j = sigma e derivation for the Vector form of Ohm’s law

In this post, we will derive the equation j = sigma e and we will do the derivation using Ohm’s law equation (V=IR). Obtaining this equation vec J = σ vec E, or deriving J = σ E helps us to get the relationship between the current density, conductivity, and electric field intensity. This J = σ E is also the vector form of Ohm’s Law.

j = sigma e derivation | derive J = σ E | Vector form of Ohm’s law

As we derive Ohm’s Law using drift velocity equations, we get the equations of resistance R and resistivity ρ. Here, using these formulas we will derive the Vector form of Ohm’s law (J = σ E).

The equation of resistance R is expressed like this:
R = ρ (L/A) where ρ is the resistivity (or specific resistance), L is the length of the conductor, and A is the area of the conductor.

ρ = RA/L ……………….. (i)

Using Ohm’s Law equation (V=IR)

ρ = (V/I) (A/L) = (VA) / (IL)=(V/L) (A/I) = E / J

[ as we know, Electric field intensity = E = V/L
and J = current density = I/A ]

ρ = E / J ……………. (ii)

E = ρ J ……………. (iii)

J = E/ρ

J = σ E ……………. (iV)

Here, σ = conductivity or specific conductance = 1/ρ

In vector form: J = σ E ……………. (V). This is the Vector form of Ohm’s Law.

Thus we have completed the j = sigma e derivation using Ohm’s law formula.

Related:

derive Ohm’s Law using drift velocity equations

resistivity

j = sigma e derivation for the Vector form of Ohm’s law
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