We are given three copper wires of different lengths and different areas of cross-section. Which one of the following would have highest resistivity ?
All the wires would have same resistivity
Understanding Resistivity of Copper Wires
The question asks about the resistivity of three different copper wires, each having a different length and a different area of cross-section (implied by diameter). We need to determine which wire would have the highest resistivity or if they would all have the same resistivity.
What is Resistivity?
Resistivity is a fundamental property of a material that quantifies how strongly it resists the flow of electric current. It is an intrinsic property, meaning it depends only on the material itself and its temperature, not on its shape or size.
The formula relating resistance ($R$) of a conductor to its resistivity ($\rho$), length ($L$), and area of cross-section ($A$) is:
\begin{equation} R = \rho \frac{L}{A} \end{equation}
From this formula, resistivity can be expressed as:
\begin{equation} \rho = R \frac{A}{L} \end{equation}
While resistance depends on the dimensions ($L$ and $A$), resistivity ($\rho$) is a constant value for a given material at a specific temperature.
Analyzing the Copper Wires
We are given three wires made of copper:
- Copper wire of 50 cm length and 1 mm diameter
- Copper wire of 25 cm length and 0.5 mm diameter
- Copper wire of 10 cm length and 2.0 mm diameter
All three wires are made of the same material, which is copper. Even though their lengths and diameters (and thus areas of cross-section) are different, the material itself remains copper.
Conclusion on Resistivity
Since resistivity is a property of the material and not the dimensions, all wires made of the same material at the same temperature will have the same resistivity. Assuming the temperature is the same for all three wires, their resistivity will be identical.
Therefore, none of the individual wires will have a higher resistivity than the others; they will all possess the same resistivity value characteristic of copper.
| Property | Depends On | Does NOT Depend On |
|---|---|---|
| Resistance (R) | Material (resistivity), Length (L), Area of Cross-section (A), Temperature | — |
| Resistivity ($\rho$) | Material Type, Temperature | Length (L), Area of Cross-section (A), Shape |
Revision Table: Resistivity vs. Resistance
| Feature | Resistivity ($\rho$) | Resistance (R) |
|---|---|---|
| Definition | Intrinsic property measuring how strongly a material opposes current flow. | Opposition to current flow in a specific conductor. |
| Units | Ohm-meter ($\Omega \cdot$m) | Ohm ($\Omega$) |
| Dependency | Material type, Temperature | Material type, Length, Area of cross-section, Temperature |
| Nature | Material constant (at constant T) | Specific to a conductor's dimensions and material |
Additional Information on Resistivity Factors
While resistivity does not depend on the dimensions of the conductor, it can be affected by other factors:
- Temperature: For most conductors, resistivity increases with increasing temperature.
- Material Composition: Different materials have vastly different resistivities (e.g., metals like copper have low resistivity, insulators like rubber have very high resistivity).
- Impurities: Adding impurities to a pure metal can increase its resistivity.
In the context of the question, since all wires are specified as copper and no mention of temperature differences is made, we assume they are at the same temperature, leading to the conclusion that their resistivity is the same.
Similar Questions
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Important Questions from Resistance and Resistivity
If the length of a conductor is doubled, then the resistance of the conductor will be: (other parameters are kept same)
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A uniform wire of resistance 9Ω is bent in the form of an equilateral triangle. Find the effective resistance across a side of the triangle.
