What is Displacement Current : Derivation & Its Properties

In electromagnetic theory, the phenomenon of the magnetic field can be explained concerning a change in the electric field. The magnetic field is produced in the surroundings of the electric current (conduction current). Since the electric current might be in the steady-state or varying state. The concept displacement current depends on the variation of time of the electric field E, developed by the British physicist James Clerk Maxwell in the 19th century. He proved that the displacement current is another kind of current, proportional to the rate of change of electric fields and also explained mathematically. Let’s discuss the displacement current formula and necessity in this article.


What is the Displacement Current?

The displacement current is defined as, the type of current produced due to the rate of electric displacement field D. It is a time-varying quantity introduced in Maxwell’s equations. It is explained in the units of the density of electric current. It is introduced in the law of Ampere circuits.
The SI unit of displacement current is Ampere (Amp). The dimension of this can be measured in the unit of length, which can be the max, min or equal to the actual distance traveled from an initial point to endpoint.

Derivation

The displacement current formula, dimensions, and derivation of displacement current can be explained by considering the basic circuit, that gives the displacement current in a capacitor.

Consider a parallel plate capacitor with a required power supply. When the supply is given the capacitor, it starts charging and there will be no conduction of current initially. With the increase in time, the capacitor charges continuously and accumulates above the plates. During the charging of a capacitor with time, there will be a change in the electric field between the plates which induces the displacement current.

From the given circuit, consider the area of the parallel plate capacitor = S

Displacement current = Id

Jd = displacement current density

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d= €E ie., related to electric field E

€ = permittivity of the medium between the plates of a capacitor

The displacement current formula of a capacitor is given as,


Id= Jd × S = S [dD / dt]

Since Jd = dD/dt

From Maxwell’s equation, we can conclude that the displacement current will have the same unit and effect on the magnetic field of the conduction current.

▽×H=J+Jd

Where,

H = magnetic field B as B=μH

μ = permeability of the medium in between the plates of a capacitor

J = conducting current density.

Jd =displacement current density.

As we know that ▽(▽×H)=0 and ▽.J=−∂ρ/∂t=−▽(∂D/∂t)

By using Gauss’s law that is ▽.D=ρ

Here, ρ = electric charge density.

Hence we can conclude that, Jd=∂D/∂t displacement current density and it is necessary to balance RHS with LHS of the equation.

The Necessity of Displacement Current 

There is no flow of charge carriers through the two plates of a capacitor and the conduction current doesn’t take place through this insulation. The continuous magnetic field effects between the plates give the displacement current. The size of this can be calculated from the charging and discharging current of a circuit which is equal to the size of the conduction current of a conducting wire that connects a capacitor (starting point to ending point)

The necessity of this can be explained by considering the following factors,

  • In electromagnetic radiation like light waves and radio waves are propagated into space.
  • When the varying magnetic field is directly proportional to the rate of change of the electric field.
  • The displacement current is necessary to produce the magnetic field between the two plates of a capacitor.
  • Used in Amperes circuit.
  • The displacement current is made possible to understand how the electromagnetic waves propagate through empty spaces.

Displacement Current in a Capacitor

A capacitor always depends on the displacement current and not on the conduction current when there is a potential difference is below the max voltage between the plates. Since we know that, the flow of electrons gives the conduction current. While this current in a capacitor is due to the rate of change of electric field which is equivalent to the current flowing through the plates.

Displacement Current in a Capacitor
Displacement Current in a Capacitor

When the maximum voltage is applied to the capacitor, it starts charging and conducting. When the voltage exceeds, then it acts like a conductor and results in a conduction current. At this stage, it is called as breaking down of a capacitor.

Difference between Conduction Current and Displacement Current

The difference between conduction current and displacement current include the following.

Conduction Current

Displacement Current

It is defined as the actual current produced in the circuit due to the flow of electrons at an applied voltage. It is defined as the rate of change of the electric field between the plates of a capacitor at an applied voltage.
It is produced due to the flow of charge carriers (electrons) uniformly whereas the electric field is constant with time It is produced due to the movement of electrons with the rate of change of electric field
It accepts ohm’s law It doesn’t accept ohm’s law
It is given as I = V/R It is given as Id = Jd x S
It is represented as actual current It is represented as apparent current produced due to the electric field in a varying time

Properties

The properties of displacement current are mentioned below,

  • It is a vector quantity and obeys the property of continuity in a closed path.
  • It changes with the rate of change of current in an electric density field.
  • It gives zero magnitudes when the current in an electric field of a wire is steady
  • It depends on the varying time of an electric field.
  • It had both direction and magnitude, which can be a value of positive, negative, or zero
  • The length of this can be taken as the minimum distance from the starting point to the ending point regardless of the path.
  • It can be measured in a unit of length
  • It has a minimum or maximum or equal magnitude of displacement for a given time to the actual distance from the point.
  • It depends on an electromagnetic field.
  • It gives zero value when the starting point and the ending point is the same

Thus, this is all about an overview of the displacement current – formula, derivation, significance, necessity, and displacement current in a capacitor. Here is a qi for you, ” What is conduction current in a capacitor? “

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