Temperature Dependence of Resistivity 🔥
Metals sit at the “easy-current” end of the scale with resistivities around 10-8 Ω m – 10-6 Ω m, while ceramic, rubber and plastics live all the way up to values about 1018 times larger 👀. In the middle are semiconductors whose resistivity actually drops when they warm up—one of the secrets behind modern electronics :contentReference[oaicite:0]{index=0}.
1 . The “alpha” formula 📝
Over a moderate temperature window, a metal’s resistivity follows a neat straight-line rule:
$$\rho_T=\rho_0\,[\,1+\alpha\,(T-T_0)\,] \tag{3.26}$$
- ρT – resistivity at temperature T.
- ρ0 – resistivity at reference temperature T0.
- α – temperature coefficient of resistivity (unit = K-1).
- For metals, α > 0, so resistivity climbs when the wire heats up 😊.
The straight-line picture works well near T0, but at very low temperatures (below 0 °C for copper) the graph bends away from the line :contentReference[oaicite:1]{index=1}.
2 . Metals, Alloys & Semiconductors 🚦
| Material | What the graph does | Why it matters |
|---|---|---|
| Copper | Straight rise with T (Fig 3.8). | Great for everyday wiring; expect resistance to grow as it heats. |
| Nichrome (Ni + Cr + Fe) | Very gentle slope — almost flat (Fig 3.9). | Perfect for heater coils & lab resistors because its value hardly changes with temperature :contentReference[oaicite:2]{index=2}. |
| Manganin, Constantan | Behave like nichrome (tiny α). | Standard-resistor favourites. |
| Semiconductors | Sharp drop with temperature (Fig 3.10). | Heating frees more charge carriers, so devices conduct better when warm 🌡️ :contentReference[oaicite:3]{index=3}. |
3 . Microscopic story 🔬
A deeper look ties resistivity to the charge-carrier crowd and their traffic jams:
$$\rho=\frac{m}{n\,e^{2}\,\tau} \tag{3.27}$$
- n – free-electron density.
- τ – average time between collisions.
- m, e – electron mass & charge.
Raising the temperature makes electrons dash faster, so collisions come more often and τ falls. In metals, n barely budges, so ρ goes up. In semiconductors, n skyrockets and more than cancels the shorter τ, so ρ slides down ⤵️ .
4 . Real-world snapshots 📷
- Toaster element (nichrome): initial R = 75.3 Ω at 27 °C. With 230 V applied, R rises to 85.8 Ω and the wire settles at a sizzling 847 °C 🔥 (using α ≈ 1.70 × 10-4 °C-1) :contentReference[oaicite:4]{index=4}.
- Platinum thermometer: R jumps from 5.00 Ω (ice) to 5.23 Ω (steam). If R reaches 5.795 Ω in a bath, the bath is about 346 °C ♨️ :contentReference[oaicite:5]{index=5}.
5 . Quick Tips for Calculations ✏️
- Pick a convenient T0 (often 0 °C or 20 °C) and grab ρ0 from tables.
- Use α for the material’s temperature range; alloys have tiny α, metals have moderate α, semiconductors have negative “effective α”.
- For a wire: R ∝ ρ, so the same formula works for resistance.
High-Yield Ideas for NEET 🎯
- The linear law $$\rho_T=\rho_0\,[\,1+\alpha\,(T-T_0)\,]$$ and how to switch it to resistances.
- Sign of α: positive for metals, nearly zero for nichrome-type alloys, effectively negative for semiconductors.
- The microscopic link $$\rho=\dfrac{m}{n e^{2}\tau}$$ explaining why metals and semiconductors behave oppositely with temperature.
- Typical exam numerics: finding the final temperature of a heating element or bath using resistance data.
- Choosing materials: nichrome/manganin for stable resistors, copper for wiring, semiconductors for sensors and devices.
Keep practicing and stay curious 🚀!
