Current electricity class 12 notes: Temperature Dependence of Resistance

.ast-archive-entry-banner[data-post-type="post"][data-banner-background-type="custom"]{--wpr-bg-866f2994-2128-4951-8dc1-ae62e0e1be38: url('https://asteriskclasses.com/wp-content/uploads/2024/05/pexels-photo-4144101.jpeg');}

Current electricity class 12 notes: Temperature Dependence of Resistance

Unit 2 • Current Electricity

Understanding how resistance changes with temperature in conductors, semiconductors, and special materials

Temperature Effects on Resistance

Fundamental Principle:

The resistance of materials changes with temperature due to variations in electron mobility and charge carrier concentration, with different behavior for conductors vs semiconductors.

Conductors (Metals)

\[ R_T = R_0[1 + \alpha(T – T_0)] \]

Resistance increases with temperature
α = positive temperature coefficient (~0.004/°C for Cu)
Caused by increased lattice vibrations scattering electrons
Linear approximation valid over moderate temperature ranges

Semiconductors

\[ R_T = R_0 e^{E_g/2kT} \]

Resistance decreases with temperature
Negative temperature coefficient
Caused by exponential increase in charge carriers
E_g = band gap energy, k = Boltzmann constant

Key Observations

Superconductors:

Resistance drops abruptly to zero below critical temperature T_c
Hg: T_c = 4.2K, YBCO: T_c = 93K
Applications: MRI machines, maglev trains

Alloys:

Minimal temperature dependence (e.g., manganin, constantan)
Used in precision resistors and measuring instruments
α ≈ 0.00001/°C for manganin

Temperature Coefficient of Resistance (α)

Definition

\[ \alpha = \frac{R_T – R_0}{R_0(T – T_0)} \]

Units: °C^-1 or K^-1

For conductors:

α ≈ 0.003 to 0.006/°C (positive)

For semiconductors:

α ≈ -0.05 to -0.1/°C (negative)

Material Comparison

Material	α at 20°C (/°C)	Type
Copper	0.00404	Conductor
Aluminum	0.00429	Conductor
Silicon	-0.07	Semiconductor
Manganin	0.00001	Alloy

Practical Implications

Copper wire resistance increases ~40% from 20°C to 120°C
Semiconductor thermistors used as temperature sensors
Alloy resistors maintain stable resistance in varying temperatures
Overheating can significantly increase circuit resistance

Worked Example

Problem:

An aluminum wire has resistance 50Ω at 20°C (α = 0.00429/°C). Calculate:

Resistance at 100°C
Temperature when resistance is 55Ω
% change in resistance from 20°C to 200°C

Solution:

(1) Resistance at 100°C:

\[ R_{100} = R_{20}[1 + \alpha(100 – 20)] \] \[ = 50[1 + 0.00429 \times 80] \approx 67.16 \, Ω \]

(2) Temperature for 55Ω:

\[ 55 = 50[1 + 0.00429(T – 20)] \] \[ 1.1 = 1 + 0.00429(T – 20) \] \[ T \approx 43.3 \, °C \]

(3) % change (20°C to 200°C):

\[ R_{200} = 50[1 + 0.00429 \times 180] \approx 88.61 \, Ω \] \[ \% \text{ increase} = \frac{88.61 – 50}{50} \times 100 \approx 77.2\% \]

Practice Problems

Problem 1

A copper wire (α = 0.00404/°C) has 10Ω resistance at 20°C. Find its resistance at 80°C.

Problem 2

A tungsten filament has 100Ω resistance at 20°C and 200Ω at operating temperature. If α = 0.0045/°C, find operating temperature.

Problem 3

A silicon resistor has 1kΩ at 25°C with α = -0.07/°C. Find its resistance at 100°C.

Problem 4

A platinum wire has α = 0.00392/°C. By what percentage does its resistance increase from 20°C to 220°C?

Problem 5

A manganin resistor (α ≈ 0) has 500Ω at 25°C. What will be its resistance at 300°C?

Key Takeaways

Essential Formulas

Conductors: \( R_T = R_0[1 + \alpha(T – T_0)] \)
Semiconductors: \( R_T = R_0 e^{E_g/2kT} \)
Temperature coefficient: \( \alpha = \frac{\Delta R}{R_0 \Delta T} \)
Superconductors: \( R = 0 \) below T_c

Practical Considerations

Always check units (°C vs K)
Linear approximation valid only for moderate ΔT
Semiconductors need exponential model for large ΔT
Alloy resistors used when stability is critical

Next Topic: Internal Resistance of a Cell

In the next section, we’ll explore how real energy sources have internal resistance that affects circuit performance.

Continue to Next Topic →