The number of free electrons per unit volume in a material is n and when electric field E is applied, these electrons are found to aquire a drift velocity of `v_(d)`. What will be the resistivity of the material?

Options

A. `(E)/(n ev_(d))`
B. `(nE)/(ev_(d))`
C. `(eE)/(nv_(d))`
D. `(v_(d))/(n eE)`

Explanation

To find the resistivity of the material given the number of free electrons per unit volume \( n \), the electric field \( E \), and the drift velocity \( v_d \), we can follow these steps:

### Step 1: Understand the relationship between drift velocity and electric field
The drift velocity \( v_d \) of electrons in a conductor can be expressed using the formula:
\[ v_d = \frac{eE}{m} \tau \]
where:
- \( e \) is the charge of an electron,
- \( E \) is the electric field,
- \( m \) is the mass of an electron,
- \( \tau \) is the relaxation time.

### Step 2: Rearranging the formula for relaxation time
From the equation above, we can express \( \tau \) as:
\[ \tau = \frac{m v_d}{e E} \]

### Step 3: Write the formula for resistivity
The resistivity \( \rho \) of a material is given by the formula:
\[ \rho = \frac{m}{n e^2 \tau} \]
where:
- \( n \) is the number of free electrons per unit volume,
- \( e \) is the charge of an electron.

### Step 4: Substitute the expression for \( \tau \) into the resistivity formula
Now, we can substitute the expression for \( \tau \) into the resistivity formula:
\[ \rho = \frac{m}{n e^2 \left(\frac{m v_d}{e E}\right)} \]

### Step 5: Simplify the expression
Now, simplify the equation:
\[ \rho = \frac{m \cdot e E}{n e^2 v_d} \]
The \( m \) cancels out:
\[ \rho = \frac{E}{n e v_d} \]

### Final Result
Thus, the resistivity \( \rho \) of the material is given by:
\[ \rho = \frac{E}{n e v_d} \]