Answer any three questions: a) What is zener diode? Explain its use as a voltage regulator. b) Discuss photoelectric effect and derive Einstein's photoelectric equation. What is stopping potential?…

a) Zener Diode and Its Use as a Voltage Regulator

What is a Zener Diode?

A Zener diode is a specially designed semiconductor diode that allows current to flow in the reverse direction when a certain reverse voltage (called the Zener breakdown voltage, $V_Z$) is applied across it. Unlike a regular diode, which is damaged by reverse voltages beyond its breakdown voltage, a Zener diode is operated in the reverse-bias breakdown region for voltage regulation.

Construction and Symbol

• It is constructed by heavily doping the p-type and n-type regions.
• The depletion region is very thin, allowing electrons to tunnel through under reverse bias.
• The symbol is identical to a regular diode but is often labeled with a Zener voltage (e.g., $V_Z=5.1\text{V}$).

V-I Characteristics of a Zener Diode

The current-voltage (I-V) characteristic of a Zener diode shows:

1. Forward bias: Acts like a normal diode (conducts after $V_{on}$).
2. Reverse bias:
   • Initially, a small reverse saturation current ($I_s$) flows.
   • Beyond $V_Z$, the current suddenly increases due to Zener breakdown (for $V_Z<5\text{V}$) or avalanche breakdown (for $V_Z>5\text{V}$).

Zener Diode as a Voltage Regulator

A Zener diode maintains a constant output voltage across a load, even when the input voltage or load current varies. This is achieved using a Zener diode in parallel with the load and a resistor in series (called a current-limiting resistor, $R_s$).

Circuit Diagram

Working Principle

1. Input voltage ($V_{in}$) is applied across $R_s$ and the Zener diode.
2. When $V_{in}$ exceeds $V_Z$, the Zener diode conducts in reverse, maintaining $V_{out}=V_Z$.
3. If $V_{in}$ increases, the excess voltage is dropped across $R_s$, keeping $V_{out}$ constant.
4. If the load current ($I_L$) changes, the Zener diode adjusts its current to compensate, ensuring $V_{out}$ remains stable.

Conditions for Proper Regulation

1. Minimum input voltage ($V_{in(min)}$): $$V_{in(min)}=V_Z+I_L\cdot R_s$$ (Ensures enough voltage to maintain $V_Z$ even at maximum load current.)

2. Maximum input voltage ($V_{in(max)}$): $$V_{in(max)}=V_Z+(I_Z+I_L{)}_{max}\cdot R_s$$ (Prevents excessive power dissipation in the Zener diode.)

3. Power dissipation in Zener diode ($P_Z$): $$P_Z=(V_{in}-V_Z)\cdot I_Z$$ (Must be within the maximum power rating of the diode.)

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b) Photoelectric Effect and Einstein’s Photoelectric Equation

Photoelectric Effect

The photoelectric effect is the emission of electrons (called photoelectrons) from a metal surface when light (or electromagnetic radiation) of sufficient frequency is incident on it.

Key Observations (Experimental Findings)

1. Threshold Frequency ($f_0$):

   • No electrons are emitted if the frequency of light is below a certain minimum value ($f_0$).
   • This depends on the material of the metal.

2. Instantaneous Emission:

   • Photoelectrons are emitted instantaneously (within $10^{-9}$ s), even at very low light intensity.

3. Kinetic Energy of Photoelectrons:

   • The maximum kinetic energy ($K_{max}$) of emitted electrons depends on the frequency of light, not its intensity.
   • Given by: $$K_{max}=\frac{1}{2}m{v}_{max}^2=hf-\varphi$$ where:
     • $h$ = Planck’s constant ($6.626\times 10^{-34}\text{J s}$)
     • $f$ = frequency of incident light
     • $\varphi$ = work function of the metal (minimum energy required to remove an electron)

4. Intensity of Light:

   • Increases the number of photoelectrons but does not affect their maximum kinetic energy.

Einstein’s Photoelectric Equation

Einstein explained the photoelectric effect by proposing that light consists of discrete packets of energy called photons, each with energy: $$E=hf$$ where:

• $h$ = Planck’s constant
• $f$ = frequency of light

When a photon strikes the metal surface:

• If $hf\ge \varphi$, the electron is ejected with maximum kinetic energy: $$K_{max}=hf-\varphi$$
• If $hf<\varphi$, no emission occurs (regardless of light intensity).

Derivation of Einstein’s Photoelectric Equation

1. Work Function ($\varphi$): The minimum energy required to remove an electron from the metal surface is called the work function: $$\varphi =h{f}_0$$ where $f_0$ is the threshold frequency.

2. Energy Conservation: The energy of the incident photon ($hf$) is used to:

   • Overcome the work function ($\varphi$).
   • Provide the maximum kinetic energy to the ejected electron ($K_{max}$). $$hf=\varphi +K_{max}$$ $$K_{max}=hf-\varphi$$

3. Stopping Potential ($V_0$): The stopping potential is the minimum negative potential applied to the anode that stops the most energetic photoelectrons from reaching it.

   • At stopping potential, the maximum kinetic energy of the photoelectrons is just enough to overcome the potential barrier: $$e{V}_0=K_{max}$$ $$V_0=\frac{K_{max}}{e}=\frac{hf-\varphi }{e}$$
   • Measuring $V_0$ allows us to determine $K_{max}$ and verify Einstein’s equation.

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c) Mass Defect and Binding Energy of a Nucleus

Mass Defect

When a nucleus is formed from its constituent protons and neutrons, the actual mass of the nucleus ($m_n$) is less than the sum of the masses of its individual nucleons. This missing mass is called the mass defect ($\Delta m$).

$$\Delta m=[Z{m}_p+(A-Z)m_n]-m_n$$ where:

• $Z$ = number of protons
• $A$ = mass number (total nucleons)
• $m_p$ = mass of a proton ($1.007276\text{u}$)
• $m_n$ = mass of a neutron ($1.008665\text{u}$)
• $m_n$ = actual mass of the nucleus

Binding Energy

The mass defect is converted into binding energy ($E_b$), which is the energy required to disassemble the nucleus into its individual protons and neutrons. According to Einstein’s mass-energy equivalence: $$E_b=\Delta m\cdot c^2$$ where:

• $c$ = speed of light ($3\times 10^8\text{m/s}$)
• $1\text{u}=931.5{\text{MeV/c}}^2$

Thus, binding energy can also be expressed in MeV: $$E_b(\text{MeV})=\Delta m(\text{u})\times 931.5$$

Binding Energy per Nucleon

The binding energy per nucleon ($E_b/A$) is obtained by dividing the total binding energy by the mass number ($A$): $$\frac{E_b}{A}=\frac{\Delta m\cdot c^2}{A}$$ This quantity helps compare the stability of different nuclei.

Graph of Binding Energy per Nucleon vs. Atomic Number

The graph shows how $E_b/A$ varies with atomic number ($A$) for different nuclei.

Interpretation of the Graph

1. Peak at Iron ($A\approx 56$):

   • Nuclei around iron (⁵⁶Fe) have the highest binding energy per nucleon (~8.8 MeV).
   • This means they are the most stable (require the most energy to break apart).

2. Light Nuclei (H, He, Li):

   • Low $E_b/A$ (~1–7 MeV).
   • Fusion (combining light nuclei) releases energy (e.g., H → He in stars).

3. Heavy Nuclei (U, Pu, Th):

   • Lower $E_b/A$ (~7–8 MeV).
   • Fission (splitting heavy nuclei) releases energy (e.g., U-235 → smaller nuclei + energy).

4. Nuclei with Very Low $E_b/A$:

   • Unstable and undergo radioactive decay to move toward higher stability.

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d) Renewable and Non-Renewable Sources of Energy

Comparison of Renewable and Non-Renewable Energy Sources

Examples of Renewable and Non-Renewable Sources

Energy Consumption Scenario in Nepal

Nepal’s energy consumption is heavily dependent on non-renewable and traditional sources, but there is a growing shift toward renewables due to limited fossil fuel reserves and environmental concerns.

Current Energy Mix in Nepal (Approximate)

Challenges in Nepal’s Energy Sector

1. Hydropower Dependence:

   • Seasonal variability (droughts reduce generation).
   • Transmission losses (~20–30% due to poor infrastructure).
   • Political and financial delays in project completion.

2. Reliance on Firewood:

   • Deforestation and environmental degradation.
   • Indoor air pollution (major health hazard).

3. Limited Access to Electricity:

   • ~85% rural electrification, but frequent blackouts.
   • Remote areas still depend on kerosene and biomass.

4. Fossil Fuel Imports:

   • Nepal imports ~90% of its oil, causing foreign exchange drain.
   • Price volatility affects the economy.

Future Prospects and Renewable Energy Potential

Nepal has enormous potential for renewable energy, particularly:

1. Hydropower Expansion:

   • Untapped potential: ~60,000 MW (only ~2,000 MW utilized).
   • Projects: West Seti, Pancheshwar, Budhi Gandaki (under construction).

2. Solar Energy:

   • High solar insolation (~5–6 kWh/m²/day).
   • Government incentives for solar home systems.

3. Wind Energy:

   • Potential in Tarai and hilly regions (~10,000 MW).
   • First wind farm: Lower Mustang (5 MW).

4. Biomass and Biogas:

   • Agro-waste and animal dung can be converted to biogas.
   • Reduces firewood dependence.

5. Geothermal Energy:

   • Himalayan region has high geothermal potential (~7,000 MW).
   • Pilot projects in Chautara and Jorpati.

Government Initiatives

• Alternative Energy Promotion Centre (AEPC) promotes renewables.
• National Energy Policy (2019) aims for 30% renewable energy by 2030.
• Subsidies for solar panels, biogas plants, and electric vehicles.

Conclusion

Nepal’s energy future lies in diversifying from hydropower and firewood to solar, wind, and geothermal. While hydropower remains dominant, solar and wind are the most immediately scalable solutions. Policy support, infrastructure development, and foreign investment are crucial for achieving sustainable and reliable energy access for all Nepalis.