Question

Find new resistivity when radius doubles and length reduces by half of its original resistivity is $\rho$.

• A. halved
• B. unchanged
• C. doubled
• D. tripled
• E. quadrupled

Hint:

Read carefully! "Resistance" changes with dimensions. "Resistivity" (specific resistance) only changes if you change the material itself or its temperature.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
It is crucial to distinguish between resistance (\(R\)) and resistivity (\(\rho\)).
- Resistance is an \textit{extensive} property; it depends on the physical dimensions of the object (length, cross-sectional area).
- Resistivity is an \textit{intensive} property; it is an intrinsic characteristic of the material itself (e.g., copper vs. iron) and depends only on the temperature and the microscopic structure of the material.
Step 2: Key Formula or Approach:
The formula relating the two is \(R = \rho \frac{L}{A}\). While resistance \(R\) changes with length \(L\) and area \(A\), the resistivity \(\rho\) is the constant of proportionality that describes the material.
Step 3: Detailed Explanation:
The problem states that the physical dimensions of the wire are altered:
- Radius doubles (\(r \to 2r\))
- Length reduces by half (\(L \to L/2\))
If the question asked for the new resistance, we would calculate it using the new dimensions.
However, the question specifically asks for the new resistivity.
Because cutting, stretching, or reshaping a material does not change what the material is made of, its intrinsic properties remain identical (assuming temperature stays constant). A piece of copper has the same resistivity whether it is a long thin wire or a short thick block.
Therefore, the resistivity \(\rho\) does not change with changes in physical dimensions.
Step 4: Final Answer:
The resistivity remains unchanged.