If a wire having resistance 80 Ω is stretched to double the length, its resistance becomes

Question

Accepted Answer

When a wire is stretched to double its original length, its resistance changes due to the alteration in its physical dimensions. The resistance $ R $ of a wire is given by the formula:

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

where:
- $ \rho $ is the resistivity of the material,
- $ L $ is the length of the wire,
- $ A $ is the cross-sectional area of the wire.

Initially, the wire has a resistance of 80 Ω. When the wire is stretched to double its length, the new length becomes $ 2L $. Stretching the wire also affects its cross-sectional area. Assuming the volume of the wire remains constant, the cross-sectional area $ A $ is inversely proportional to the length $ L $. Therefore, when the length is doubled, the cross-sectional area is halved, i.e., the new area becomes $ \frac{A}{2} $.

Substituting these changes into the resistance formula, the new resistance $ R' $ is:

$$ R' = \rho \frac{2L}{\frac{A}{2}} = \rho \frac{4L}{A} $$

This shows that the new resistance is four times the original resistance. Therefore, the new resistance is:

$$ R' = 4 \times 80 \, \Omega = 320 \, \Omega $$

Thus, the correct answer is Option 3: 320 Ω.

Explanation for why other options are incorrect:

1. **Option 1: 80 Ω** - This option is incorrect because it assumes that the resistance remains unchanged when the wire is stretched. However, stretching the wire to double its length increases the resistance due to the change in dimensions.

2. **Option 2: 160 Ω** - This option is incorrect because it suggests that the resistance only doubles. However, as explained, the resistance actually quadruples due to both the doubling of the length and the halving of the cross-sectional area.

3. **Option 4: 640 Ω** - This option is incorrect because it overestimates the increase in resistance. The resistance quadruples, not increases eightfold, when the wire is stretched to double its length.

Therefore, the correct answer is indeed Option 3: 320 Ω.