Mobilities of electorns and holes in a sample of intrinsic germanium at room temperature are `0.54m^(2)V^(-1)s^(-1)` and `0.18m^(2)V^(-1)s^(-1)` respectively.
If the electron and hole densities are equal to `3.6xx10^(19)m^(-3)` calculate the germanium conductivity.

Options

A. `4.14 S m^(-1)`
B. `2.12 S m^(-1)`
C. `1.13 S m^(-1)`
D. `5.6 S m^(-1)`

Explanation

To calculate the conductivity of intrinsic germanium, we can use the formula for conductivity (\( \sigma \)) in terms of the mobilities of electrons and holes and their respective densities:

\[ \sigma = q (n \mu_n + p \mu_p) \]

Where:
- \( \sigma \) = conductivity
- \( q \) = charge of an electron (approximately \( 1.6 \times 10^{-19} \, \text{C} \))
- \( n \) = density of electrons
- \( p \) = density of holes
- \( \mu_n \) = mobility of electrons
- \( \mu_p \) = mobility of holes

### Step-by-Step Solution:

1. **Identify Given Values:**
- Mobility of electrons, \( \mu_n = 0.54 \, \text{m}^2/\text{V}\cdot\text{s} \)
- Mobility of holes, \( \mu_p = 0.18 \, \text{m}^2/\text{V}\cdot\text{s} \)
- Density of electrons, \( n = 3.6 \times 10^{19} \, \text{m}^{-3} \)
- Density of holes, \( p = 3.6 \times 10^{19} \, \text{m}^{-3} \)

2. **Substitute Values into the Conductivity Formula:**
\[ \sigma = q (n \mu_n + p \mu_p) \]
\[ \sigma = (1.6 \times 10^{-19}) \left( (3.6 \times 10^{19}) \times 0.54 + (3.6 \times 10^{19}) \times 0.18 \right) \]

3. **Factor Out Common Terms:**
Since \( n = p \), we can factor out \( n \):
\[ \sigma = q n (\mu_n + \mu_p) \]
\[ \sigma = (1.6 \times 10^{-19}) (3.6 \times 10^{19}) \left( 0.54 + 0.18 \right) \]

4. **Calculate the Sum of Mobilities:**
\[ \mu_n + \mu_p = 0.54 + 0.18 = 0.72 \, \text{m}^2/\text{V}\cdot\text{s} \]

5. **Substitute Back into the Formula:**
\[ \sigma = (1.6 \times 10^{-19}) (3.6 \times 10^{19}) (0.72) \]

6. **Calculate the Product:**
\[ \sigma = (1.6 \times 3.6 \times 0.72) \times 10^{-19 + 19} \]
\[ \sigma = (3.6864) \, \text{S/m} \]

7. **Final Calculation:**
\[ \sigma \approx 3.69 \, \text{S/m} \]

### Final Answer:
The conductivity of intrinsic germanium at room temperature is approximately \( 3.69 \, \text{S/m} \).