Electric Displacement

What is Electric Displacement?

Electric displacement, denoted by D, is the charge per unit area that would be displaced across a layer of conductor placed across an electric field. It is also known as electric flux density.

Electric displacement is used to in dielectric material to find the response of the materials on the application of an electric field E. In Maxwell’s equation, it appears as a vector field.

The SI unit of electric displacement is Coulomb per metre square (C m-2).

The mathematical representation is as follows:

\(D\equiv \epsilon _{0}E+P\)

Where,

  • ϵ0: Vacuum permittivity
  • P: Polarization density
  • E: Electric field
  • D: Electric displacement field

It was found that if electric field E is applied on the dielectric material, the bound charges inside the material responds to the electric field such that the distribution of positive changes take place in the direction of the electric field while the distribution of negative charges takes place in the opposite direction of the electric field. Before the application of the electric field, the material is electrically neutral and there is a formation of electric dipoles.

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What is Dielectric Material?

A dielectric material is an electrical insulator such that when an electric field is applied it can be polarised. There is no flow of electric charges when placed in an electric field but they do move from their average equilibrium position resulting in dielectric polarisation.

These materials find applications in capacitors, radios and transmission lines for radio frequency. The dielectric material in a capacitor is given as:

\(C=\frac{\epsilon_{r} \epsilon_{0} A}{d}\)

Where,

  • C: Capacitance
  • ϵ0: Vacuum permittivity
  • ϵr: Relative permittivity of the material
  • A: Area of the plates used in the capacitor
  • d: Distance between the plates

What is the Vacuum Permittivity?

Vacuum permittivity is the value of absolute dielectric permittivity ie; it is the capability of the vacuum to permit electric field lines. Following is the approximate value and formula of vacuum permittivity which is represented by ϵ0:

ϵ0 = 8.854187  …*10-12 F.m-1

\(\epsilon _{0} = \frac{1}{\mu _{0}c^{2}}\)

What is Polarisation Density?

Polarisation density is used in classical electromagnetism to express the density of induced electric dipole moments in a dielectric material. The electric polarisation of the dielectric is the electric dipole moment induced per unit volume of the dielectric material. The mathematical representation is:

\(P=\frac{\Delta p}{\Delta V}\)

Where,

  • P: polarisation density
  • Δp: dipole moment
  • ΔV: volume element

Displacement Field in Capacitor

Consider a parallel-plate capacitor with no free charges except on the capacitor placed in space. Let A be the area of the capacitor with thickness t and separation d. To calculate the electric field between the conductors, the potential difference between the plates is given as:

\(C=\frac{Q}{V}\)

 

Using Gaussian cylinder that encloses some charge on one plate, we can find E and D:

\(\oint D.da=4\pi Q_{f}\) \(D.a=4\pi \sigma _{f}a=4\pi a\frac{Q}{A}\) \(D=-4\pi \frac{Q}{A}\hat{z}\)

 

The electric field outside and inside the plate are:

EV = D

\(E_{d}=\frac{D}{\epsilon }\)

EV (d-t) + Edt = V

\(E_{V}=\frac{V}{d-t+\frac{t}{\epsilon }}=D=\frac{4\pi Q}{A}\)

Therefore,

\(C=\frac{Q}{V}\)

=\(\frac{A}{4\pi [d-t-\frac{t}{\epsilon }]}\) \(C_{empty}=\frac{A}{4\pi d}\) \(C_{full}=\frac{\epsilon A}{4\pi d}=\epsilon C_{empty}\)

Sine ϵ ≥ 1, the capacitance is increased for fixed plate size and separation.

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