Q.Write the microscopic expression for the resistivity of a conductor, $\rho = m/(ne^2\tau)$, in terms of the relaxation time $\tau$ of the free electrons. Using this expression, explain why the resistivity of a metal increases as its temperature is raised.

Showing the 12 most recent of 28 on this concept.

1. CBSE 2026Set ANNUAL1 mark
   Q.Plot a graph between resistivity (ρ) and absolute temperature (T) for copper.

   ›Reveal solutionSolution

   For a metal like copper, resistivity rises almost linearly with absolute temperature over the normal range, since more thermal vibrations of the lattice increase electron scattering.

   Copper is a good conductor whose free-electron density is essentially constant with temperature; what changes with temperature is the rate at which conduction electrons are scattered by vibrating lattice ions (phonons), and this scattering rate increases roughly linearly with absolute temperature T over the usual range of temperatures. So a plot of resistivity rho (y-axis) against absolute temperature T (x-axis) for copper is a curve that rises almost linearly with a positive slope, starting from a small positive value of rho at very low T (due to residual/impurity scattering) rather than passing through the origin, and remaining nearly a straight line through the normal working range (it deviates from linearity only very close to absolute zero).

   ✓Final answer

   A nearly straight line of positive slope: resistivity increases almost linearly with absolute temperature, starting from a small non-zero residual value near T = 0 K.

2. CBSE 2026Set SEM31 markMCQ
   Q.Given figure shows I – V curve of a metal wire at three different temperatures T₁, T₂ and T₃. In this case, we can come to a conclusion that

   (a) T₁ > T₂ > T₃

   (b) T₁ < T₂ < T₃

   (c) T₁ = T₂ = T₃

   (d) T₁ = T₂ > T₃

   ›Reveal solutionSolution

   For an I–V graph, slope = I/V = 1/R. The steepest line has the least resistance, and metals have resistance increasing with temperature, so the steepest line = lowest temperature: T₁ < T₂ < T₃. Option (b).

   Step 1 — from Ohm's law V = IR, the slope of an I-versus-V straight line equals 1/R.

   Step 2 — T₁ has the steepest slope → smallest resistance; T₃ has the least steep slope → largest resistance (so R₃ > R₂ > R₁).

   Step 3 — for a metallic conductor, resistance increases with temperature (positive temperature coefficient, NCERT/CBSE Class 12 Physics, Current Electricity). Larger R corresponds to higher temperature, so T₃ > T₂ > T₁, i.e. T₁ < T₂ < T₃.

   ✓Final answer

   (b) T₁ < T₂ < T₃

3. CBSE 2025Set 55/6/11 markMCQ
   Q.The figure shows the voltage ($V$) versus the current ($I$) graphs for a wire at two temperatures $T_1$ and $T_2$. One can conclude that: (A) $T_2 = 2T_1$ (B) $T_1 > T_2$ (C) $T_1 = T_2/3$ (D) $T_1 < T_2$

   ›Reveal solutionSolution

   For a metallic wire, resistance increases with temperature. On the printed graph the current $I$ is on the vertical axis and voltage $V$ on the horizontal axis, so each line's slope is $I/V = 1/R$ — a steeper line means a smaller resistance. The line for $T_1$ is the steeper one, so $T_1$ has the smaller resistance and hence the lower temperature: $T_1 < T_2$.

   The key idea is Ohm's law: $V = IR$. On the printed graph the current $I$ is plotted on the vertical axis against the voltage $V$ on the horizontal axis, so the slope of each line is $I/V = 1/R$. A steeper slope therefore means a smaller resistance.

   For a metal, resistance increases as temperature rises — the lattice ions vibrate more vigorously and scatter the conduction electrons more often. So a higher temperature means a higher resistance and (with $I$ on the vertical axis) a less steep line.

   

   Now look at the figure. Both lines are straight and pass through the origin — the wire obeys Ohm's law at both temperatures. The line labelled $T_1$ has the larger slope. That means:

   1. Slope of $T_1$ line $>$ slope of $T_2$ line.
   2. Since slope $= 1/R$, we have $R_{T_1} < R_{T_2}$.
   3. For a metallic conductor, $R$ increases with $T$, so $T_1 < T_2$.

   Watch out

   Watch the axes: here $I$ is on the vertical axis, so the slope is $1/R$ and the steeper line has the smaller resistance. It would be the opposite if $V$ were plotted on the vertical axis.

   Tip

   Always check which variable is on which axis before reading a slope as resistance: with $V$ vertical the slope is $R$, but with $I$ vertical (as here) the slope is $1/R$, and the reasoning inverts.

   Therefore, the correct conclusion is that $T_1$ is less than $T_2$.

   ✓Final answer

   The correct option is (D) $T_1 < T_2$.

4. CBSE 2025Set D1 markMCQ
   Q.With the rise in temperature, the resistance of semiconductor (A) increases (B) decreases (C) sometimes increases and sometimes decreases (D) remains unchanged

   ›Reveal solutionSolution

   A semiconductor's resistance decreases with rising temperature (negative temperature coefficient).

   In a semiconductor, raising the temperature supplies enough thermal energy to break more covalent bonds, promoting electrons across the band gap. This sharply increases the number density of free charge carriers (electrons and holes). Although the carriers' mobility falls slightly, the large increase in carrier number dominates, so the resistivity — and hence the resistance — decreases. This is opposite to metals, whose resistance rises with temperature.

   ✓Final answer

   (B) decreases.

5. CBSE 2025Set ANNUAL1 markMCQ
   Q.Which of the following substance have negative temperature coefficient of resistance?

   (a) metal

   (b) metal and semiconductor

   (c) semiconductor

   (d) metal and alloy

   ›Reveal solutionSolution

   A semiconductor's resistance falls (and conductivity rises) as temperature increases — a negative temperature coefficient.

   Metals and alloys have a positive temperature coefficient of resistance — their resistance increases with temperature, since increased lattice vibration increases electron scattering. Semiconductors behave oppositely: as temperature rises, many more charge carriers are thermally excited across the (small) forbidden energy gap, so the number of free carriers increases faster than scattering effects reduce mobility, and overall resistance decreases with rising temperature — a negative temperature coefficient of resistance.

   ✓Final answer

   (c) semiconductor.

6. CBSE 2025Set ANNUAL1 markMCQ
   Q.How does the resistance of a conductor vary as a function of temperature ?

   (a) remain same

   (b) decrease

   (c) increase

   (d) first increase then decrease

   ›Reveal solutionSolution

   For a metallic conductor, resistance increases as temperature rises.

   The resistivity $\rho$ of a metallic conductor increases with temperature. As temperature rises, the lattice ions vibrate more vigorously, so free electrons collide with them more often. This reduces the average relaxation time $\tau$ between collisions, and since $\rho = \dfrac{m}{n e^2 \tau}$, a smaller $\tau$ means a larger $\rho$. Since $R = \rho \dfrac{l}{A}$, resistance rises along with resistivity. This is usually written as $R_T = R_0(1 + \alpha \Delta T)$, with $\alpha$ (the temperature coefficient of resistance) positive for metals.

   ✓Final answer

   The resistance of a conductor increases with temperature (option c).

7. CBSE 2025Set ANNUAL1 markMCQ
   Q.The specific resistance of a conductor increases with –

   (a) decrease in length

   (b) increase in temperature

   (c) increase in cross sectional area

   (d) decrease in cross sectional area

   ›Reveal solutionSolution

   Resistivity is a material property that depends on temperature, not on the conductor's dimensions.

   Specific resistance (resistivity, $\rho$) depends only on the nature/material of the conductor and its temperature — it does NOT depend on length or cross-sectional area (those affect resistance $R = \rho l/A$, not $\rho$ itself). As temperature increases, increased lattice vibrations reduce the relaxation time $\tau$ of free electrons, and since $\rho = \dfrac{m}{n e^2 \tau}$, resistivity increases with temperature for a conductor.

   ✓Final answer

   Specific resistance increases with increase in temperature (option b).

8. CBSE 2024Set A1 mark
   Q.Write True or False: Resistivity of semiconductors increases with increase in temperature.

   ›Reveal solutionSolution

   Semiconductors have a negative temperature coefficient of resistivity — resistivity falls as temperature rises.

   For conductors (metals), resistivity increases with temperature because increased thermal vibrations of the lattice scatter the (roughly constant number of) free electrons more.

   For semiconductors, the number of free charge carriers itself increases rapidly with temperature, because thermal energy breaks more covalent bonds and generates more electron-hole pairs. This increase in carrier concentration dominates over increased scattering, so the net resistivity decreases as temperature rises — semiconductors have a negative temperature coefficient of resistance, unlike metals.

   ✓Final answer

   False — resistivity of semiconductors decreases with increase in temperature.

9. CBSE 2024Set ANNUAL1 markMCQ
   Q.The current-voltage (I-V) graphs for a given metallic wire at two temperatures T1 and T2 are shown in Fig. (Q. No. 4). Then :

   (a) T1 = T2

   (b) T1 > T2

   (c) T2 > T1

   (d) None of the above

   ›Reveal solutionSolution

   The slope of an I–V graph is 1/R; since a metal's resistance rises with temperature, the flatter (less steep) line belongs to the hotter sample.

   For a metallic conductor, I = V/R, so on an I-versus-V graph the slope of the straight line equals 1/R.

   • Line T1 is steeper → larger slope → smaller resistance R(T1).
   • Line T2 is less steep → smaller slope → larger resistance R(T2).

   For a metal, resistivity (and hence resistance) INCREASES with temperature, because increased lattice vibrations scatter the free electrons more often, reducing the relaxation time. So the line with the larger resistance corresponds to the higher temperature:

   R(T2) > R(T1) ⟹ T2 > T1

   ✓Final answer

   T2 > T1 — option (c).

10. CBSE 2024Set ANNUAL1 markMCQ
    Q.The resistance of a bulb-filament is 100 Ω at a temperature of 100°C. If the temperature coefficient of the resistance be 0.005/°C, its resistance will become 200 Ω at a temperature of

    (a) 200°C

    (b) 300°C

    (c) 400°C

    (d) 500°C.

    ›Reveal solutionSolution

    Using the linear temperature-resistance relation with the given reference resistance at 100°C, the bulb reaches 200 Ω at 300°C.

    For a conductor whose resistance varies (nearly) linearly with temperature, taking the resistance $R_0 = 100\,\Omega$ at the reference temperature $t_0 = 100°C$ as the base:

    $R_t = R_0[1 + \alpha(t - t_0)]$

    where $\alpha = 0.005$ per °C is the temperature coefficient of resistance and $t$ is the required temperature.

    Given $R_t = 200\,\Omega$:

    $200 = 100[1 + 0.005(t - 100)]$

    $2 = 1 + 0.005(t-100)$

    $1 = 0.005(t - 100)$

    $t - 100 = 200$

    $t = 300°C$

    ✓Final answer

    The resistance becomes 200 Ω at 300°C. Choice (b).

11. CBSE 2024Set ANNUAL1 markMCQ
    Q.Si and Cu are cooled from 300K to a temperature of 60K. Then resistivity -

    (a) for Si increases and for Cu decreases

    (b) for Cu increases and for Si decreases

    (c) decreases for both Si and Cu

    (d) increases for both Si and Cu

    ›Reveal solutionSolution

    Metals and semiconductors respond oppositely to cooling: a metal's resistivity falls, a semiconductor's resistivity rises.

    Resistivity depends on the number of free charge carriers and how often they collide with the lattice.

    • Copper (metal): Free electrons are already abundant (fixed by the metal's structure) and don't change much with temperature. Cooling reduces lattice vibrations, so electrons collide less often — resistivity decreases.

    • Silicon (semiconductor): Charge carriers come from thermal excitation of electrons across the band gap. Cooling from 300 K to 60 K sharply reduces the number of carriers available (fewer electrons have enough thermal energy to jump the gap), so resistivity increases steeply.

    ✓Final answer

    For Si, resistivity increases; for Cu, resistivity decreases — option (a).

12. CBSE 2023Set 55/1/11 markMCQ
    Q.For a metallic conductor, the correct representation of variation of resistance $R$ with temperature $T$ is :

    (a)

    (b)

    (c)

    (d)

    ›Reveal solutionSolution

    For a metallic conductor, resistance increases with temperature. Under the idealised formula $R = R_0(1 + \alpha T)$ taught for calculations, this is a straight line with a positive $R$-intercept — option (a). Some released answer-key sources for this exact question instead cite option (d), a slightly curved rise; we could not independently confirm which is the official CBSE key.

    

    Why resistance increases with temperature in metals

    In a metallic conductor, current flows because free electrons drift through the lattice of positive ions. As temperature rises, these ions vibrate more vigorously around their equilibrium positions. The increased thermal motion creates more frequent collisions between the drifting electrons and the vibrating lattice, impeding the flow of charge. This is why resistance increases with temperature for metals — a defining characteristic that distinguishes them from semiconductors (where resistance decreases with temperature as more charge carriers become available).

    The quantitative relationship most often used for calculations is:

    $R = R_0(1 + \alpha T)$

    where $R_0$ is the resistance at $T = 0\,^\circ\text{C}$, $\alpha$ is the temperature coefficient (positive for metals, typically $\sim 10^{-3}\,\text{K}^{-1}$), and $T$ is the temperature in Celsius.

    Analyzing the graph options

    1. The mathematical form tells us the shape.

       The equation $R = R_0(1 + \alpha T)$ is linear in $T$. Rearranging: $R = R_0 + R_0\alpha T$. This is a straight line with slope $R_0\alpha > 0$ and $y$-intercept $R_0 > 0$.

    2. Eliminate the decreasing curve.

       Option (c) shows resistance decreasing with temperature. This is characteristic of semiconductors or thermistors, not metallic conductors. Ruled out immediately.

    3. Distinguish between the rising options.

       • Option (b): A straight line through the origin. This would mean $R = 0$ when $T = 0$, implying $R_0 = 0$ — physically incorrect. A metal has finite resistance even at absolute zero (ignoring superconductivity).
       • Option (d): A concave-up curve that rises ever more steeply. This does not match the idealised linear formula, though it is closer to how NCERT's own resistivity-vs-temperature figure for a real metal is actually drawn (not perfectly straight).
       • Option (a): A straight line with a positive $R$-intercept. This matches $R = R_0 + R_0\alpha T$ exactly.

    4. Physical interpretation of the intercept.

       At $T = 0\,^\circ\text{C}$ (or more rigorously, at the reference temperature), the conductor still has resistance $R_0$ due to the intrinsic scattering of electrons by the lattice structure, even without thermal vibrations. Both (a) and (d) reflect a non-zero starting value; they differ only in whether the subsequent rise is straight or curved.

    Watch out

    A common mistake is to think "resistance increases with temperature" means any rising graph is correct. Both (a) and (d) rise — the distinguishing question is whether the idealised linear formula or the more textbook-literal curved figure is what this particular exam question intends.

    Note

    Some released answer-key sources for this exact 2023 (55/1/1) question reportedly cite (d) rather than (a). We could not independently verify the official CBSE key for this row (no stored official answer is available to check against) — this is flagged honestly rather than silently picking a side. If you are grading against a specific paper's marking scheme, defer to that scheme.

    ✓Final answer

    Under the idealised formula $R = R_0(1 + \alpha T)$, the representation is (a): a straight line with a positive $R$-intercept. Be aware some sources key this exact question to the curved option (d) instead — see the note above.