Current Density Formula

An electrical current is a stream of electrons or ions that are passing through the electrical conductors or space. The current is the scalar quantity. They can be measured by calculating the net flow of electric charges passing through a surface. According to the ohms law, the current (I) across the conductor is directly proportional to the potential difference (V) between the two points and inversely proportional to the resistance (R) of the conductor 

$I = \dfrac{V}{R}$

What is Current Density? 

Using the current density equation, we can calculate the amount of electric charge flowing per unit area of the conductor’s cross-section in the unit time. The current density is a vector quantity. So, the current density has both magnitude and direction. The charge density can be denoted by J. 

The Electric current density formula is,

$J = \dfrac{I}{A}$

Here, 

J denotes the current density

I denote the current passing through a conductor

A denotes the cross-sectional area of the conductor

As the SI unit of current (I) is ampere and the unit of the cross-sectional area of the conductor is in $m^2$. The SI unit of conduction current density formula is A/m\[^{2}\]. 

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Diagram shows the flow of electrons in a conductor, whose cross-sectional area is A. 

Displacement Current

According to the Maxwell's equation, the displacement current has the same effect and unit on the magnetic field as for conduction current. The SI unit of displacement current density formula is C/m\[^{2}\]

▽×H=J+J\[_{D}\]

Here, J\[_{D}\] represents the displacement current density formula

H is related to the magnetic field around the conductor B 

B=μH

Here, μ  represents permeability of the medium in between the two plates

Surface Current Density

The surface charge density is a measurement of electric charges accumulated over a surface. The surface charge density can be calculated by charges per unit surface area.  The SI unit of the surface current density formula is Cm\[^{-2}\] or C/m\[^{2}\]. And surface current density formula is σ=qA

Here, 

q represents the charge

A represents the surface area. 

Conduction Current Density

The conduction current density termed the amount of current or charges that flow through the conduction surface within a time t. Here, the conduction surface is always parallel to the current flow. 

Conduction current density formula J\[_{c}\] = σE.

J\[_{c}\] denotes the conduction current density.

E denotes the electric field

σ denotes the electrical conductivity. 

Corrosion Current

A corrosion current is the current, which is produced in an electrochemical cell while occurring corrosion. The loss of electrons from the anodes in electrochemical cells subsequently triggers the oxidation reaction. This may lead to corrosion of the anode and the cathode in the electrochemical cell remains unaffected. 

Corrosion current CR = 3.27x10\[^{-3}\]i\[_{corr}\]E\[_{w}\]/p

i\[_{corr}\] represents corrosion current density. And corrosion current density formula i\[_{corr}\] =  10\[^{6}\] B/R\[_{p}\].  

E\[_{w}\] represents the equivalent weight of the corroding metals in grams

p represents the density of the corroding metal 

B represents the Stern-Geary coefficient 

R\[_{p}\] shows the polarization resistance in ohm-cm\[^{2}\]. 

Solved Problems

1. A copper wire with an area of $3\,mm^2$ has a current of $2\,mA$ running through it. Find out its current density.

Ans: The quantities that are given;

Area of the wire, $d=3\,mm^2 = 3 \times 10^{-3}\,m^2$

Current, $I=2\,mA=2 \times 10^{-3}\,A$

Now applying the formula of current density;

$J=\dfrac{I}{A}$

Upon putting the values of quantities, we get;

$J=\dfrac{2 \times 10^{-3}}{3 \times 10^{-3}}$

$J=0.66\, A/m^2$

Hence, the current density of the wire is $0.66\, A/m^2$.

2. What is the surface charge density of a conductor having charge of $3\,C$ in an area of $8\,m^2$?

Ans: According to the question, the given quantities are;

Charge, $q=3\,C$

Area of the conductor, $A=8\,m^2$

The expression of surface charge density is given as;

$\sigma=\dfrac{q}{A}$

Upon putting the values of quantities, we get;

$\sigma=\dfrac{3}{8}$

$\sigma= 0.375\,C/m^2$

Hence, the value of surface charge density is $0.375\,C/m^2$.

Summary

In this post we have discussed the concept of current density and came to know that it is the ratio of current flowing through the conductor to the area of the conductor. We have also mentioned about the concept of displacement current which is responsible for the continuity of current between the plates of the capacitor. We have also mentioned surface charge density and in the end discussed the numericals based on these concepts for better understanding.