Answer
397.2k+ views
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 $ .
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 $ .
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)