Question

The dimensional formula of current density is
$A)\text{ }[{{M}^{0}}{{L}^{-2}}{{T}^{-1}}Q]$
$B)\text{ }[{{M}^{0}}{{L}^{2}}{{T}^{1}}{{Q}^{-1}}]$
$C)\text{ }[ML{{T}^{-1}}Q]$
$A)\text{ }[M{{L}^{-2}}{{T}^{-1}}{{Q}^{2}}]$

Answer 1

Hint: This problem can be solved by using the mathematical formula for the current density as a function of the current passing through the cross section of a conductor with the cross sectional area. By using the dimensional formula for the individual physical quantities in the mathematical definition, we can get the dimensional formula for the current density.

Formula used:
$J=\dfrac{I}{A}$

Complete step-by-step answer:
We will solve this problem by using the mathematical formula for current density and use the dimensional formulae for the individual quantities to get the dimensional formula for current density. Hence, let us proceed to do that.
The current density $J$ passing through a cross section of a conductor is given by
$J=\dfrac{I}{A}$ --(1)
where $I$ is the net current passing through the cross section of the conductor with cross sectional area $A$.
Therefore, using (1), we get,
$\left[ J \right]=\left[ \dfrac{I}{A} \right]=\dfrac{\left[ I \right]}{\left[ A \right]}$ --(2)
Now, the dimensional formula for current is $\left[ {{M}^{0}}{{L}^{0}}{{T}^{-1}}Q \right]$.
The dimensional formula for cross sectional area is $\left[ {{M}^{0}}{{L}^{2}}{{T}^{0}}{{Q}^{0}} \right]$.
Putting these values in (2), we get,
$\left[ J \right]=\dfrac{\left[ {{M}^{0}}{{L}^{0}}{{T}^{-1}}Q \right]}{\left[ {{M}^{0}}{{L}^{2}}{{T}^{0}}{{Q}^{0}} \right]}=\left[ {{M}^{0}}{{L}^{-2}}{{T}^{-1}}Q \right]$
Hence, the dimensional formula for electric current density is $\left[ {{M}^{0}}{{L}^{-2}}{{T}^{-1}}Q \right]$.
Therefore, the correct option is $A)\text{ }[{{M}^{0}}{{L}^{-2}}{{T}^{-1}}Q]$.

Note: Students must remember that even though current density can be written as the ratio of two scalar quantities (current and area), it is defined as a vector quantity. The direction of current density is perpendicular to the cross sectional area in the direction of the flow of electric current. Current density can also be written as the ratio of the electric field to the resistivity of a material. Then the fact that the current density is a vector is clearer since the electric field is a vector. Current density will be in the direction of the electric field.