Answer :
Certainly! Let's go through each part of the question step by step.
### 11.a) Hubble's Law and its Significance
Hubble's Law:
Hubble's Law is a fundamental observation in cosmology which states that the velocity at which a galaxy is receding from the Earth is directly proportional to its distance from us. Mathematically, it is expressed as:
[tex]\[ v = H_0 \cdot d \][/tex]
where:
- [tex]\( v \)[/tex] is the recessional velocity of the galaxy,
- [tex]\( H_0 \)[/tex] is the Hubble constant (with units of velocity per unit distance, e.g., km/s/Mpc),
- [tex]\( d \)[/tex] is the distance to the galaxy.
Significance:
Hubble's Law is significant because it provides strong evidence for the expansion of the universe. The law suggests that galaxies are moving away from us, and the universe is expanding in all directions uniformly. This observation supports the Big Bang theory, which posits that the universe began from an extremely dense and hot state and has been expanding over time.
### 11.b) Doping and N-type Semiconductor
Doping:
Doping is the process of intentionally introducing impurities into an intrinsic (pure) semiconductor to modify its electrical properties. The impurities added are called dopants. Depending on the type of dopant, the semiconductor can become either an N-type or a P-type semiconductor.
- N-type Semiconductor: An N-type semiconductor is formed by doping an intrinsic semiconductor (like silicon) with a pentavalent impurity (like phosphorus, arsenic, or antimony), which has five valence electrons. The additional electron provided by the pentavalent atom becomes a free carrier of electric charge, increasing the conductivity of the semiconductor.
Crystal Lattice of N-type Semiconductor:
In an N-type semiconductor, the structure would look like the intrinsic silicon lattice but with some silicon atoms replaced by pentavalent dopant atoms.
```
Si Si Si
| | |
Si — P — Si — Si
| | |
Si Si Si
```
N-type Semiconductor Lattice
Energy Band Diagram of an N-type Semiconductor at Room Temperature:
```
Conduction Band (CB)
| |
|-----------|--------------------> Donor Level (close to CB)
| |
|-----------|--------------------> Fermi Level (somewhere below CB but above intrinsic level)
| |
| |
|-----------|--------------------> Valence Band (VB)
```
### 11.c) Critical Density of the Universe
What Happens When the Average Density of the Universe Reaches the Critical Density:
When the average density of the universe reaches the critical density, the universe is said to have a flat geometry (Euclidean geometry). This implies that the universe would expand forever at a rate that asymptotically approaches zero, and it would not re-collapse on itself nor expand indefinitely at an accelerating rate.
Derivation of the Expression for the Critical Density:
The critical density ([tex]\( \rho_c \)[/tex]) is derived from the Friedmann equation, which describes the expansion of the universe. The Friedmann equation is given by:
[tex]\[ H^2 = \frac{8 \pi G}{3} \rho - \frac{k}{a^2} \][/tex]
where:
- [tex]\( H \)[/tex] is the Hubble constant,
- [tex]\( G \)[/tex] is the gravitational constant,
- [tex]\( \rho \)[/tex] is the density of the universe,
- [tex]\( k \)[/tex] is the curvature parameter,
- [tex]\( a \)[/tex] is the scale factor of the universe.
For a flat universe ([tex]\( k = 0 \)[/tex]),
[tex]\[ H^2 = \frac{8 \pi G}{3} \rho \][/tex]
The critical density ([tex]\( \rho_c \)[/tex]) is defined as the density needed for a flat universe, so:
[tex]\[ H^2 = \frac{8 \pi G}{3} \rho_c \][/tex]
Solving for [tex]\( \rho_c \)[/tex],
[tex]\[ \rho_c = \frac{3 H^2}{8 \pi G} \][/tex]
This critical density is the density at which the universe is perfectly balanced between continued expansion and potential collapse.
### 11.a) Hubble's Law and its Significance
Hubble's Law:
Hubble's Law is a fundamental observation in cosmology which states that the velocity at which a galaxy is receding from the Earth is directly proportional to its distance from us. Mathematically, it is expressed as:
[tex]\[ v = H_0 \cdot d \][/tex]
where:
- [tex]\( v \)[/tex] is the recessional velocity of the galaxy,
- [tex]\( H_0 \)[/tex] is the Hubble constant (with units of velocity per unit distance, e.g., km/s/Mpc),
- [tex]\( d \)[/tex] is the distance to the galaxy.
Significance:
Hubble's Law is significant because it provides strong evidence for the expansion of the universe. The law suggests that galaxies are moving away from us, and the universe is expanding in all directions uniformly. This observation supports the Big Bang theory, which posits that the universe began from an extremely dense and hot state and has been expanding over time.
### 11.b) Doping and N-type Semiconductor
Doping:
Doping is the process of intentionally introducing impurities into an intrinsic (pure) semiconductor to modify its electrical properties. The impurities added are called dopants. Depending on the type of dopant, the semiconductor can become either an N-type or a P-type semiconductor.
- N-type Semiconductor: An N-type semiconductor is formed by doping an intrinsic semiconductor (like silicon) with a pentavalent impurity (like phosphorus, arsenic, or antimony), which has five valence electrons. The additional electron provided by the pentavalent atom becomes a free carrier of electric charge, increasing the conductivity of the semiconductor.
Crystal Lattice of N-type Semiconductor:
In an N-type semiconductor, the structure would look like the intrinsic silicon lattice but with some silicon atoms replaced by pentavalent dopant atoms.
```
Si Si Si
| | |
Si — P — Si — Si
| | |
Si Si Si
```
N-type Semiconductor Lattice
Energy Band Diagram of an N-type Semiconductor at Room Temperature:
```
Conduction Band (CB)
| |
|-----------|--------------------> Donor Level (close to CB)
| |
|-----------|--------------------> Fermi Level (somewhere below CB but above intrinsic level)
| |
| |
|-----------|--------------------> Valence Band (VB)
```
### 11.c) Critical Density of the Universe
What Happens When the Average Density of the Universe Reaches the Critical Density:
When the average density of the universe reaches the critical density, the universe is said to have a flat geometry (Euclidean geometry). This implies that the universe would expand forever at a rate that asymptotically approaches zero, and it would not re-collapse on itself nor expand indefinitely at an accelerating rate.
Derivation of the Expression for the Critical Density:
The critical density ([tex]\( \rho_c \)[/tex]) is derived from the Friedmann equation, which describes the expansion of the universe. The Friedmann equation is given by:
[tex]\[ H^2 = \frac{8 \pi G}{3} \rho - \frac{k}{a^2} \][/tex]
where:
- [tex]\( H \)[/tex] is the Hubble constant,
- [tex]\( G \)[/tex] is the gravitational constant,
- [tex]\( \rho \)[/tex] is the density of the universe,
- [tex]\( k \)[/tex] is the curvature parameter,
- [tex]\( a \)[/tex] is the scale factor of the universe.
For a flat universe ([tex]\( k = 0 \)[/tex]),
[tex]\[ H^2 = \frac{8 \pi G}{3} \rho \][/tex]
The critical density ([tex]\( \rho_c \)[/tex]) is defined as the density needed for a flat universe, so:
[tex]\[ H^2 = \frac{8 \pi G}{3} \rho_c \][/tex]
Solving for [tex]\( \rho_c \)[/tex],
[tex]\[ \rho_c = \frac{3 H^2}{8 \pi G} \][/tex]
This critical density is the density at which the universe is perfectly balanced between continued expansion and potential collapse.
