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Wize University Physics Textbook (Master) > DC Circuits

Resistivity and Conductivity

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Resistivity and Conductivity



The resistivity ρ\bcth\rho of a material is a measure of its resistance to electrons flowing through it.

The conductivity σ\bcth\sigma of a material is a measure of how easily electrons flow through it. It's defined to be:

 σ=ne2τm \boxed{\ \sigma=\frac{ne^2\tau}{m} \ }
  • nn is the electron density (number of electrons per unit volume)
  • ee is the charge of the electron
  • τ\tau is the mean free time (time between collisions)
  • mm is the mass of the electron


The resistivity is defined as:
 ρ=1σ \boxed{ \ \rho=\dfrac{1}{\sigma} \ }
The units of resistivity are Ωm\Omega m.


Exam Tip
Resistivity and conductivity are reciprocals of each other.


Wize Concept
The resistivity/conductivity are intrinsic properties of the material, and don't depend on the size or shape of the conductor.



We can now relate the current density to the conductivity/resistivity of the material and the electric field inside:

 J⃗=σE⃗ \boxed{\ \vec{J}=\sigma\vec{E} \ }

where J⃗\vec{J} is the current density and E⃗\vec{E} is the electric field.


Wize Concept
The current density J⃗\vec{J} and electric field E⃗\vec{E} are proportional to each other (and pointing in the same direction, with the conductivity σ\sigma being the constant of proportionality.


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Example: Different Thicknesses


A cylindrical conductor is made out of copper as shown below. The two ends of it are connected to a battery. Compare the current, current density, electric field, conductivity, and resistivity in the regions A and B.




The current is the same everywhere inside, so we have IA=IBI_A=I_B.

The current density is defined to be J=IAJ=\dfrac{I}{A}, therefore inversely proportional to the area. Since the areas are AA>ABA_A>A_B, we have JA<JBJ_A<J_B.

Both pieces are made of the same material, so their resistivity and conductivity are the same: ρA=ρB\rho_A=\rho_B and σA=σB\sigma_A=\sigma_B.

The electric field is related to current density as J=σEJ=\sigma E, therefore they are proportional and since σ\sigma is the same for both, the same relationship holds for the field as for the density: EA<EBE_A<E_B.


Practice: Find Electric Field


A piece of copper wire has diameter of 2.42.4 mm and is connected to a battery. Find the magnitude of the electric field which is generating a current of 55 mA inside the wire. The resistivity of copper is 1.72×10−81.72\times 10^{-8} Ωm.