Question
By explaining the (Electric) current density, derive Ohm's law in vector form.
✓
Answer
$\rightarrow$ Current density : "The electric current flowing per unit cross sectional area perpendicular to the current is called the (electric) current density $(\vec{j}) . "$
$\rightarrow$ Suppose current $I$ is flowing through the cross sectional area $A $ then electric current density
$\therefore j=\frac{ I }{ A }$
$\rightarrow$ The $SI$ unit of current density is $\frac{ A }{ m ^2}$.
$\rightarrow$ It is a vector quantity and the dimensional formula is $M ^0 L^{-2} T^0 A^1$.
$\rightarrow $ If the magnitude of uniform electric field in the conductor of length $l$ is $E ,$
then the potential difference across its ends, $V = E l$
$\rightarrow $ According to the ohm's law
$V = IR$
$\text { but } R =\frac{ \rho l}{A}$
$ V =\frac{ I\rho l}{A}$
$\therefore E l =\frac{ I\rho l}{A}$
$\therefore E =\frac{ I\rho }{ A } (\because V = E l)$
$\rightarrow$ Using equation $(1),$
$\therefore E =j \rho $
$\rightarrow$ The reciprocal of resistivity is called the conductivity of material.
$\therefore \sigma=\frac{1}{\rho }$
$\rightarrow$ where $\sigma=$ conductivity of material from equation $(2)$ and $(3),$
$\therefore E =\frac{j}{\sigma}$
$\therefore j=\sigma E$
$\rightarrow $ This equation can be written in vector form
$\therefore \vec{j}=\sigma \overrightarrow{ E }$
$\rightarrow$ This equation is called vector form of Ohm's law.
$\rightarrow$ Suppose current $I$ is flowing through the cross sectional area $A $ then electric current density
$\therefore j=\frac{ I }{ A }$
$\rightarrow$ The $SI$ unit of current density is $\frac{ A }{ m ^2}$.
$\rightarrow$ It is a vector quantity and the dimensional formula is $M ^0 L^{-2} T^0 A^1$.
$\rightarrow $ If the magnitude of uniform electric field in the conductor of length $l$ is $E ,$
then the potential difference across its ends, $V = E l$
$\rightarrow $ According to the ohm's law
$V = IR$
$\text { but } R =\frac{ \rho l}{A}$
$ V =\frac{ I\rho l}{A}$
$\therefore E l =\frac{ I\rho l}{A}$
$\therefore E =\frac{ I\rho }{ A } (\because V = E l)$
$\rightarrow$ Using equation $(1),$
$\therefore E =j \rho $
$\rightarrow$ The reciprocal of resistivity is called the conductivity of material.
$\therefore \sigma=\frac{1}{\rho }$
$\rightarrow$ where $\sigma=$ conductivity of material from equation $(2)$ and $(3),$
$\therefore E =\frac{j}{\sigma}$
$\therefore j=\sigma E$
$\rightarrow $ This equation can be written in vector form
$\therefore \vec{j}=\sigma \overrightarrow{ E }$
$\rightarrow$ This equation is called vector form of Ohm's law.
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