The electric resistance of a certain wire of iron is R . If its length and radius are both doubled, then

Options

A. Will be doubled and the specific resistance will be halved
B. Will be halved and the specific resistance will remain unchanged
C. Will be halved and the specific resistance will be doubled
D. And the specific resistance both will remain unchanged

Explanation

To solve the problem, we need to analyze how the resistance of a wire changes when both its length and radius are doubled. The resistance \( R \) of a 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 1: Initial Resistance Calculation
Let's denote the initial length as \( L \) and the initial radius as \( r \). The cross-sectional area \( A \) of the wire can be expressed as:

\[ A = \pi r^2 \]

Thus, the initial resistance can be expressed as:

\[ R = \frac{\rho L}{\pi r^2} \]

### Step 2: New Dimensions
According to the problem, both the length and radius are doubled. Therefore, the new length \( L' \) and new radius \( r' \) are:

\[ L' = 2L \]
\[ r' = 2r \]

### Step 3: New Cross-Sectional Area
Now, we need to calculate the new cross-sectional area \( A' \):

\[ A' = \pi (r')^2 = \pi (2r)^2 = \pi (4r^2) = 4\pi r^2 \]

### Step 4: New Resistance Calculation
Now we can calculate the new resistance \( R' \):

\[ R' = \frac{\rho L'}{A'} = \frac{\rho (2L)}{4\pi r^2} \]

Simplifying this expression gives:

\[ R' = \frac{2\rho L}{4\pi r^2} = \frac{\rho L}{2\pi r^2} = \frac{R}{2} \]

### Conclusion
Thus, the new resistance \( R' \) is half of the initial resistance \( R \):

\[ R' = \frac{R}{2} \]

### Summary of Results
- The resistance of the wire when both the length and radius are doubled is \( \frac{R}{2} \).
- The resistivity \( \rho \) remains unchanged as it is a property of the material.

### Final Answer
The resistance of the wire after doubling both the length and the radius is \( \frac{R}{2} \).

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