If n, e, `tau`, m, are representing electron density charge, relaxation time and mass of an electron respectively then the resistance of wire of length l and cross sectional area A is given by

Options

A. `(ml)/(n e^(2) tauA)`
B. `(m tau^(2)A)/("ne"^(2)l)`
C. `("ne"^(2)tauA)/(2ml)`
D. `("ne"^(2)A)/(2m taul)`

Answer

To find the resistance \( R \) of a wire of length \( l \) and cross-sectional area \( A \), given the parameters \( n \) (electron density), \( e \) (charge of an electron), \( \tau \) (relaxation time), and \( m \) (mass of an electron), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Relationship**:
The resistance \( R \) of a conductor is defined as the ratio of the potential difference \( V \) across the conductor to the current \( I \) flowing through it:
\[ R = \frac{V}{I} \]

2. **Expression for Current**:
The current \( I \) can be expressed in terms of the electron density \( n \), charge \( e \), cross-sectional area \( A \), and drift velocity \( V_D \):
\[ I = n \cdot e \cdot A \cdot V_D \]

3. **Finding Drift Velocity**:
The drift velocity \( V_D \) can be expressed as:
\[ V_D = \frac{eE\tau}{m} \]
where \( E \) is the electric field. Here, \( E \) can be related to the potential difference \( V \) and length \( l \) of the wire:
\[ E = \frac{V}{l} \]

4. **Substituting Drift Velocity**:
Substituting \( E \) into the expression for \( V_D \):
\[ V_D = \frac{e \cdot \frac{V}{l} \cdot \tau}{m} \]

5. **Substituting Current**:
Now substitute \( V_D \) back into the current equation:
\[ I = n \cdot e \cdot A \cdot \left(\frac{e \cdot \frac{V}{l} \cdot \tau}{m}\right) \]
Simplifying this gives:
\[ I = \frac{n \cdot e^2 \cdot A \cdot V \cdot \tau}{m \cdot l} \]

6. **Finding Resistance**:
Now substituting this expression for \( I \) back into the resistance formula:
\[ R = \frac{V}{I} = \frac{V}{\frac{n \cdot e^2 \cdot A \cdot V \cdot \tau}{m \cdot l}} \]
This simplifies to:
\[ R = \frac{m \cdot l}{n \cdot e^2 \cdot A \cdot \tau} \]

7. **Final Expression**:
Thus, the resistance \( R \) of the wire is given by:
\[ R = \frac{m \cdot l}{n \cdot e^2 \cdot \tau \cdot A} \]

### Final Answer:
The resistance of the wire is:
\[ R = \frac{m \cdot l}{n \cdot e^2 \cdot \tau \cdot A} \]