Question

What happens to the resistance of a wire if its radius is halved?

• A. It becomes half
• B. It becomes double
• C. It increases 4 times
• D. It increases 16 times

Hint:

Resistance is inversely proportional to the square of the radius. Small changes in radius cause large changes in resistance. If radius is halved, resistance becomes \(4\) times; if radius is doubled, resistance becomes \(\frac{1}{4}\).

Answer 1

The Correct Option is C

Concept: The resistance of a wire is given by the formula: \[ R = \rho \frac{L}{A} \] where:
• \(R\) = resistance
• \(\rho\) = resistivity of the material
• \(L\) = length of the wire
• \(A\) = cross-sectional area For a cylindrical wire: \[ A = \pi r^2 \] Thus, resistance is inversely proportional to the square of the radius: \[ R \propto \frac{1}{r^2} \]

Step 1: Express resistance in terms of radius. \[ R \propto \frac{1}{r^2} \]

Step 2: Apply the condition that radius is halved. If the new radius is: \[ r' = \frac{r}{2} \] Then, \[ R' \propto \frac{1}{(r/2)^2} = \frac{1}{r^2/4} = \frac{4}{r^2} \]

Step 3: Compare new and original resistance. \[ \frac{R'}{R} = \frac{4}{1} = 4 \] Thus, the resistance becomes 4 times the original value. \[ \boxed{\text{Resistance increases 4 times}} \]