Question

The voltage-current graph for a metal wire of uniform area of cross section at two different temperatures $T$ and $T'$ is shown. Then choose the correct statement:

• A. Resistance of the conductor at temperature $T$ is greater than resistance of the conductor at temperature $T'$
• B. Temperature $T$ is greater than temperature $T'$
• C. Temperature $T'$ is greater than temperature $T$
• D. Resistivity is independent of temperature

Hint:

Always double-check the axis labels! On a V-I graph (V on y-axis), a steeper slope means more resistance. On an I-V graph (I on y-axis), a steeper slope means less resistance. Here, $T$ has a steeper slope, so it has less resistance and must be the cooler temperature!

Answer 1

The Correct Option is C

Concept: According to Ohm's Law, the relationship between voltage and current is given by $V = IR$, which can be rewritten as $I = \left(\frac{1}{R}\right)V$. On an $I-V$ coordinate chart where Current ($I$) is on the vertical axis and Voltage ($V$) is on the horizontal axis, the slope of the line equals the inverse of the resistance: \[ \text{Slope} = \frac{I}{V} = \frac{1}{R} \] For metallic conductors, as temperature increases, thermal vibrations of the metal lattice increase, causing more frequent collisions for migrating electrons. This causes resistance to increase with temperature.

Step 1: Analyze the slopes to compare resistance values.
From standard $I-V$ plots where the line for $T$ climbs steeper than the line for $T'$: \[ \text{Slope}(T) > \text{Slope}(T') \] Since slope is equal to $\frac{1}{R}$: \[ \frac{1}{R_T} > \frac{1}{R_{T'}} \implies R_{T'} > R_T \] This shows that the wire exhibits higher electrical resistance at temperature $T'$ than it does at temperature $T$.

Step 2: Relate resistance behavior back to temperature.
Since the electrical resistance of a metallic conductor increases linearly with an increase in temperature ($\Delta R \propto \Delta T$), a higher resistance directly points to a higher thermal environment: \[ R_{T'} > R_T \implies T' > T \]