Electrical conductivity (σ) is a measure of a material's ability to conduct electricity. It is defined as the reciprocal of resistivity (ρ):
σ = 1/ρ
To derive the equation for electrical conductivity, we need to understand the relationship between current (I), voltage (V), and resistance (R) in a material. This relationship is described by Ohm's Law:
V = IR
Where:
* V is the voltage across the material
* I is the current flowing through the material
* R is the resistance of the material
Resistance, in turn, is dependent on the material's resistivity (ρ), length (L), and cross-sectional area (A):
R = ρL/A
Now, combining these equations, we get:
V = I(ρL/A)
Rearranging to solve for current density (J = I/A):
J = V/(ρL)
Since the electric field (E) is defined as the voltage difference per unit length (E = V/L), we can rewrite the above equation as:
J = E/ρ
Finally, substituting the definition of conductivity (σ = 1/ρ), we arrive at the equation for electrical conductivity:
σ = J/E
Therefore, electrical conductivity is defined as the ratio of current density to electric field strength.
In summary, the derivation of the electrical conductivity equation can be summarized as follows:
1. Start with Ohm's Law: V = IR
2. Relate resistance to resistivity: R = ρL/A
3. Substitute resistance into Ohm's Law: V = I(ρL/A)
4. Rearrange to get current density: J = V/(ρL)
5. Express voltage difference in terms of electric field: J = E/ρ
6. Substitute conductivity for resistivity: σ = J/E
This derivation shows that electrical conductivity is a fundamental property of a material that governs its ability to conduct electricity under an applied electric field.
