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
$\text{V = IR}$
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
$\text{V = IR}$
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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