Question

Obtain the equation $\overrightarrow{J}=\sigma \overrightarrow{E}$ of Ohm's law on the basis of drift velocity.

Answer 1

Step 1: Ohm's law and its parameters

1. Ohm's law states that the voltage across a resistance is proportional to the current flowing through the resistance when the temperature and other physical states are constant with time.
2. Ohm's law is defined by the form, $V=IR$, where, I is the current flowing through the resistance, R is the constant of proportionality and V is the voltage across the resistance.
3. $\overrightarrow{J}=\sigma \overrightarrow{E}$ is the vector form of Ohm's law, where, J is the current density, which is defined as $J=\frac{Currentflow\left(I\right)}{crosssectionalarea\left(A\right)}$ , $E$ is the electric field, and $\sigma$ is the current conductance of the resistance.

Step 2: Resistance and conductivity

The resistance, $R=\rho \frac{L}{A}$, where, $\rho$ is the resistivity of the resistance, L and A are the length and area of the conductor.

Conductivity is the reciprocal of resistivity, i.e, $\sigma =\frac{1}{\rho }$, where, $\sigma$ is the conductivity.

Step3: Finding the expression $\overrightarrow{J}=\sigma \overrightarrow{E}$

As we know, $J=\frac{Currentflow\left(I\right)}{crosssectionalarea\left(A\right)}$

$orJ=\frac{I}{A}$

$orJ=\frac{V}{RA}$ (From ohm's law, $V=IR$)

$orJ=\frac{V}{A\times \left(\rho {\displaystyle \frac{L}{A}}\right)}=\frac{V}{\rho L}$ (since, resistance, $R=\rho \frac{L}{A}$).

$orJ=\frac{EL}{L\times \rho }=\frac{E}{\rho }$ (since we know $V=EL$).

$orJ=E\sigma$ ($\sigma =\frac{1}{\rho }$, conductance is the reciprocal of resistivity).

So, as a vector form, $\overrightarrow{J}=\overrightarrow{E}\sigma$ (proved).