What is current density?

The amount of charge flowing per unit area of a chosen cross-section is called a current density in electromagnetism. The current density vector is the electric current per unit cross-sectional area at a given point in space. Its direction is the same as that of the motion of the positive charges at this point. In SI unit system the current density is measured as amperes per square metre or $A/m^2$. Its dimensions are $I{L}^{-2}$ .

At a given point M, assume that A is a small surface area and perpendicular to the motion of the charges at point M. Consider $I_A$ as the current flowing through the surface area A and electric current density j at point M is given by the following equation- 

$j = \underset{A\to 0}{\lim } \frac{I_A}{A} = \frac{\partial I}{\partial A}{|}_{A=0}$

Surface A is centered at point M and A is perpendicular to the motion of the charges during the limit process.

The current density vector has the magnitude same as the electric current density. Its direction is the same as the motion of the positive charges at M.

If v is the velocity of the charges and $\partial A$ is the small surface centered at point M and perpendicular to v then only the charge present in the volume formed by $\partial A$ at given time dt and $I = \frac{dq}{dt}$ will flow through small area dA. This type of charge is given by $\rho ||v||\mathrm{dt} \mathrm{dA}$ . The current density vector can be given by, $j = \mathrm{\rho v}$. Over the time duration $t_1$ to $t_2$, the surface integral of j over a surface S gives the total amount of charge which is flowing through the surface in that time $(t_2-t_1)$ : 

$q = \int {\int }_S j . \hat{n} dA dt$

Between $t_1$ and $t_2$, this is the integral of the flux of j across S. The area required to calculate the flux is flat, curved, real, imaginary, either as a cross-sectional area or a surface area. For instance, when the charge carriers pass through a conductor, the area is the cross-sectional area of the conductor. The vector area is the area of the conductor through which the charge carriers pass. Consider A as the unit vector and it is normal to the area $\hat{n}$ and the relation is given by, $A = A\hat{n}$ 

Also, the differential vector area is given by the equation similarly as, $dA = dA\hat{n}$. 

Consider if the current density j passes through an area at an angle $\theta$ to the area normal $\hat{n}$ then it is given by, $j . \hat{n} = j \cos \left(\theta \right)$

where "." represents the dot products of the vectors and the component of current density passing tangentially to the area is $j \sin \left(\theta \right)$ and the component of current density passing through the surface is $j \cos \left(\theta \right)$. There is no component of current density passing tangentially through the area and the only component of current density passing normal to the area is the cosine component. 

Importance of current density

To design the electrical and electronic system, the current density is an important factor. The designer current level is the factor on which the circuit performance depends and with the help of the dimensions of the conducting current the current density is then determined. For instance, despite the lower current demanded by smaller devices as integrated circuits are reduced in size, there is a type of trend in achieving the higher device number in even smaller chip areas. The current density is increased in this region at higher frequencies because the conducting region in a wire becomes confined and this is known as the skin effect. The consequences increase as the current densities become higher.

To dissipate power in the form of heat most of the electrical conductors have finite, positive resistance. To prevent the conductor from melting, burning, or insulating material falling, the current density must be kept low. A phenomenon called electromigration happens because at high current densities the material forming the interconnections actually moves. To cause spontaneous loss of the superconductive property, the current density may generate a strong magnetic field in superconductors. In Ampere's circuital law, the current density is one of the major factors because it relates current density to the magnetic field. Charge and current are combined into a 4-vector in the case of special relativity theory.

Calculation of current densities in matter

Free currents

A free current density is one that consists of charge carriers. The average quantity that tells us what is happening in the entire wire is the electric current. The current density describes the distribution of charge flowing at position r at time t.

$j(r, t) = \rho (r, t) v_d(r, t)$

where j(r, t) = current density vector, $v_d$(r, t) = particle average drift velocity and $\rho (r, t) = q n (r, t)$ is the charge density where n(r, t) = number of particles/volume and q = charge of the individual particle with density n. 

The electric current is simply proportional to the electric field and it is the common approximation to the current density and is given by -

$j = \mathrm{\sigma E}$

where E = electric field and $\sigma$ = electrical conductivity.

The reciprocal of resistivity is called the conductivity which is denoted by $\sigma$ and its SI units is S.$m^{-1}$ and the SI unit of electric field E is newton per unit coulomb or  N$C^{-1}$. 

Polarization and Magnetization currents

When there is a non-uniform distribution of charge, the current arises in materials. The polarization P is the net movement of electric dipole moment per unit volume.

$j_{P = \frac{\partial P}{\partial t}}$

The magnetization M is the circulations of the magnetic dipole moment per unit volume in magnetic materials and it leads to magnetization currents. 

$j_{M = \nabla X M}$

The bound current density in the material is formed by adding these terms together. 

$j_{b }= j_P+ j_M$

Total currents in materials

The sum of free and bound currents is called the total currents.

$j = j_r+ j_b$

Displacement current

Corresponding to the time-varying electric displacement field D, there is also a displacement current which is given by,

$j_D = \frac{\partial D}{\partial t}$

and this is one of the most important terms in Ampere's circuital law and one of the equations of Maxwell's. The absence of this term would not predict the time evolution of electric fields and electromagnetic waves to propagate in general. 

Context and Applications

This topic is significant in the professional exam for undergraduate, graduate, and postgraduate courses.

• Bachelors of Technology (Electrical Engineering)
• Bachelors of Technology (Electronics and Telecommunication Engineering)
• Masters of Technology (Electrical Engineering)
• Masters of Technology (Electronics and Telecommunication Engineering)

Practice Problems

Question 1: Which is the correct symbol for current density?

1. $\beta$
2. $\xi$
3. j
4. P

Answer: Correct option c

Explanation: Current density is represented by the symbol "j".

Question 2: In the SI unit system how current density is measured?

1. N/$m^2$
2. A/$m^2$
3. N/C
4. None of the above

Answer: Correct option b

Explanation: In the SI unit system the current density is measured as amperes per square metre or A/$m^2$.

Question 3: What are the dimensions of current density?

1. $I{L}^{-2}$
2. $I{L}^{-1}$
3. $I{L}^2$
4. None of the above

Answer: Correct option a

Explanation: The dimensions of current density are $I{L}^{-2}$.

Question 4: What does free current density consist of?

1. Potential difference
2. magnetic field
3. charge carriers
4. macroscopic quantity

Answer: Correct option c

Explanation: A free current density consists of charge carriers.

Question 5: Which component of current density passes through the area?

1. $j \sin \left(\theta \right)$
2. $j \cos \left(\theta \right)$
3. Both 1 and 2
4. None of the above

Answer: Correct option b

Explanation: There is no component of current density passing tangentially through the area and the only component of current density passing normal to the area is the cosine component. 

Related Concepts

• Hall effect
• Superconductivity
• Electron mobility
• Drift velocity
• Electrical resistance
• Speed of electricity