Electrostatics - Dielectric Polarization

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Dielectric Polarization

When a non – polar dielectric is held in an external electric filed  ~Eo  the centre of positive charge (protons) in each molecule is pulled in the direction of  ~Eo  (towards negative plate) and the centre of negative charge (electrons) is pulled in a direction opposite to  ~Eo  (towards positive plate) therefore the two centers of positive and negative charges in the molecule are separated. The molecule gets distorted and is said to be polarized or a tiny dipole moment is imparted to each molecule. In fact, the force due to electric field pulling the two centers apart and the force of mutual attraction between the centers of positive and negative charges reach a balance or equilibrium and the molecule gets polarized. 

A dielectric with polar molecules also develops a net dipole moment in an external field but for a different reason. When no external field is applied the different permanent dipoles of such a dielectric are oriented randomly due to thermal agitation so the total dipole moment is zero. When an external electric fields is applied the individual dipole moments tend to all in with the field when summed over all the molecules, thee appears some net dipole moment in the direction of the external field the dielectric is polarized. The extent of polarization depends on the relative strength of two mutually opposing factors. The applied external electric field tending to align the dipoles with field and thermal energy tending to disrupt the alignment.

Thus each molecule becomes a tiny electric dipole, with a dipole moment parallel to the external field, and proportional to it. 

Induced dipole moment p acquired by the molecule may be written as 

P = a ∈₀ ~E_o

Where a is a constant of proportionality and is called atomic/molecular polarizability.

From (22), we can write the dimensions of a:

A = dimensions of p / (dimensions of ∈₀) (dimensions of ~E_o) = C m / (C₂N^-1m^-2)(NC^-1) = m³

Thus atomic polarizability has the dimensions of volume. For most of the atoms a is of the order of 10^-29 to 10^-30 m³; which is also the order of atomic volume. 

Let us now consider a non- polar dielectric slab ABCD placed in an electric field E₀ maintained between the two plates.

Suppose all its atoms are uniformly polarized in the direction of E₀ and z is the displacement between the centers of the ± charges in the atom. Dipole moment of each atom p = q x. 

If N is the number of atoms per unit volume then dipole moment per unit volume = total dipole moment density

P = Np or P = Nq x 

This dipole moment density p is called electric polarization. The units of p are 

C^-m (m^-3) = C-m^-2

If we consider any small volume element in the interior of the slab, each volume element has no net charge thought it has net dipole moment. This is because positive charge of one dipole sits close to the negative charge of the adjacent dipole. However, at the surfaces of the dielectric normal to the electric field there is net charge density. As seen in the negative ends of the dipoles remain neutralized at the surface AB and positive ends of the dipoles remain neutralized at the surface CD. These are the induced charges or polarization charges due to external field. They set up an electric field E opposite to ~E_o.

Dielectic is E = E₀ – E_p (2)

Here E is called the reduced value of the electric filed further,

E_p = σ_i / ∈ = 1 / ∈ (Q_i / A) = p / ∈ (3)

Where p is total dipole moment density 

Obviously the value of E depends upon the nature of the dielectric slab introduced. It is found that the ratio of external electric field applied (E₀) to the reduced value f electric field (E) is constant of the material. This constant is denoted by K and is called dielectric constant of the material of the slab 

~E_o/E= K

Electric susceptibility it is found that the electric polarization p is directly proportional to the reduced valued of electric filed (E)

P ∝ E

Or P = ∈yE

Where y is a constant, called electric susceptibility of the dielectric here ∈ has been used to keep y dimensionless. The electric susceptibility describes the electrical behaviour of a dielectric. It has deferent value for different dielectrics. For vacuum y = 0 

From 2, 3, and 5

E = E₀ – P / ∈ = E₀ - ∈yE / ∈

Or E = E₀ – yE

Or E₀ = E + yE = E (1 + y)

Or E₀ / E = 1 + y

K = 1 + y 

Using (25) 

This is the relation between dielectric content and electrical susceptibility of the material. 

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