Mobilities of electrons and holes in a sample of intrinsic germanium at room temperature are `0.36 m^(2)//Vs` and `0.17 m^(2)//Vs`. The electron and hole densities are each equal to `2.5 xx 10^(19) m^(-3)`. The electrical conductivity of germanium is.

Options

A. `0.47 S//m`
B. `5.18 S//m`
C. `2.12 S//m`
D. `1.09 S//m`

Answer

To find the electrical conductivity of intrinsic germanium, we can use the formula for conductivity (\( \sigma \)) in terms of electron and hole mobilities and their respective densities:

\[ \sigma = q (n_e \mu_e + n_h \mu_h) \]

Where:
- \( \sigma \) = electrical conductivity
- \( q \) = charge of an electron (\( 1.6 \times 10^{-19} \, C \))
- \( n_e \) = density of electrons
- \( n_h \) = density of holes
- \( \mu_e \) = mobility of electrons
- \( \mu_h \) = mobility of holes

### Step-by-Step Solution:

1. **Identify Given Values**:
- Mobility of electrons (\( \mu_e \)) = \( 0.36 \, m^2/Vs \)
- Mobility of holes (\( \mu_h \)) = \( 0.17 \, m^2/Vs \)
- Density of electrons (\( n_e \)) = \( 2.5 \times 10^{19} \, m^{-3} \)
- Density of holes (\( n_h \)) = \( 2.5 \times 10^{19} \, m^{-3} \)
- Charge of an electron (\( q \)) = \( 1.6 \times 10^{-19} \, C \)

2. **Substitute Values into the Conductivity Formula**:
\[ \sigma = q (n_e \mu_e + n_h \mu_h) \]
Substitute the known values:
\[ \sigma = (1.6 \times 10^{-19}) \left( (2.5 \times 10^{19} \times 0.36) + (2.5 \times 10^{19} \times 0.17) \right) \]

3. **Calculate Each Term**:
- Calculate \( n_e \mu_e \):
\[ n_e \mu_e = 2.5 \times 10^{19} \times 0.36 = 9.0 \times 10^{18} \, m^{-1} \cdot s^{-1} \]
- Calculate \( n_h \mu_h \):
\[ n_h \mu_h = 2.5 \times 10^{19} \times 0.17 = 4.25 \times 10^{18} \, m^{-1} \cdot s^{-1} \]

4. **Add the Two Terms**:
\[ n_e \mu_e + n_h \mu_h = 9.0 \times 10^{18} + 4.25 \times 10^{18} = 13.25 \times 10^{18} \, m^{-1} \cdot s^{-1} \]

5. **Multiply by Charge of Electron**:
\[ \sigma = (1.6 \times 10^{-19}) \times (13.25 \times 10^{18}) \]
\[ \sigma = 2.12 \, S/m \]

### Final Answer:
The electrical conductivity of germanium is \( \sigma = 2.12 \, S/m \).