If the ratio of the concentration of electron to that of holes in a semiconductor is `(7)/(5)` and the ratio of current is `(7)/(4)` then what is the ratio of their drift velocities ?

Options

A. `4//7`
B. `5//8`
C. `4//5`
D. `5//4`

Answer

To solve the problem, we need to find the ratio of the drift velocities of electrons and holes in a semiconductor given the ratios of their concentrations and currents.

### Step-by-Step Solution:

1. **Understand the Given Ratios:**
- The ratio of the concentration of electrons (n_e) to that of holes (n_h) is given as:
\[ \frac{n_e}{n_h} = \frac{7}{5} \]
- The ratio of the current due to electrons (I_e) to the current due to holes (I_h) is given as:
\[ \frac{I_e}{I_h} = \frac{7}{4} \]

2. **Write the Current Equations:**
- The current due to electrons can be expressed as:
\[ I_e = n_e \cdot e \cdot A \cdot v_{de} \]
- The current due to holes can be expressed as:
\[ I_h = n_h \cdot e \cdot A \cdot v_{dh} \]
- Here, \(e\) is the charge of an electron, \(A\) is the cross-sectional area, and \(v_{de}\) and \(v_{dh}\) are the drift velocities of electrons and holes, respectively.

3. **Formulate the Ratio of Currents:**
- Now, we can write the ratio of the currents:
\[ \frac{I_e}{I_h} = \frac{n_e \cdot e \cdot A \cdot v_{de}}{n_h \cdot e \cdot A \cdot v_{dh}} = \frac{n_e \cdot v_{de}}{n_h \cdot v_{dh}} \]

4. **Substituting the Given Ratios:**
- From the given ratios, we substitute:
\[ \frac{7}{4} = \frac{n_e \cdot v_{de}}{n_h \cdot v_{dh}} \]
- We also know that:
\[ \frac{n_e}{n_h} = \frac{7}{5} \]
- Therefore, we can substitute \(n_e\) in terms of \(n_h\):
\[ n_e = \frac{7}{5} n_h \]
- Substitute this into the current ratio:
\[ \frac{7}{4} = \frac{\left(\frac{7}{5} n_h\right) \cdot v_{de}}{n_h \cdot v_{dh}} \]

5. **Simplifying the Equation:**
- Cancel \(n_h\) from both sides:
\[ \frac{7}{4} = \frac{\frac{7}{5} \cdot v_{de}}{v_{dh}} \]
- Rearranging gives:
\[ \frac{7}{4} = \frac{7 \cdot v_{de}}{5 \cdot v_{dh}} \]

6. **Cross-Multiplying to Solve for the Drift Velocities:**
- Cross-multiplying gives:
\[ 7 \cdot v_{dh} = 4 \cdot \frac{7}{5} \cdot v_{de} \]
- Simplifying further:
\[ v_{dh} = \frac{4}{5} v_{de} \]

7. **Finding the Ratio of Drift Velocities:**
- Therefore, the ratio of drift velocities is:
\[ \frac{v_{de}}{v_{dh}} = \frac{5}{4} \]

### Final Answer:
The ratio of the drift velocities of electrons to holes is:
\[ \frac{v_{de}}{v_{dh}} = \frac{5}{4} \]