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Ohm’s law in vector form is:
(A) $ V = I.R $
(B) $ \vec J = \sigma \vec E $
(C) $ \vec J = \rho \vec E $
(D) $ \vec E = \sigma \vec J $

Last updated date: 18th Jun 2024
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Hint: The quantities used in Ohm’s law are all scalar. To convert them into vector form, the resistance is written in the form of resistivity, length, and area of cross-section. The quantities are then rearranged to convert the potential difference into the potential gradient and the current into current density, which is vector quantities.

Complete step by step solution
Ohm’s law states that the current passing through a conductor is directly proportional to the potential difference between the two points. The constant is known as Resistance and it is the property of a material to resist the flow of electric current through a material.
It can be written as,
 $ V = IR $
where $ V $ is the potential difference between the two ends of the wire
 $ I $ is the current
and $ R $ is the resistance of the wire.
The resistance of a wire can be given by the formula-
 $ R = \dfrac{{\rho L}}{A} $
Here, $ \rho $ is the resistivity of the material,
 $ L $ is the length of the wire,
And $ A $ is the cross-section area of the wire.
Conductivity $ \left( \sigma \right) $ is the reciprocal of resistivity or,
 $ \rho = \dfrac{1}{\sigma } $
This value of resistivity can be substituted in the formula of resistance to give-
 $ R = \dfrac{L}{{\sigma A}} $
Now we substitute this value of resistance in Ohm’s law,
It becomes-
 $ V = \dfrac{{IL}}{{\sigma A}} $
If we rearrange this equation by shifting $ L $ and $ \sigma $ on the LHS, we get-
 $ \sigma \dfrac{V}{L} = \dfrac{I}{A} $
We know that Potential Gradient,
 $ \vec E = \dfrac{V}{L} $ is a vector quantity.
Also, the current density,
 $ \vec J = \dfrac{I}{A} $ is also a vector quantity.
Thus the ohm’s law can be rewritten as-
 $ \sigma \vec E = \vec J $
Or $ \vec J = \sigma \vec E $
Thus, option (B) is the correct answer.

The terms current density and the potential gradient are vectors because the area and the length of the wire are assumed to have a particular direction. The formula $ \vec J = \sigma \vec E $ can also be written in terms of resistivity, $ \rho \vec J = \vec E $ .