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Wize University Physics Textbook (Master) > DC Circuits

Temperature Dependent Resistors

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Temperature Dependent Resistors


The resistivity of a material may depend on its temperature. For most conductive metals, this relationship may be written as follows:

 ρ=ρ0 [1+α(T−T0)] \boxed{\ \rho=\rho_0 \ [1+\alpha(T-T_0)] \ }


Since the resistance of a wire and the resistivity are directly proportional, the resistance follows the same relationship:

 R=R0 [1+α(T−T0)] \boxed{\ R=R_0 \ [1+\alpha(T-T_0)] \ }
  • ρ\rho and RR are the resistivity and resistance
  • TT is the temperature
  • ρ0\rho_0 and R0R_0 and the reference resistivity and resistance at the reference temperature T0T_0
  • α\alpha is the temperature coefficient (specific to the material)



Some materials have a resistivity that actually becomes exactly zero below a certain temperature. These are called superconductors. Currents in a superconductor (called supercurrents) can flow forever without dissipation.

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Example: Silver Wire


At 20°20\degree C, silver has a resistivity of 1.59×10−81.59 \times10^{-8} Ωm and it has a temperature coefficient of 3.8×10−33.8\times 10^{-3} K-1. What must be the cross-sectional area of a 1010 m silver wire so that it has a resistance of 1.5×10−31.5 \times 10^{-3} Ω at 100°100\degree C?


Let's use the temperature-dependent resistivity equation:

ρ=ρ0 [1+α(T−To)]\rho=\rho_0\ [1+\alpha(T-T_o)]

Combine it with the resistance of a wire:

R=ρLA   →   ρ=RALR=\dfrac{\rho L}{A} \ \ \ \to \ \ \ \rho=\dfrac{RA}{L}

So we get:

RAL=ρ0 [1+α(T−To)]\dfrac{RA}{L}=\rho_0\ [1+\alpha(T-T_o)]


Isolate the area:

A=ρ0LR [1+α(T−To)]A=\dfrac{\rho_0L}{R}\ [1+\alpha(T-T_o)]

=(1.59×10−8)(10)1.5×10−3 [1+3.8×10−3(100−20)]=\dfrac{(1.59 \times10^{-8})(10)}{1.5 \times 10^{-3}}\ [1+3.8\times 10^{-3}(100-20)]

=1.38×10−4=1.38\times10^{-4} (m2)


Practice: Tungsten Filament


At 15°15\degreeC, you want a tungsten filament (in a light bulb) to have a resistance of 0.20.2 Ω. The filament has a diameter of 0.10.1 mm. What should be the length of this filament? Answer in millimeters.

The resistivity of tungsten is 52.8×10−852.8 \times 10^{-8} Ωm at 20°20\degreeC, and it is 48.05×10−8 48.05 \times 10^{-8 } Ωm at 0°0\degreeC.