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