Question

The electric field intensity $E$ , current density $J$ and specific resistance $k$ are related to each other through the relation:
(A) $E = \dfrac{J}{k}$
(B) $E = Jk$
(C) $E = \dfrac{k}{J}$
(D) $k = JE$

Answer 1

We need to define each term given in the question separately. Specific conductance is inversely proportional to each other.

Formula used: The formulae used in the solution are given here.
Current density $J = \dfrac{I}{A}$ where $I$ is the current and $A$ is the area of the cross section.
The specific resistance $k = \dfrac{{RA}}{L}$ where $R$ is the resistance, $A$ is the area of the cross section and $L$ is the length of the substance.
Specific conductance $\sigma = \dfrac{L}{{RA}}$ .
The electric field intensity $E = \dfrac{1}{{4\pi \varepsilon }}\dfrac{q}{{{d^2}}}$ where $\dfrac{1}{{4\pi \varepsilon }}$ is a constant, $q$ is the unit charge and $d$ is the distance between the charge and point of measurement.
Also, $E = \dfrac{I}{{\sigma A}}$ where $\sigma$ is the specific conductance.

Complete step by step solution:
The amount of electric current traveling per unit cross-section area is called current density and expressed in amperes per square meter. More the current in a conductor, the higher will be the current density. However, the current density alters in different parts of an electrical conductor and the effect takes place with alternating currents at higher frequencies.
Current density is a vector quantity having both a direction and a scalar magnitude. The electric current flowing through a solid having units of charge per unit time is calculated towards the direction perpendicular to the flow of direction.
Current density $J = \dfrac{I}{A}$ where $I$ is the current and $A$ is the area of the cross section.
Specific resistance is defined as the resistance offered per unit length and unit cross-sectional area when a known amount of voltage is applied.
Mathematically, specific resistance $k = \dfrac{{RA}}{L}$ where $R$ is the resistance, $A$ is the area of the cross section and $L$ is the length of the substance.
Specific resistance is the reciprocal of specific conductance, which is defined as a measure of a material’s ability to conduct electricity.
Thus, specific conductance $\sigma = \dfrac{L}{{RA}}$ .
The space around an electric charge in which its influence can be felt is known as the electric field. The electric field Intensity at a point is the force experienced by a unit positive charge placed at that point.
Mathematically, electric field intensity $E = \dfrac{1}{{4\pi \varepsilon }}\dfrac{q}{{{d^2}}}$ where $\dfrac{1}{{4\pi \varepsilon }}$ is a constant, $q$ is the unit charge and $d$ is the distance between the charge and point of measurement.
The intensity of the electric field at any point due to a number of charges is equal to the vector sum of the intensities produced by the separate charges.
Now, electric field intensity in terms of $E$ in terms of current $I$ by the relationship, $E = \dfrac{I}{{\sigma A}}$ where $\sigma$ is the specific conductance.
We already know that, $J = \dfrac{I}{A}$ .
Thus, we can write, $E = \dfrac{J}{\sigma }$ . Since the specific resistance is the reciprocal of specific conductance, we get,
$E = Jk$ .
Hence the correct answer is Option B.

Note:
Electric field is an elegant way of characterising the electrical environment of a system of charges. Electric field at a point in the space around a system of charges tells you the force a unit positive test charge would experience if placed at that point (without disturbing the system). Electric field is a characteristic of the system of charges and is independent of the test charge that you place at a point to determine the field.