Question

I. Define resistivity.
II. Justify with an example why resistivity is an intrinsic property of a material.

Hint:

Start from \(R = \rho L / A\) and see what happens to \(RA/L\) when the wire size changes.

Answer 1

Step 1: Part I: Define resistivity:
The resistance of a wire depends on its length \(L\) and cross-sectional area \(A\). It is given by
\[ R = \frac{\rho L}{A} \]
where \(\rho\) is a constant for the material. This \(\rho\) is called the resistivity. It is the resistance of a wire of the material that has unit length and unit cross-sectional area.

Step 2: Find the unit:
Rearranging gives \(\rho = \dfrac{RA}{L}\). So the unit is ohm times metre squared divided by metre, which is ohm metre (\(\Omega\,\text{m}\)).

Step 3: Part II: Take an example:
Take two copper wires. Wire 1 has \(L = 1\) m and \(A = 1\) mm\(^2\) = \(10^{-6}\) m\(^2\), and its resistance is about \(1.7 \times 10^{-2}\ \Omega\). Wire 2 has \(L = 2\) m and the same area, and its resistance is about \(3.4 \times 10^{-2}\ \Omega\).

Step 4: Calculate rho for both wires:
Wire 1: \(\rho = \dfrac{RA}{L} = \dfrac{1.7 \times 10^{-2} \times 10^{-6}}{1} = 1.7 \times 10^{-8}\ \Omega\,\text{m}\).
Wire 2: \(\rho = \dfrac{3.4 \times 10^{-2} \times 10^{-6}}{2} = 1.7 \times 10^{-8}\ \Omega\,\text{m}\).

Step 5: Conclusion:
The resistance changed when the length changed, but the resistivity stayed the same. So resistivity depends only on the material (and on temperature), not on the size or shape of the wire. That is why it is an intrinsic property of a material.

Final Answer:
Resistivity \(\rho = \dfrac{RA}{L}\) depends only on the material, so it is an intrinsic property. \[ \boxed{\rho = \dfrac{RA}{L}} \]