Temperature Dependence, Thermistors & Superconductors-Lite

Every lithium battery pack in a phone, laptop or electric car carries NTC thermistors pressed against its cells; the battery-management chip reads their falling resistance and will refuse to fast-charge a cold pack or keep charging a hot one — this module's plunging red curve, deployed billions of times over as a safety instrument. At the precision end, PT100 platinum sensors are the workhorse thermometers of industrial process control, from pharmaceutical cold chains to jet-engine test rigs, all running equation (1) backwards with a quadratic correction bolted on. And the cliff earns its keep at scale: an MRI scanner's field coil is niobium–titanium wire at $4\ \mathrm{K}$ carrying hundreds of amperes with zero resistive heating — switch off the power supply and the current keeps circulating, a persistent-mode magnet running on Onnes' 1911 physics. The Large Hadron Collider bends its proton beams with 27 kilometres of the same idea. The unfinished business is the temperature: room-temperature superconductivity remains one of physics' great open hunts, because a zero-loss power grid would rewrite the energy economy.

Go back inside the wire, to the picture built in Drift Velocity & the Microscopic Picture. Free electrons drift through a lattice of ions, gaining speed from the field, losing it in collisions. Resistivity came out of that story with two knobs in the denominator: $\rho = m/(ne^2\tau)$ — how many carriers there are per cubic metre ($n$), and how long each gets to run between collisions ($\tau$). Temperature grabs both knobs, but with very different force in different materials.

In a metal, the electron sea is already full — every copper atom has donated its electron, and no amount of school-lab heating adds more. So $n$ is frozen. But the lattice ions vibrate more violently as the metal warms, presenting bigger, more erratic targets: collisions come sooner, $\tau$ shrinks, and resistivity climbs. Near room temperature the climb is nearly linear — the basis of the $\alpha$ formula and of resistance thermometry.

In a semiconductor, the situation inverts. At low temperature almost every electron is locked into bonds: carriers are scarce, and scarce carriers is the definition of a poor conductor. Heat is the liberator — each rise in temperature shakes loose exponentially more electrons into conduction. The collision knob still turns the wrong way ($\tau$ still falls), but it fights a losing battle against a carrier population that can double over tens of degrees. Net result: resistance falls, steeply, as a semiconductor warms.

Hold that graph in your head — it is the module. A rising, gently sloped line for the metal; a plunging curve for the semiconductor. (What current does when you sweep the voltage across such components at various temperatures is graph craft that belongs to Ohmic & Non-Ohmic: I–V Characteristics; here every graph has temperature on the horizontal axis.) The one behaviour the graph cannot show at this scale is the strangest: for some materials, cool far enough and the metal line does not glide smoothly to a small value — it falls off a cliff to exactly zero.

For metals over everyday ranges, experiment hands us a mercifully simple law. Define a reference resistance $R_0$ at a reference temperature $T_0$ (usually $20\ \mathrm{^\circ C}$, sometimes $0\ \mathrm{^\circ C}$ — check the question), and the resistance a temperature change $\Delta T = T - T_0$ away is:

where: $R_0$ — resistance at the reference temperature; $\alpha$ — the temperature coefficient of resistance, in $\mathrm{^\circ C^{-1}}$ (equivalently $\mathrm{K^{-1}}$ — a change of one degree Celsius is a change of one kelvin, so $\Delta T$ is numerically identical in both); $\Delta T$ — how far you moved from the reference, positive or negative. Read $\alpha$ as a percentage machine: copper's $\alpha = 3.9\times10^{-3}\ \mathrm{^\circ C^{-1}}$ means each degree adds $0.39\%$ of the reference resistance. Sixty degrees adds $23\%$ — noticeable, which is why precision circuits specify resistor temperature ratings. Because geometry barely changes with temperature (thermal expansion is a $10^{-5}$-per-degree effect, a hundred times smaller than $\alpha$), the same law with the same $\alpha$ governs the material property itself:

where: $\rho_0$ — resistivity at the reference temperature (the table values in Resistivity & Geometry are $20\ \mathrm{^\circ C}$ values for exactly this reason). Equations (1) and (2) carry identical content for a fixed-shape conductor; exams use them interchangeably. Two honesty clauses before the derivation. First, the law is a linear approximation — excellent for metals within a hundred or so degrees of the reference, drifting off beyond that (platinum thermometry uses a quadratic correction at high precision). Second, $\alpha$ itself is quoted at the reference temperature; using a $20\ \mathrm{^\circ C}$ coefficient with a $0\ \mathrm{^\circ C}$ reference resistance mixes conventions — the classic trap dissected in the competitive worked example.

The table rewards a designer's read. Pure metals cluster near $4\times10^{-3}$; alloys are dramatically flatter — manganin was invented precisely so that standard resistors would not drift with room temperature. And nichrome's flatness is the quiet second reason (after its high $\rho$) that it makes good heating elements: its resistance, and hence its power draw, barely changes between cold start and red heat.

Run the law backwards and resistance becomes temperature. A platinum resistance thermometer is a coil of fine platinum wire whose $R_0$ is measured once at a reference temperature. Later, wherever the coil sits — a furnace, a freezer, a vaccine cold-chain crate — measure its resistance and solve equation (1) for $\Delta T$. Platinum earns the job on three counts: chemically inert (the calibration survives), high melting point (works to $600\ \mathrm{^\circ C}$ and beyond), and an unusually linear $R$–$T$ line. The industrial standard "PT100" is exactly this: platinum, $R_0 = 100\ \Omega$ at $0\ \mathrm{^\circ C}$. The board-level worked example below is a PT-thermometer question in kettle clothing. The NTC thermistor (negative temperature coefficient) is the semiconductor counterpart — a bead whose resistance might fall from $10\ \mathrm{k\Omega}$ at room temperature to $1\ \mathrm{k\Omega}$ near $80\ \mathrm{^\circ C}$. Ten-to-one over a modest range: far more sensitive than platinum's percent-per-degree, at the price of a strongly curved calibration. It answers "has the battery pack crossed $45\ \mathrm{^\circ C}$?" superbly and "is the furnace at $1180$ or $1200\ \mathrm{^\circ C}$?" not at all. How a thermistor's falling resistance gets converted into a switching voltage — the night-light and thermostat circuits — is the business of Potential Dividers, later in this chapter.

Then the cliff. Cool mercury, lead, niobium — many metals, though famously not copper — and at a sharp critical temperature $T_c$ the resistance drops to zero. Exactly zero: this is not a very good conductor but a different electrical state of matter, in which the electron sea moves without any dissipation at all. Start a current in a superconducting ring and it persists — experiments have watched such currents for years and set lower bounds of tens of thousands of years on their decay. Mercury's $T_c$ is $4.2\ \mathrm{K}$; niobium alloys used in magnet wire sit near $10$–$18\ \mathrm{K}$; the 1986–87 "high-temperature" ceramic superconductors leapt to $92\ \mathrm{K}$ and beyond — crucially above the $77\ \mathrm{K}$ boiling point of cheap liquid nitrogen. Why superconductivity happens (paired electrons, a genuinely quantum story) is far beyond the HS syllabus — what boards test is the phenomenology: the sharp cliff at $T_c$, the exactly-zero resistance, persistent currents, and the flagship applications: MRI magnets and particle-accelerator magnets carry hundreds of amperes in superconducting coils with zero heating loss.

Board level — a copper winding warms up (CBSE/ICSE pattern, 3 marks)

A copper coil has resistance $4.00\ \Omega$ at $20\ \mathrm{^\circ C}$. The temperature coefficient of resistance of copper is $\alpha = 3.9\times10^{-3}\ \mathrm{^\circ C^{-1}}$ (referred to $20\ \mathrm{^\circ C}$). (a) Find its resistance at $80\ \mathrm{^\circ C}$. (b) During operation the coil's resistance is measured as $5.20\ \Omega$. What is its temperature?

Step 1. Part (a): $\Delta T = 80 - 20 = 60\ \mathrm{^\circ C}$. Apply equation (1): $R = 4.00\,(1 + 3.9\times10^{-3} \times 60) = 4.00 \times 1.234 = \boxed{4.94\ \Omega}$. A $23\%$ rise — motor windings really do this, which is why their rated currents assume the hot resistance.

Step 2. Part (b) is the thermometer direction: solve (1) for $\Delta T$. $\Delta T = \dfrac{R/R_0 - 1}{\alpha} = \dfrac{5.20/4.00 - 1}{3.9\times10^{-3}} = \dfrac{0.300}{0.0039} = 76.9\ \mathrm{^\circ C}$.

Step 3. Add back the reference: $T = 20 + 76.9 \approx \boxed{97\ \mathrm{^\circ C}}$. The winding is near boiling-water temperature — plausible for a hard-working motor, and the plausibility check is worth one written sentence.

Step 4. Marks-scheme habits: state $\Delta T$ explicitly before substituting; keep $R/R_0$ as a clean ratio ($1.300$) rather than juggling absolute resistances; quote the final temperature to the data's precision (3 significant figures here).

Competitive level — alpha from two readings, and the reference trap (JEE-Main pattern, ~3 minutes)

A metal wire has resistance $10.0\ \Omega$ at $20\ \mathrm{^\circ C}$ and $12.0\ \Omega$ at $90\ \mathrm{^\circ C}$. (a) Find the temperature coefficient $\alpha$ referred to $20\ \mathrm{^\circ C}$. (b) Predict the resistance at $150\ \mathrm{^\circ C}$. (c) A classmate computes $\alpha$ "referred to $0\ \mathrm{^\circ C}$" from the same data and gets a slightly different number. Who is wrong?

Step 1. Part (a): equation (1) with $R_0 = 10.0\ \Omega$ at $T_0 = 20\ \mathrm{^\circ C}$: $12.0 = 10.0\,(1 + \alpha \times 70)$, so $\alpha = \dfrac{12.0/10.0 - 1}{70} = \dfrac{0.200}{70} = \boxed{2.86\times10^{-3}\ \mathrm{^\circ C^{-1}}}$.

Step 2. Part (b): $\Delta T = 150 - 20 = 130\ \mathrm{^\circ C}$ from the same reference: $R = 10.0\,(1 + 2.86\times10^{-3} \times 130) = 10.0 \times 1.371 = \boxed{13.7\ \Omega}$. (Bait: using $\Delta T = 150 - 90 = 60$ from the second data point with the 20-degree-referred $\alpha$ mixes references and lands near $13.7$ only by luck in some numbers — here it gives $12.0 \times 1.171 = 14.06\ \Omega$, wrong.)

Step 3. Part (c): neither is wrong — $\alpha$ is convention-dependent. Referred to $0\ \mathrm{^\circ C}$, the same physical line has a different fractional slope because the denominator $R_0$ changed. The physical content (the line itself) is identical; the coefficient is a bookkeeping choice. Exams signal the convention in words ("at $20\ \mathrm{^\circ C}$", "referred to $0\ \mathrm{^\circ C}$") — match your $R_0$, $T_0$ and $\alpha$ to the same reference and no trap can fire. Timing note: 2.5–3 minutes. The α extraction is 30 seconds; the marks separate on part (b)'s discipline — one reference, one ΔT. If an MCQ offers both 13.7 Ω and 14.1 Ω, the setter is testing exactly the mixed-reference slip.

Mixing references. $R_0$, $T_0$ and $\alpha$ must belong to the same reference temperature. The formula is not wrong when references differ — you are. Anchor once, compute $\Delta T$ from that anchor, never hop mid-problem.

Kelvin panic. $\Delta T$ in $\mathrm{^\circ C}$ and in K are numerically identical (a degree step is the same size), so equation (1) needs no conversion. Converting $T$ itself to kelvin and then subtracting a Celsius reference is the error; differences are safe, absolute values are not interchangeable.

The sign fork. Metals up, semiconductors down. One unqualified sentence ("resistance increases with temperature") in a long answer about thermistors forfeits the concept mark.

Linearity overreach. Equation (1) within ±100 °C of the reference: trustworthy. Extrapolating a room-temperature $\alpha$ to a $2500\ \mathrm{^\circ C}$ tungsten filament: numerically wild (the real hot resistance is ~10× cold, not the ~1×+α×ΔT the naive line gives at that range with quoted α — the law is local). State the approximation when a question pushes far.

Thermistor direction in circuits. "Thermistor warms → resistance falls" is only half an answer in a circuit question; the mark usually hangs on what that does downstream (more current, or a shifted divider voltage — the latter belongs to Potential Dividers, where this trap resurfaces with teeth).

AP-2 scope note. AP Physics 2 demands only the qualitative trend — "resistivity of conductors typically increases with temperature" — not $\alpha$ numerics or thermistor curves. CBSE, ISC, JEE-Main, NEET and A-Level all run the full quantitative law; JEE-Advanced dropped explicit R(T) from its syllabus, so treat it there as background physics rather than a target topic.

Superconductor phrasing. "Very low resistance" is wrong; below $T_c$ the resistance is exactly zero, and "persistent currents" is the two-word evidence examiners want. Above $T_c$ the material is ordinary — the cliff is sharp, not a slope.

Texts. NCERT Physics Class 12, Ch. 3 (temperature dependence of resistivity) · HC Verma, Concepts of Physics Vol. 2, Ch. 32 · CIE 9702 (resistance thermometry practicals) · ISC Class 12 syllabus, Current Electricity unit (α numericals).

Provenance. All results are standard. A verifier should recompute: copper coil $4.00 \times (1 + 3.9\times10^{-3}\times60) = 4.936 \approx 4.94\ \Omega$; thermometer inversion $\Delta T = 0.300/0.0039 = 76.9$, $T \approx 97\ \mathrm{^\circ C}$; two-reading coefficient $\alpha = 0.200/70 = 2.86\times10^{-3}$, prediction $10.0\times(1+2.86\times10^{-3}\times130) = 13.71 \approx 13.7\ \Omega$; quiz item $8.0\times(1+3.9\times10^{-3}\times100) = 11.12 \approx 11.1\ \Omega$; mercury $T_c = 4.2\ \mathrm{K}$, YBCO $92\ \mathrm{K}$ vs liquid nitrogen $77\ \mathrm{K}$.

Prerequisites. Resistivity & Geometry · Ohmic & Non-Ohmic: I–V Characteristics. Leads to. Potential Dividers (thermistor sensor circuits) · Semiconductors (future Electronics track) · Current, Resistance and Resistivity (UG deep dive).