Temperature Dependence of Resistance

1.0What Is Temperature Dependence on Resistance?

Temperature dependence of resistance describes how a material's electrical resistance changes with its temperature. This relationship is crucial for understanding how circuits behave under varying thermal conditions and is a key concept in both fundamental physics and engineering. The resistance of a conductor is determined by the collisions of electrons with the atoms in the material's lattice. As temperature changes, the frequency and nature of these collisions are altered, thereby affecting the material's resistance.

2.0Temperature Dependence of Resistance Formula

The relationship between resistance and temperature is typically linear over a limited temperature range. The temperature dependence of resistance formula is given by:

Resistance corresponding to temperature difference (ΔT) is given as

$R_T=R_0(1+\alpha \Delta T)$

Where,  

$(R_T)$ = Resistance at T°C,

$(R_0)$ = Resistance at 0°C

ΔT = Change in temperature

$(\alpha )$= Temperature coefficient of resistance

For metals: $(\alpha )$ is positive
For semiconductors and insulators: $(\alpha )$ is negative

Resistance of the conductor decreases linearly with decrease in temperature and becomes zero at a specific temperature. This temperature is called critical temperature. Below this temperature a conductor becomes a superconductor.

Illustration 20

The temperature coefficient of resistance for a wire is 0.00125/°C. At 300 K its resistance is 1 ohm. The temperature at which the resistance becomes 2 ohm is
(1) 1154 K  (2) 1100 K  (3) 1400 K  (4) 1127 K

Solution: (4)
R = R₀ (1 + αt)
t = 300 − 273 = 27°C
$\frac{R_1}{R_2}=\frac{(1+\alpha t_1)}{(1+\alpha t_2)}\Rightarrow \frac{1}{2}=\frac{(1+0.00125\times 27)}{(1+0.00125\times t)}\Rightarrow t=854°C\Rightarrow T=1127K$
$R_T=R_0[1+\alpha (T-T_0)]$

Here:

• $R_T$ is the resistance at a given temperature T.
• $R_0$ is the resistance at a reference temperature T​ (often 0∘C or 20∘C).
• T is the final temperature.
• $T_0$ is the reference temperature.
• α is the temperature coefficient of resistance. This coefficient is a material-specific property that quantifies how much a material's resistance changes per degree Celsius (or Kelvin). Its unit is per degree Celsius or $K-1$.

The relation between temperature and resistances is often plotted as a straight line, showing a positive or negative slope depending on the material.

3.0Why Does Resistance Depend on Temperature?

The temperature effect on resistance can be explained at the microscopic level. In a conductor, a free electron moves and collides with the positive ions of the material's crystal lattice. These collisions hinder the flow of electrons, which is what we call resistance.

As temperature increases, the thermal energy of the atoms in the lattice also increases, causing them to vibrate with greater amplitude. These increased vibrations lead to more frequent collisions with the free electrons. Consequently, the average drift velocity of the electrons decreases, and the resistance of the material increases.

4.0Temperature Dependence of Resistance in Metals and Semiconductors

The relationship between temperature and resistance varies significantly between different classes of materials.

• In Metals (Conductors): For most metals, the number of free electrons remains relatively constant with temperature. The dominant effect is the increased vibration of the lattice atoms. As temperature rises, collisions become more frequent, so the resistance of metals increases with temperature. This gives them a positive temperature coefficient of resistance (α>0). Common examples include copper and aluminum.
• In Semiconductors: In semiconductors like silicon and germanium, the number of free charge carriers (electrons and holes) increases significantly with a rise in temperature. This effect far outweighs the increased lattice vibrations. Therefore, as temperature rises, the resistance of semiconductors decreases with temperature. This gives them a negative temperature coefficient of resistance (α<0).

5.0Factors Affecting Resistance

Resistance depends on:

1. Material of the conductor – metals vs semiconductors
2. Length of conductor (LL) – longer conductors have higher resistance
3. Cross-sectional area (AA) – thicker conductors have lower resistance
4. Temperature – increases for metals, decreases for semiconductors

Illustration 14

The current–voltage graphs for a given metallic wire at two different temperature T₁ and T₂ are shown in the figure. Which one is higher, T₁ or T₂.

Solution:

$$\begin{gathered} SlopeofI–Vcurve=1/R \\ \left(\frac{1}{R_1}\right)>\left(\frac{1}{R_2}\right)\Rightarrow R_2>R_1\Rightarrow T_2>T_1 \end{gathered}$$

Illustration 15

The resistance of a thin silver wire is 1.0 Ω at 20°C. The wire is placed in liquid bath and its resistance rises to 1.2 Ω. What is the temperature of the bath? (Here α = 10⁻² /°C)

Solution:
Here change in resistance is small so we can apply
R = R₀ (1 + α Δθ)

⇒ 1.2 = 1 × (1 + 10⁻² Δθ)
⇒ Δθ = 20°C
⇒ θ − 20 = 20
⇒ θ = 40°C Ans.