Dielectrics and Molecular Theory of Induced Charges

Dielectrics and Molecular Theory of Induced Charges

A material which cannot conduct electric current but can transmit electric field is called dielectric. Glass, mica, paper etc are some of the examples of dielectric. There are two types of dielectric. They are:

1. Non-Polar dielectrics
2. Polar dielectrics

Non-polar dielectrics

If the center of positive charge of a molecule coincides with the center of the negative charge, the molecule is said to be non-polar molecule. It has zero dipole moment as the distribution of negative charge and positive charge is symmetrical. Benzene, oxygen, nitrogen, methane are some non-polar dielectrics.

Polar dielectrics

If a center of positive charge and negative charge in a molecule do not coincide, the molecule is said to be polar molecule. The dielectric mode of such molecule is said to be polar dielectric. It has permanent dipole moment as the distribution of negative charge and positive charge is asymmetrical.

Polarization

When the non-polar molecules are placed in an electric field E_o, the positively charged nucleus is pulled along the direction of the electric field and the negative charges pushed in opposite direction. As a result, the positive and the negative charges are separated by a small distance. The process of separation of the positive and negative charges of a non-polar molecules in the presence of an electric field is called polarization.

The dipole moment induced in molecule in the molecules is proportional to the applied field. So,

$$p \propto \epsilon _o \alpha E_o$$

$$\text {or,} p = \epsilon \alpha E_o $$

where α is a constant called molecular polarizability. Its unit is m³.

When a dielectric slab ABCD is placed in an electric field E_o as shown in the figure. If x is the distance between the positive and negative charges, the dipole moment on each molecule is p = Qx. If n number of molecules per unit volume is

P = np = n Qx.

This is called electric polarization or polarization vector. Its unit is Cm^-2 and so, it is equal to the induced charge per unit volume.

Is a small volume, such as A'B'C'D' is taken inside the dielectric, the net charge in it is zero because there are equal and opposite charges and charge density is zero, However on the surface of this volume, there are positive charges on the side A'D' and equal negative changes on the side A'D' and equal negative charge on the side B'C'. The polarization surface on A'D' and C'B' induces an electric field E_p in opposite E_o. Therefore, the net electric field in the polarized dielectric,

$$E = E_o -E_p$$

Let +Q_i and -Q_i be the induced charges on the two surfaces separated by a distance d, which is equal to the thickness of the slab. The dipole moment of the induced surface charge, p = Q₁d. The volume of the dielectric is Ad where A is its surface area. The induced dipole moment per unit volume, called as polarization, is

$$P = \frac pV = \frac {Q_1d}{Ad} = \frac {Q_1}{A} = \sigma _1 $$

whereσ_i is induced surface charge density. So, induced electric field

$$E_p = \frac {\sigma _i}{\epsilon _o} = \left (\frac {Q_1}{A} \right ) \frac {1}{\epsilon _o} = \frac {P}{\epsilon _o} $$

Dielectric Constant

The induced electric field E_p in a dielectric depends on the material of the dielectric and is proportional to the applied field E_o. It is observed that the ratioE_o/E is constant for a material and the ratio is called the dielectric constant or relative permittivity.

$$\frac {E_o}{E} = K $$

K represents the ability of a dielectric to decrease the electric field inside it. For most of the solids, K ranges from 2 to 10, while for metals it is infinity.

Electric Susceptibility

If E_o is not very strong, the field inside the dielectric E_p is not much strong and in such condition, the polarization is proportional to E,

$$P \propto E$$

$$\text {or,} P =\chi \epsilon _o E$$

where \(\chi\) is constant called the electric susceptibility. It is dimensionless quantity whose value depends on material of dielectric. For vacuum, \(\chi = 0\) as there are no molecules.

Relation between \(\chi \text {and} K\)

The electric field in the dielectric is

$$E = E_o -E_p $$

But, \(E_o = \frac {\sigma}{\epsilon _o}\) where \(\sigma\) is the surface charge density of capacitor plates before introducing the dielectric and \(E_p = \frac {\sigma _i}{\epsilon _o}\). Then,

$$E = E_o -E_p = \frac {\sigma }{\epsilon _o} -\frac {\sigma _i}{\epsilon _o} = E_o - \frac {\sigma _i}{\epsilon _o}$$

$$=E_o - \frac {P}{\epsilon _o}$$

But \(P =\chi \epsilon _o E\). Then

$$E = E_o - \frac {\chi \epsilon _o E}{\epsilon _o} = E_o - \chi E$$

$$\text {or,} E_o = E + \chi E = E(1 + \chi) $$

$$\frac {E_o}{E} = (1 + \chi)$$

But \(\frac {E_o}{ E} =K\), then

$$\boxed {K = (1 + \chi)}$$

This is the relation between\(\chi \text {and} K\)

Displacement Vector

The field E_o due to the free charge can be denoted by \(D/ \epsilon _o \) where D is called displacement vector. The field inside the dielectric is then,

$$E = E_o -E_p = \frac {D}{\epsilon _o} -\frac {D}{\epsilon _o}$$

$$\text {or,} D = \epsilon _o E + P $$

The displacement vector D is related to the free charges on the capacitor plates and is equal to \(\sigma\). the above equation can be written as

$$ E = \frac {D - P}{\epsilon_o} $$

This field is 1\k times \(\frac {D}{\epsilon_o}) where K is the dielectric constant of the slab. then we have,

$$\frac {D}{K \epsilon_o} = \frac {D - p}{\epsilon _o}$$

$$\text {or,} D \left (1 - \frac 1K \right ) = P $$

$$\text {which gives as}\: \sigma \left (1 - \frac 1K \right ) = \sigma _i $$

as the directions of D and P are the same.

Precautions

1. The dimensions of E is V/m² while the dimensions of D and P are those of ε_oE or ofσ i.e. C/m².
2. The sources of D are the free charges, (in this case +σ and -σ).
3. The sources of Pare the polarized charges, (in this case of +σ_i and -σ_i).
4. The sources of E are the free charges and polarized charges.

Dielectric Breakdown

When the charge in capacitor is gradually increased electric field between the plates also increases and a situation comes at which the electrons of dielectric are forcefully taken out from its molecules and the dielectric can no more act as insulator. This situation is called dielectric breakdown.

Dielectric Strength

The maximum applied electric field up to which the dielectric material can withstand without dielectric breakdown is called electric strength.

The maximum applied electric field up to which the dielectric material can withstand without dielectric breakdown is called electric strength of that material. It's unit is volt/meter.The dielectric strength of a vacuum is supposed to be infinity whereas the dielectric constant of a vacuum is 1.