High resistance until breakdown voltage ($V_Z$).

Energy Band Diagram:

Full-Wave Rectifier

A full-wave rectifier converts AC to pulsating DC using two diodes (center-tapped transformer) or four diodes (bridge rectifier). The bridge rectifier (shown below) is more efficient.

Circuit Diagram:

flowchart TD
    A["AC Input"] --> B["D1 (Anode)"]
    A --> C["D2 (Anode)"]
    B --> D["Load (R_L)"]
    C --> D
    D --> E["Ground"]
    B --> F["D3 (Cathode)"]
    C --> G["D4 (Cathode)"]
    F --> E
    G --> E

Working Principle:

1. First Half-Cycle: D1 and D4 conduct; current flows through $R_L$.
2. Second Half-Cycle: D2 and D3 conduct; current again flows through $R_L$ (polarity preserved).
3. Output: Pulsating DC with double the frequency of input AC.

Advantages:

• Higher efficiency (~81% vs. 40% for half-wave).
• No center-tap transformer needed.

Applications:

• Power supplies (chargers, adapters).
• Signal demodulation in radios.
• Battery charging circuits.

(c) Laws of Radioactive Disintegration and Derivation of $N=N_0{e}^{-\lambda t}$

Laws of Radioactive Disintegration

1. Law of Proportionality: The decay rate ($-\frac{dN}{dt}$) is proportional to the number of undecayed nuclei ($N$): $$-\frac{dN}{dt}=\lambda N,$$ where $\lambda$ is the decay constant.
2. Law of Exponential Decay: The number of nuclei decays exponentially with time: $$N=N_0{e}^{-\lambda t}.$$
3. Law of Probability: The probability of decay per nucleus per unit time is constant ($\lambda$).

Derivation of $N=N_0{e}^{-\lambda t}$

Starting from the decay rate: $$-\frac{dN}{dt}=\lambda N.$$ Separate variables and integrate: $${\int }_{N_0}^N\frac{dN}{N}=-\lambda {\int }_0^{t}dt.$$ $$\ln N-\ln N_0=-\lambda t\Rightarrow \ln \left(\frac{N}{N_0}\right)=-\lambda t.$$ Exponentiating both sides: $$\frac{N}{N_0}=e^{-\lambda t}\Rightarrow N=N_0{e}^{-\lambda t}.$$

Key Terms:

• Half-Life ($t_{1/2}$): Time for half the nuclei to decay: $$t_{1/2}=\frac{\ln 2}{\lambda }.$$
• Activity ($A$): Decay rate: $$A=\lambda N.$$

(d) Sources of Energy and Global Consumption Patterns

Sources of Energy

Energy sources are classified into conventional and non-conventional (renewable):

Global Energy Consumption Patterns

1. Primary Consumers (2022 Data):

   • Oil: 33% (transportation, industry).
   • Coal: 27% (electricity generation).
   • Natural Gas: 24% (heating, power).
   • Renewables: 13% (solar, wind, hydro).
   • Nuclear: 4%.

2. Regional Trends:

   • Developed Nations: High per-capita consumption (e.g., USA: 7.5 toe/capita), reliance on gas/nuclear.
   • Developing Nations: Rapid growth (e.g., India/China: 1.5–2.5 toe/capita), coal-dominated.
   • Africa: Low consumption (~0.5 toe/capita) but high potential for solar/hydro.

3. Demand Drivers:

   • Industrialization: Steel, cement, and chemical industries drive fossil fuel demand.
   • Urbanization: Increased electricity use for HVAC, appliances.
   • Transportation: Global vehicle fleet growth (1.4B cars by 2040).
   • Climate Policies: Shift toward renewables (e.g., EU’s 2050 net-zero target).

4. Challenges:

   • Energy Poverty: 770M lack access to electricity (WHO).
   • Grid Stability: Intermittency of renewables requires storage (batteries, pumped hydro).
   • Geopolitical Risks: Oil/gas dependencies (e.g., Russia-Ukraine conflict).

Future Projections:

• Renewables to supply 40% of global energy by 2040 (IEA).
• Hydrogen and fusion as long-term alternatives.
• Energy Efficiency: Critical to meet Paris Agreement goals (limit warming to 1.5°C).