Temperature Dependence of Resistivity

Resistivity is a property of a material that tells how strongly it opposes the flow of electric current.

For many materials, resistivity changes when temperature changes.

In metals, resistivity usually increases with an increase in temperature over a limited temperature range.

In semiconductors and insulators, resistivity usually decreases with an increase in temperature.

Resistivity, denoted by ρ, is the intrinsic property of a material that determines how much it resists current flow.

The temperature coefficient of resistivity, denoted by α, measures the fractional change in resistivity per degree change in temperature in the linear range.

• Unit: per degree Celsius or per kelvin.
• For metals, α > 0.
• For semiconductors, α < 0.

ρ_T ​= ρ₀​[1 + α(T − T₀​)]

Here:

• ρ_T​ = resistivity at temperature T.
• ρ₀​ = resistivity at reference temperature T₀.
• α = temperature coefficient of resistivity.

R_T ​= R₀​(1 + αΔT)

where ΔT = T − T₀​.

The microscopic relation for resistivity is:

where m is the electron mass, n is the number density of free electrons, e is the electronic charge, and τ is the average relaxation time.

In metals

As the temperature rises, lattice vibrations increase, so electrons experience more frequent collisions.

This reduces the relaxation time τ, thereby increasing resistivity.

In semiconductors

When the temperature rises, the number of charge carriers increases significantly.

This increase in carrier density dominates, so resistivity decreases.

A nichrome toaster element has a resistance of 75.3 ohms at 27 degrees Celsius and 85.8 ohms at its operating temperature. If α = 1.7 × 10⁻⁴ per degree Celsius, find the steady temperature.

Using

Substituting the values gives the operating temperature approximately equal to 847 degrees Celsius.

Final answer: 847 degrees Celsius.

A platinum resistance thermometer has a resistance of 3 ohms at 0 degrees Celsius and 3.75 ohm at 100 degrees Celsius. If its resistance in a hot bath is 5.591 ohms, find the bath temperature.

Using the linear thermometer relation,

The bath temperature is approximately 345.65 degrees Celsius.

Final answer: 345.65 degrees Celsius.

• Toaster coil: A nichrome heating element becomes hot in operation, and its resistance changes with temperature.
• Resistance thermometer: Platinum resistance changes predictably with temperature, so it is used to measure temperature.
• Electronic devices: Semiconductor behaviour with temperature is important in sensors and circuits.

Resistivity and Temperature:

\[\rho_T=\rho_0[1+\alpha(T-T_0)]\]

Resistance and Temperature:

\[R_T=R_0(1+\alpha\Delta T)\]

Temperature Coefficient (α):

• Unit: °C⁻¹ (or K⁻¹)
• Metals: α > 0→ resistivity increases with temperature

Semiconductors & insulators:

α < 0 → resistivity decreases with temperature