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

Options

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

Explanation

To find the ratio of drift velocities of electrons and holes in a semiconductor, we can use the relationship between current, charge, concentration, and drift velocity.

### Step-by-Step Solution:

1. **Understanding the Current Formula**:
The current (I) in a semiconductor can be expressed as:
\[ I = n \cdot A \cdot e \cdot v_d \]
where:
- \( n \) is the concentration of charge carriers (electrons or holes),
- \( A \) is the cross-sectional area,
- \( e \) is the charge of the carriers,
- \( v_d \) is the drift velocity of the carriers.

2. **Setting Up Ratios**:
Given:
- The ratio of the concentration of electrons (\( n_e \)) to holes (\( n_h \)) is \( \frac{n_e}{n_h} = \frac{7}{5} \).
- The ratio of the currents (\( I_e \) for electrons and \( I_h \) for holes) is \( \frac{I_e}{I_h} = \frac{7}{4} \).

3. **Expressing Current for Electrons and Holes**:
For electrons:
\[ I_e = n_e \cdot A \cdot e \cdot v_{de} \]
For holes:
\[ I_h = n_h \cdot A \cdot e \cdot v_{dh} \]

4. **Taking the Ratio of Currents**:
The ratio of the currents can be expressed as:
\[ \frac{I_e}{I_h} = \frac{n_e \cdot v_{de}}{n_h \cdot v_{dh}} \]

5. **Substituting Known Ratios**:
Substitute the known ratios into the equation:
\[ \frac{7}{4} = \frac{n_e \cdot v_{de}}{n_h \cdot v_{dh}} \]
Using \( \frac{n_e}{n_h} = \frac{7}{5} \), we can substitute:
\[ \frac{7}{4} = \frac{\frac{7}{5} \cdot v_{de}}{v_{dh}} \]

6. **Cross-Multiplying**:
Cross-multiply to solve for the ratio of drift velocities:
\[ 7 \cdot v_{dh} = 4 \cdot \frac{7}{5} \cdot v_{de} \]
Simplifying gives:
\[ 7 \cdot v_{dh} = \frac{28}{5} \cdot v_{de} \]

7. **Finding the Ratio of Drift Velocities**:
Rearranging the equation gives:
\[ \frac{v_{de}}{v_{dh}} = \frac{5}{4} \]

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