Current Density

Current Density is defined as the amount of current flowing through the per unit area of the wire. It is measured in ampere/m². Before learning more about current density we have to learn about current. 

Current is defined as the flow of the electron in a wire under a voltage difference. Current is the most common type of energy used in our daily life it provides electrical energy which is responsible for the working of all the electrical appliances which we used in our daily life.

In this article, we will learn about current density, its formula, derivation with brief introduction of electric current.

Current Definition

The flow of electrons or holes in the conductor is defined as the electric current. Electric current flows because of the electro-potential force generated at the end of the conductor by battery or AC sources. The current is defined by the symbol “I “and is measured in the Ampere. Current is classified as Alternating current and direct current depending upon the direction of flow.

The following image shows current flowing through the circuit which lights up the bulb.

Current is the fundamental quantity and its dimensional symbol is [I].

The formula for calculating the current is, 

i = q/t

where,
i is the current,
q is charge following in the conductor,
t is the time taken

One Ampere is defined as one Coulomb of the charge flowing through the conductor in one sec. I amp current is defined as 6.24×10¹⁸ electrons flowing through the conductor in one sec.

What is Current Density?

The rate of flow of current across the unit cross-section area of a conductor is referred to as current density. 

A flow of electrons is what an electric current is commonly thought to be. Electrons flow out of one end of the battery, through the wire, and enter the other end of the battery when two ends of the battery are connected by metal wire. If the amount and direction of the current are both constant, the current produced is usually considered steady.

The density of current flow in per unit area of conductor is defined as current density. The letter J is used to represent it. Current Density and its measurement are particularly essential in the study of electromagnetism. The charge per unit time per unit area of an electrical conductor is called the current Density and it is a vector quantity as it has both magnitude and direction.

Formula for Current Density 

The current density formula is as follows:

J = I/A

where,

• J is the Current Density (A/m²)
• I is the flow of current through the conductor (A)
• A is the Cross sectional area (m²)

Unit and Dimension of Current Density

• The SI unit for measuring the current density is Ampere/m²
• The dimensional formula for density is [M⁰L^-2T⁰I¹]

Derivation of Current Density Formula

The current density formula is derived from the definition of the current density. We know that the current density is defined as the current flowing through the per unit area of the conductor then,

J = dI/dA

where,
dI/dA is the Current change with the unit area.

which implies,

dI = J × dA

dI = J × dAcosθ

when 
dA is perpendicular to J,   (if θ = 90^o so, cos θ = 1).

Therefore, 

dI = J × dA

integrating both sides

∫₀ ^J dI = J × ∫₀ ^A dA

I = J×A

J = I/A

Do Check,

• What is Alternating Current?
• Electric Current in Conductors
• Difference Between Electric Potential and Potential Difference

Current Density Formula Examples

Example 1: What is the current density in a wire with a cross-section area of 23 mm² and a current of 8 mA running through it?

Solution:

Given,

A = 23 mm²
   = 0.000023 m², 

I = 8 mA = 0.008 A

Since,

J = I/A

   = 0.008 / 0.000023

   = 347.82 A/m²

Example 2: A copper wire with a diameter of 3 mm² has a current of 9 mA flowing through it. Find the current density of the conductor.

Solution:

Given,

A = 3 mm²
   = 0.000003 m², 

I = 9 mA 
  = 0.009 A

Since,

J = I/A

  = 0.009 / 0.000003

  = 3000 A/m²

Example 3: If 42 A of current flows through the battery in an 8 m² region, what is the current density of the conductor?

Solution:

Given,

A = 8 m², 

I = 42 A

Since,

J = I/A

  = 42 / 8

  = 5.25 A/m²

Example 4: If the current density is 4 A/m² and the current through the conductor is 24 A. Find the cross-sectional area.

Solution:

Given,

J = 4 A/m²

I = 24 A

Since,

J = I/A

A = I/J

   = 24/4

  = 6 m²

Current Density Frequently Asked Questions

What is current density?

The rate of charge flow across any cross-section of any conductor is referred to as current density of the conductor.

Is current density scalar or vector quantity?

Current Density is a vector quantity as it has both magnitude and direction.

What is the current density formula?

The current density of any material is measured with the formula, 

J = I/A

Why current density is a vector density?

Current density has both magnitude and direction and thus it is a vector quantity as vector quantities have direction.

What is current?

The rate of flow of electric charge through a conductor is defined as the electric current. It can be classified as,

• Direct Current
• Alternating Current