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.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
With the help of suitable diagram explain the real depth and apparent depth.
Figure, shows two coherent sources $S_1$ and $S_2$ which emit sound of wavelength $\lambda$ in phase. The separation between the sources is $3\lambda.$ A circular wire of large radius is placed in such a way that $S_1S_2$ lies in its plane and the middle point of $S_1S_2$ is at the centre of the wire. Find the angular positions $\theta$ on the wire for which constructive interference takes place.
→In RL and RC series AC circuits the resonance phenomenon isn't possible. Why?
A long wire is bent as shown in the figure. Find the magnitude and direction of magnetic field at center $O$ of the circular part when a current $i$ ampere flows through it.
→- The peak voltage of an ac supply is 300V. What is the rms voltage?
- The rms value of current in an ac circuit is 10A. What is the peak current?
Rain is falling vertically. A man running on the road keeps his umbrella tilted but a man standing on the street keeps his umbrella vertical to protect himself from the rain. But both of them keep their umbrella vertical to avoid the vertical sun-rays. Explain.
An electric field $\vec{\text{E}}=\vec{\text{i}}\text{Ax}$ exists in the space, where $A= 10Vm^{-2}$. Take the potential at (10m, 20m) to be zero. Find the potential at the origin.
→A charge ‘q’ moving along the X-axis with a velocity v is subjected to a uniform magnetic field B acting along the Z-axis as it crosses the origin O.
- Trace its trajectory.
- Does the charge gain kinetic energy as it enters the magnetic field? Justify your answer.
A simple microscope using a single lens often shows coloured image of a white source. Why?
A wire is bent in the form of a regular hexagon and a total charge q is distributed uniformly on it. What is the electric field at the centre? You may answer this part without making any numerical calculations.