If length of a metallic wire becomes n times, its resistance becomes

Options

A. `n ^(2)` times
B. `sqrtn` times
C. `((1)/(sqrtn)) ` times
D. `((1)/(n ^(2)))` times

Explanation

To solve the problem of how the resistance of a metallic wire changes when its length becomes n times longer, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for Resistance**:
The resistance \( R \) of a metallic wire is given by the formula:
\[ R = \frac{\rho L}{A} \]
where \( \rho \) is the resistivity of the material, \( L \) is the length of the wire, and \( A \) is the cross-sectional area of the wire.

2. **Initial Parameters**:
Let the initial length of the wire be \( L \) and the initial cross-sectional area be \( A \). Therefore, the initial resistance \( R \) can be expressed as:
\[ R = \frac{\rho L}{A} \]

3. **Change in Length**:
According to the problem, the length of the wire is increased to \( nL \) (where \( n \) is a given factor).

4. **Volume Conservation**:
When the length of the wire increases, the volume of the wire remains constant. The volume \( V \) of the wire can be expressed as:
\[ V = A \times L \]
When the length becomes \( nL \), let the new cross-sectional area be \( A' \). The volume can also be expressed as:
\[ V = A' \times (nL) \]
Since the volume is constant, we can equate the two expressions:
\[ A \times L = A' \times (nL) \]

5. **Solving for New Area**:
From the equation above, we can solve for \( A' \):
\[ A' = \frac{A}{n} \]
This shows that when the length increases by a factor of \( n \), the cross-sectional area decreases by the same factor.

6. **New Resistance Calculation**:
Now, we can find the new resistance \( R' \) using the new length and new area:
\[ R' = \frac{\rho (nL)}{A'} \]
Substituting \( A' \):
\[ R' = \frac{\rho (nL)}{\frac{A}{n}} = \frac{\rho (n^2 L)}{A} = n^2 \left(\frac{\rho L}{A}\right) = n^2 R \]

7. **Conclusion**:
Therefore, the new resistance \( R' \) when the length of the wire becomes \( n \) times is:
\[ R' = n^2 R \]

### Final Answer:
If the length of a metallic wire becomes \( n \) times, its resistance becomes \( n^2 \) times the original resistance.