CLASS 12 PHYSICS, MODULE – 01, ELECTROSTATIC,

CHAPTER – 05, Electrostatic in Materials

Electric Conductor : The substance in which the free charges can move through out the substance, is called as electric conductor. When it is placed inside an electric field, there is a large scale of physical movement of free electrons within the conductor and they move out only if we make arrangement for it. Due to movement of free charges it can conduct electricity.

Insulators : The substance in which the charges can not move freely through out the substance, is called as the insulators. It can not conduct electricity.

Permittivity of the material : It is defined as the ability of a material to permit to pass the electric lines of force through it. It affects the coulomb’s force between two point charges. It relates the electric field with the electric displacement.

Properties of Conductor :

i ) Inside an electric conductor, electrostatic field is zero.

Explanation : When a conductor is placed in an electrostatic field, it causes the electrons to move in opposite direction of the field and induced positive and negative charges are produced. This movement continuous till the field due to the induced charge is balanced by the external electric field. Thus the net electric field inside the conductor is zero.

ii ) At the surface of a charged conductor, electrostatic field must be normal to the surface at every point.

Explanation : If the electric field is not normal to the surface of the conductor, then it would have same non zero component along the surface. The free charges on the surface would then experience a force and move. In static situation, electrostatic field should not have tangential component which in turn implies that the surface of a charged conductor must be normal to the surface at every point. If the conductor has no charge, then field is zero.

iii ) The interior of a conductor can have no excess charge in static situation.

Explanation : In static situation, when a conductor is charged, the excess charge reside only on the surface. If we consider any arbitrary volume inside a conductor, the electrostatic field on the closed surface bounding the arbitrary volume is zero. Thus the flux through the surface is zero. By Gauss’ theorem, there is no net charge enclosed by the closed surface. If the surface and volume is made very small, it shows that there is no net charge inside and the excess charge can reside on the surface.

Inside the conductor,

E = 0 —————————— ( 1 )

From Gauss theorem,

E. dS = q / ԑ

E. dS = ρ dV / ԑ

ρdV / ԑ = 0

ρ = 0 

where ρ is the volume charge density and ԑ is the permittivity of the medium 

Hence, the charge reside only on the outer surface of the conductor.

iv ) Charge distribution on the surface of the conductor may not be same.

Explanation : According to the Gauss theorem,

E. dS = q / ԑ

E. dS = ρ dV / ԑ——————- ( 1 )

where ԑ is the permittivity of the medium and ρ is the volume charge density.

From equation ( 1 ), it is clear that the charge density depends on the nature of the surface of the curvature. The charge density on the conductor surface is greater at points having more curvature that is the greatest charge density is at the region where radius of the curvature is the smallest.

Dielectric materials : These are very poor conductors of the electricity. But in the presence of the strong electric field, they may conduct the electricity.

Dielectric Strength : Dielectric strength of a dielectric is defined as the maximum value of the electric field that can be applied to the dielectric without its electric breakdown. Its unit is Volt / meter. It is the ability of the material to act as an insulator.

Non Polar Dielectric : A dielectric in the atoms and molecules of which, the center of positive and negative charge coincide, is called non polar dielectric.

Examples : O₂, H₂O and H₂ etc.

Polar Dielectric : A dielectric, in the atoms and molecules of which, the center of gravity of positive and negative charges coincide, is called polar dielectric.

Examples : NH₃, CH₄, CO₂, N₂O etc.

Electric Polarization : If a rectangular slab of a non polar dielectric is placed in uniform electric field E₀. Immediately each molecule of the dielectric gets polarized. It means that the centers of positive and negative charges are displaced from each other. On one face of the slab, a net negative charge and on other face, a net positive charge appear. There is no net charge in the interior of the dielectric. The charges +q and -q on the  two surfaces of the dielectric slab are called induced charges. The induced charges set up an electric field E_p inside the dielectric. It is called electric field due to polarization. E_p is opposite to the applied field E_0. 

This phenomenon is termed as the electric polarization.

Resultant Electric field is given by

E = E₀ –  E_p

Dielectric Constant ( K ) or Relative permittivity or Specific inductive constant : It is the ratio of the permittivity of the material to the permittivity of the free space.

K = ԑ / ԑ₀

ԑ is the permittivity of the material and ԑ₀ is the permittivity of free space.

It is always greater than 1.

Definition of Capacitor : The device which can store electrostatic energy, is called as Capacitor. It consists of two equal and opposite charged conductors, such that the potential difference between the plates is not affected by the presence of other conductor. Each conductor can be affected by the other conductor in such a way that the potential difference remains unaltered.

Capacitance:

The charge on the conductor is directly proportional to the potential difference between the two plates i.e

q ∝ V

q = CV

where C is the proportionality constant and V is the electric potential.

C = q / V

If V = 1 unit

Hence, numerically the capacitance is defined as the amount of charge required to raise the potential difference by unity.

Unit of capacitance:

1 ) Its SI unit is Farad.

2 ) Its CGS unit is Stat-Farad .

Dimension:

[ C ] = [q ] / [ V ] = M^-1 L^-2 T⁴ A²

Factors Influencing the Capacitor:

1 ) Surface area of the conductor : If surface area ( A ) of the conductor is large, its capacitance ( C ) will increase because for a given charge, its potential ( V ) will be smaller with the increase in surface area.

That is C ∝ 1 / V ————————— ( 1 )

And V α A  ———————————- ( 2 )

From equation ( 1 ) and ( 2 ) we get

C ∝ A

Hence the capacitance is proportional to the surface area of the conductor.

2 ) Proximity of the conductor : If another conductor is kept near the given conductor, its rise of potential is less du rot a charge given to it. So, its capacitance increases.

3 ) Nature of the surrounding : The capacitance of a conductor depends on dielectric constant K of the medium. Higher is the value of K, higher will be the value of C.

Potential energy of the charged conductor :

Let infinitesimally amount of charge dq is given to a conductor so that its potential remains constant. Doing so, some work is done. This work is stored in the form of potential energy.

Small work done = dw = Vdq

On integrating both sides, we get

∫ dw = ∫ Vdq

∫dw = ∫ ( q / C ) dq

W = ( 1 / C ) ∫ qdq

W = q² / 2C + A

Where A is integral constant

If q = 0 then W = 0

Therefore, A = 0

Hence, the potential energy is given by

E = q² / 2C

Or

E = CV² / 2

Combinations of capacitors :

Two or more capacitors may be combined by the following processes.

1 ) Series Combinations

2 ) Parallel Combinations

Series Combinations :

V = V₁ + V₂ + V₃ + …………………… + V_n

q / C = q [ ( 1 / C₁ ) + ( 1 / C₂ ) + ( 1 / C₃ ) +………………………… + ( 1 / C_n ) ]

1 / C = ( 1 / C₁ ) + ( 1 / C₂ ) + ( 1 / C₃ ) +………………………… + ( 1 / C_n )

Properties :

• In series combinations of capacitors, charge remains constant for each and every capacitor.
• In this combination, the electric potential gets divided.
• For two capacitors, the resultant capacitance C = 2C₁C₂ ( C₁ + C₂ )

Parallel Combinations :

q = q₁ + q₂ + q₃ + …………………… + q_n

CV = C₁V + C₂V + C₃V + ………………………. + C_nV

C = C₁ + C₂ + C₃ + ………………… + C_n

Properties:

• In parallel combinations of capacitors, electric potential remains constant for each and every capacitor.
• In this combination, the charge of the capacitors gets divided.
• For two capacitors, the resultant capacitance C = C₁ + C₂

Change in the energy on sharing charges by two capacitors :

Before connecting the two capacitors, the energy stored in them is given

U = U₁ + U₂

U = ( C₁V₁² / 2 ) + ( C₂V₂² / 2 ) ——————— ( 1 )

When the two capacitors are connecting together, then the total charge on the capacitors is given by

q = q₁ + q₂

q = C₁V₂ + C₁V₂

After connecting the two capacitors, the energy stored in the capacitor is given by

U’ = q² / 2C

U’ = ( q₁ + q₂ )² / 2C

U’ = (C₁V₂ + C₁V₂ ) ² / 2 ( C₁ + C₂ )

Change in energy is given by

ΔE = U – U’

ΔE = ( C₁V₁² / 2 ) + ( C₂V₂² / 2 ) – (C₁V₂ + C₁V₂ ) ² / 2 ( C₁ + C₂ )

On simplifying, we get

ΔE = C₁C₂ ( V₁ ~ V₂ ) ² / 2( C₁ + C₂ )

Types of capacitors :

1 ) Parallel plate capacitors

2 ) Isolated spherical capacitors ( single sphere )

3 ) Spherical capacitors ( Double spheres )

4 ) Cylindrical capacitors

5 ) Capacitor between two long thin parallel wires in air

1 ) Parallel Plate capacitors ( PPC ) :

It is formed by two identical parallel conducting plates. Let +q charge is given to plate placed at x = 0 and the plate at x = d is grounded.

Or

+ q charge is given to the plate at x = 0 and –q charge is given to the plate at x = d, where d is the separation between two plates.

Electric field directing from positive plate to the negative plate is given by

E = σ / ԑ

Where σ is the surface charge density is equal to q / A and ԑ is the absolute permittivity of the medium.

The electric potential is given by

V = ∫ E dr

V = ∫ (σ / ԑ ) dr

V = ( σ / ԑ ) ∫ dr

V = (σ / ԑ ) r + M

Where M is the integral constant

If r = 0, V = 0  then M = 0

At r = d , V = (σ / ԑ ) d = qd / A ԑ

The capacitance of the parallel plate capacitor is given by

C = q / V

C = ԑ A / d                   ( putting the value of V )

Special Case :

1 ) If the medium between the parallel capacitors is air, ԑ = ԑ₀ = 8.85 * 10^-12

C =  A / ( d / ԑ₀ )

2 ) If the there are two different dielectric medium between the parallel plate capacitors,

C =  A / { ( d₁ / ԑ₁ ) + ( d₂ / ԑ₂ ) }

C =  A ԑ₀ / { ( d₁ / K₁ ) + ( d₂ / K₂ ) }

where K₁ = ԑ₁ԑ₀ = dielectric constant of first dielectric medium having thickness d₁

K₂ = ԑ₂ԑ₀ = dielectric constant of second dielectric medium having thickness d₂

2 ) Isolated spherical capacitors ( single sphere ) : 

Let there is a spherical conductor with radius r. Let +q charge is given to the surface of the conductor so that the potential at every point of the surface is same. As a result the lines of force emerging from the sphere are normal to the sphere and they appear diverging radially from the center of the sphere. Therefore we can assume that the charge +q is concentrated at the center of the spherical conductor.

Potential at the surface of the spherical conductor is given by

V = q / 4πԑ₀ r

Thus the capacitance C is given by

C = V / q

C = 4πԑ₀ r

If there is dielectric medium of dielectric constant K, then

C = 4πԑ₀ rK

3 ) Spherical capacitors ( Double spheres ) :

Let R = outer radius of spherical capacitors

r = inner radius of the spherical capacitors

Let +q charge is given to the outer surface of the inner sphere. Due to electrostatic conduction, there is -q charge is induced at the inner surface of the outer sphere and +q charge is induced at the outer surface of the outer sphere. When it is grounded, the charge at the outer surface will be neutralized. Potential at the spherical capacitors is given by

V = ( q / 4πԑ₀ ) * ( 1/ r – 1 / R )

V = ( q / 4πԑ₀ ) * ( R – r ) / Rr

The capacitance C is given by

C = q / V

C = 4πԑ₀ rR / ( R – r )

If there is dielectric medium of dielectric constant K is given by

C = 4πԑ₀ rR K / ( R – r )