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:
Show 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!
Resistance of the conductor at temperature $T$ is greater than resistance of the conductor at temperature $T'$
Temperature $T$ is greater than temperature $T'$
Temperature $T'$ is greater than temperature $T$
Resistivity is independent of temperature
Show Solution
The Correct Option isC
Solution and Explanation
Understanding the 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:
\[
R_{T'}>R_T \implies T'>T
\]
