The Question is from neet-physics , and it was created by the author NAGA SAI

Question

Which of the following graphs best represents the variation of resistivity ($\rho$) with temperature (T) for copper?

NAGA SAI · Accepted Answer

## CORE INFORMATION
✓ **Answer**: A
## CONCEPT FRAMEWORK
 **Primary Concept**: Temperature dependence of resistivity in metals
 **Related Topics**: Electrical conductivity, thermal expansion, Matthiessen's rule
 **Prerequisites**: Understanding of resistivity and its relationship with temperature.
## SOLUTION PATHWAY
**Given Information**:
  The question asks about the relationship between resistivity ($ho$) and temperature (T) for copper.
  Copper is a metal.
**Method & Steps**:
 1. Resistivity of metals increases linearly with temperature at relatively high temperatures.
  Reasoning:  The increased thermal vibrations of the lattice atoms at higher temperatures impede the flow of electrons, leading to increased resistance and hence resistivity.
  Formula/Rule: $ho(T) = ho_0[1 + \alpha(T - T_0)]$, where $ho_0$ is the resistivity at reference temperature $T_0$, and $\alpha$ is the temperature coefficient of resistivity.
 2. The linear relationship is described by a straight line with a positive slope.
  Working: The formula above shows a direct linear relationship between $ho$ and $T$ for $\alpha > 0$, which is true for most metals at higher temperatures.
  Key Point: The linear relationship is an approximation and holds true over a limited temperature range. At very low temperatures, the behavior deviates from linearity.
 3. A straight line with a positive slope represents the linear increase of resistivity with temperature.
  Result: A straight line with a positive slope.
  Verification: The formula for resistivity as a function of temperature confirms this.
 ## OPTION ANALYSIS
A] ✓: A straight line with a positive slope correctly represents the linear increase in resistivity with temperature for copper.
B] :x: A negative slope would indicate that resistivity decreases with increasing temperature, which is not true for copper.
C] :x: A parabolic curve suggests a non-linear relationship, which is not the dominant behavior of copper's resistivity over a moderate temperature range.
D] :x: A downward-opening parabola is also incorrect, indicating a decrease in resistivity with increasing temperature.
 ## LEARNING AIDS
 **Common Mistakes**:
  1.Confusing the behavior of metals with semiconductors or insulators: Semiconductors and insulators exhibit different temperature dependencies of resistivity.
  2.Ignoring the temperature range: The linear approximation is only valid over a certain temperature range.
 **Quick Tips**:
  1.Remember that for most metals, resistivity increases with temperature.
  2.Visualize the graph: A simple sketch of a positively sloped line can help you remember the relationship.

NAGA SAI · Answer

The correct Option is "A straight line with a positive slope"

NAGA SAI · Answer

A straight line with a negative slope

NAGA SAI · Answer

A parabolic curve opening upwards

NAGA SAI · Answer

A parabolic curve opening downwards

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