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

Options

A. `n ^(2)` times
B. `((1)/( n ^(2)))` times
C. `n ^(4)` times
D. `((1)/(n ^(4)))` times

Answer

To determine how the resistance of a metallic wire changes when its radius is increased by a factor of \( n \), we can follow these steps:

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

### Step 2: Express the cross-sectional area in terms of radius
The cross-sectional area \( A \) of a circular wire can be expressed as:
\[ A = \pi r^2 \]
where \( r \) is the radius of the wire.

### Step 3: Analyze the change in radius
If the radius of the wire becomes \( n \) times its original radius, we can express the new radius as:
\[ r' = n \cdot r \]
Substituting this into the area formula gives:
\[ A' = \pi (r')^2 = \pi (n \cdot r)^2 = \pi n^2 r^2 = n^2 A \]

### Step 4: Determine the new resistance
Now, substituting the new area \( A' \) into the resistance formula, we have:
\[ R' = \frac{\rho L}{A'} = \frac{\rho L}{n^2 A} \]
This can be rewritten as:
\[ R' = \frac{R}{n^2} \]

### Step 5: Conclusion
Thus, when the radius of the wire is increased by a factor of \( n \), the new resistance \( R' \) becomes:
\[ R' = \frac{R}{n^2} \]

### Final Answer
The resistance becomes \( \frac{1}{n^2} \) times the original resistance.

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