Question

The current-voltage graphs for a given conducting sample at two different temperatures \(T_1\) and \(T_2\) are shown in the figure below. \(R_1\) is the resistance of the sample at temperature \(T_1\) and \(R_2\) is the resistance at temperature \(T_2\). Choose the correct answer.

• A. \(R_2\gt R_1,\; T_2\gt T_1\)
• B. \(R_2\gt R_1,\; T_2\lt T_1\)
• C. \(R_2\lt R_1,\; T_2\gt T_1\)
• D. \(R_2\lt R_1,\; T_2\lt T_1\)

Hint:

In an \(I\)-\(V\) graph, \[ \text{slope}=\frac{1}{R}. \] For metallic conductors, \[ R \propto T. \] So a smaller slope corresponds to a higher temperature.

Answer 1

The Correct Option is B

Step 1: Read the graph as a slope.
For an ohmic sample $I=\dfrac{V}{R}$, so on an $I$ versus $V$ plot the slope of each line is $\dfrac{1}{R}$. A steeper line means a smaller resistance.

Step 2: Compare the two lines.
The line for $T_1$ is steeper than the line for $T_2$. So its slope is bigger, meaning \[ \frac{1}{R_1}>\frac{1}{R_2}. \]

Step 3: Turn that into resistances.
Taking reciprocals reverses the inequality: \[ R_2>R_1. \]

Step 4: Bring in the temperature behaviour.
For a metallic conductor, resistance grows when temperature grows. Higher resistance goes with higher temperature.

Step 5: Match resistance to temperature.
Since $R_2>R_1$, the sample at $T_2$ is hotter, so \[ T_2>T_1. \]

Step 6: Combine the findings.
Both $R_2>R_1$ and $T_2>T_1$ hold together.
\[ \boxed{R_2>R_1,\quad T_2>T_1} \]