electricity4

B.Tech Physics Course NIT Jalandhar electrostatics Lecture 4

Dr. Arvind Kumar Physics Department

Electric field in matter:

Dielectrics: These are the insulator (which do not conduct

electricity) which are polarized in presence of Electric field . These can be solid, liquid or gases. For example glass mica, air etc.

Lets understand how dielectric get polarized:

Normally we know that atoms are electrically neutral

So we may expect that nothing will happen to these neutral atoms in presence of electric field .

The response of the dielectric to the E.F. depends upon the nature of molecules of which the dielectric is made up of. There are two kinds of molecules.

(1) Polar molecules: A polar molecule is the one in which the centre of gravity of positive charge does not coincides with the centre

of gravity of negative charge. e.g. H₂O, SO₂, CO etc. A polar Molecule possess permanent dipole moment.

(Note the centre of gravity of positive charge means the point where entire +Ve charge is supposes to be concentred. Similarly the centre of Gravity of –Ve charge means the point where entire –Ve charge is

(2) Non-polar molecules: Here the centre of gravity of positive charge coincides with the centre of gravity of –Ve charge. A non-polar molecule does not posses the permanent dipole

Response of dielectric material to external electric field:

(1) Response of Polar dielectric : We consider a dielectric material which is made of the polar molecules. In the absence of the external field these molecules will be randomly oriented and therefore the net dipole moment per unit volume will be zero. Note that although the individual molecule has the finite dipole moment but the average dipole moment of the sample is zero .

However when the E.F. is applied to such dielectric then the

molecules will be oriented in the direction of the E.F. This is because the molecules behaving as dipole will experience a torque

This torque will align the molecule in the direction of E.F.

0

E

p







_

_



The degree of alignment depends upon the strength of the applied

E.F. E₀ and also on the temperature. The degree of the alignment

Effect of the E.F. on the non-polar dielectric:

We consider a non-polar dielectric . In the absence of the E.F. these molecules do not have dipole moment. However when the E.F. is applied to non-polar dielectric then the +Ve and –Ve charges get separated. The +Ve charges get displaced in the direction of the field whereas the –Ve charges get displaced in direction opposite to the E.F. the a dipole moment is induced due to external field.

 Electric field due to Polarization of Dielectric:

When a dielectric is placed in the presence of the field then the molecules of the dielectric get aligned in the direction of the field.

This can be either due to rotation of the molecules (in case of dielectric made of polar molecules) or due to separation of the charges (in case of dielectric made of non-polar molecules).

Due to this alignment of molecules the positive charge of the molecule inside the dielectric is followed

by the negative charge of the next

There left unbalanced charge at the surface

and this lead to the polarization of the dielectric. These charges are known as the polarization charges or bound charges.

Due to these charges there is now new field denoted by E_p.

This field acts in direction opposite to the external applied field. So the net field is now,

In presence of E.F. positively charged nucleus shift in direction

of electric field and negatively charged cloud to opposite side and these are separated by finite distance.

We say that the atom has got the dipole moment which is proportional to the applied E.F.

In above α is atomic polarizability and p is induced dipole moment in atom due to external E.F.

If the dielectric material has N atoms per unit volume then total induced dipole moment is

Concept of bound charges:

Consider long string of dipoles as shown below

Head of one cancel the tail of other and we are left with two charges. Positive on right and negative on left

These net charges at the end are known as bound charges. Bound charges are known as polarization charges

The dipole moment of tiny chunk is

Charge at the end of tube is

Bound Surface charge density is,

If polarization is not uniform then there can be bound charges inside and also on the surface

Gauss Law in Presence of Dielectrics We know, bound charge densities are

E.F. due to polarization is actually the field due to above bound charges

Now we shall find the total field i.e. the field due to polarizations and also due to everything else which we call free charges.

Now according to Gauss Law

In above E is now total E.F.

We can write above eqn by taking divergence eon one side as

Where we write

Thus we have

In integral form we can write

Susceptibility, permittivity and dielectric constant

Polarization of dielectrics is proportional to the E.F.

is electric susceptibility of the material and is dimensionless quantity.

We know for linear medium we can write electric displacement vector using (1)

Where

Permittivity of the medium

---(1)

---(2)

From Eq. (3) we can define a dimensionless quantity, known as relative permittivity of the material

Local Field or Polarizing field in Dielectric:

The electric field at the site of molecule or dipole due to all sources is known as Polarizing field or local field.

To find the local filed at the site of molecule or dipole we consider an imaginary sphere of radius r such that inside the sphere

we have large number of molecules. Also outside the sphere medium behave

We consider that the dielectric is palced between the two plates of the Capacitor. The local field is thus the sum

of following fields

Now we find the various fields in Eqn. (1) as follows:

We know:

---(2)

Also

---(3)

The field E3 is due to the molecules inside the sphere. Considering each molecule behaving as dipole the E.F. is written as

---(4)

The electric field E4 due to polarization charges is calculated as follows:

Consider a small area element

(ring shaped)on the surface of cavity. It is written as

---(6)

Now the charge dq on this small area element can be written as the normal component of Polarization multiplied by the area element. So we write,

---(7) The E.F. at centre A due to the above charge element in

direction θ = 0 is written as

0 0 0 3 0    P P P E

E_L     

Thus using Eq. (2), (3), (5) and (9) the local field in eq. (1) can be written as

---(10)

Clausius Mossotti Relation:

The Clausius-Mossotti relation connects the relative permittivity ε_r

of a dielectric to the polarizability α of the atoms or molecules constituting the dielectric.

The relative permittivity is a bulk (macroscopic) property and polarizability is a microscopic property of matter and hence the

relation bridges the gap between a directly-observable macroscopic property with a microscopic molecular property.

The relation is given by

where ε₀ is the electric constant (permittivity of the vacuum)

Derivation:

We know that the induced dipole moment in an

atom is equal to and if there are N number of atoms

per unit volume then the polarization is written as

---(1)

Now using Eqns.

And

Which is solved further as below,

---(2) Also the local field is given by following definition

Using Eq. (3) in Eq. (2) we get,

Now using Eq. (4) we write following expression:

---(5)