Answer

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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.**

**Note**

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 $ .

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