A wire of length 'l' is stretched to double its length. Compare the resistance of the original and stretched wire.

Steps to Solve

1. Understand the Resistance Formula

The resistance (R) of a 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.

2. Calculate Initial Resistance

Let the initial length of the wire be ( L ) and its cross-sectional area be ( A ). The initial resistance ( R_1 ) can be calculated as:

$$ R_1 = \frac{\rho L}{A} $$

3. Determine the New Dimensions After Stretching

If the wire is stretched to double its original length, the new length ( L' ) will be:

$$ L' = 2L $$

Since the volume of the wire remains constant, the new cross-sectional area ( A' ) can be expressed as:

$$ A' = \frac{V}{L'} = \frac{AL}{2L} = \frac{A}{2} $$

4. Calculate New Resistance

Now, using the new length and new area, the new resistance ( R_2 ) can be calculated as:

$$ R_2 = \frac{\rho L'}{A'} = \frac{\rho (2L)}{\frac{A}{2}} $$

Simplifying this gives:

$$ R_2 = \frac{\rho (2L)}{\frac{A}{2}} = \frac{2\rho L \cdot 2}{A} = \frac{4\rho L}{A} $$

5. Compare Resistances

Now, we compare the new resistance ( R_2 ) with the initial resistance ( R_1 ):

$$ R_2 = 4R_1 $$

This shows that the resistance of the stretched wire is four times the original resistance.