# Conductors and Dielectrics

Conductors: The substances having free charge carriers axe called the conductors. The examples of conductors are metallic substances e.g. copper, silver, gold, aluminum, iron, mercury etc.

Insulators: The substances having no free charge carriers are called the insulators or dielectrics. The examples of insulators are glass, plastic, mica wood, cotton etc.

### # Presence of free charges and bound charges inside a conductor

The free and bound charges inside a conductor may be understood by the knowledge of structure of atom. Every substance is formed of atoms. Every atom is electrically neutral. It consists of a central, nucleus containing positive charge and negatively charged electrons revolving around the nucleus in various definite orbits. The electrons in orbits near the nucleus are tightly bound by Coulomb attractive forces; while the electrons in outermost orbit are very loosely bound. In metals these electrons are free and are not attached to individual atoms, but they can move freely throughout the volume of the metal and are often called the free electrons or free charges of the conductor / metal.

The absence of electron from a neutral atom makes it positively charged and the resulting atom is termed as positive ion. The positive ions are bound in the conductor in a regular pattern and are therefore termed as bound charges. Thus a conductor consists of free charges as well as bound charges. The free charges are free electrons and the bound charges are positive ions fixed in the lattice.

### #Dielectrics

Dielectrics are substances which do not contain free charge carriers. The examples of dielectrics are air, mica, rubber, wood, plastic etc. Each atom/molecule of a dielectric is neutral. The molecules of a dielectric may be of two types:

(i) Non-polar Molecules: If the centers of positive and negative charges in a molecule coincide; so that no electric dipole is formed, the molecule of the dielectric is said to be non-polar. The examples of non-polar molecules are $H_2 N_2 O_2$ etc

(ii) Polar Molecules: if the centers of positive and negative charges in molecules do not coincide, so that an electric dipole is formed, the molecule is said to be polar. The example of polar molecules are $NH_3 , HCL , H_2 O , CO_2$ etc.

Though each molecules of a polar dielectric has its own dipole moment; but the molecular dipoles are randomly oriented in all directions, so that the net dipole moment as a whole is zero.

Dielectric in an External Field: When dielectric is placed in an external field, the positive and negative charges of dipoles get separated in non-polar molecules or the distance between them increase in polar molecules; in each the dipole moment is included in the dielectric and the dielectric is said to be polarized.

Dielectrics

Dielectric strength: Dielectric is used in capacitor in insulate positive and negative charges of the capacitor plates. The minimum electric field, at which positive and negative charges of the molecules of the dielectric get separated, is called dielectric strength.

The dielectric strength is different for different dielectrics. For air dielectric strength is nearly $3 \times 10^6$ V/m.

### #Capacitance of a conductor

When a charge is given to a conductor, its potential increases. The ratio of charge given to a conductor and its consequent potential rise is called the capacitance of that conductor.

i.e.

$C = \dfrac{Q}{v} \text{coul/volt or farad(F)}$

### # Capacitance of an isolated sphere

Consider a charged spherical conductor of radius R and carrying charge Q. The surface of a conductor is equipotential. The electric lines emerge normally from the surface and appear to come from the center.

Capacitance of Isolated Surface

$\text{Potential at surface :} V = \dfrac{1Q}{4 \pi \epsilon ^0 R}$

$\circledast Capacitance C = \dfrac{Q}{V} = \dfrac{Q}{\dfrac{1Q}{4 \pi \epsilon ^0 R}}$

Ie.

Capacitance of spherical conductor= $C = 4 \pi \epsilon ^0 R$

### #Capacitor (or Condenser)

An arrangement which is capable of collecting charge and whose capacitance may be varied without altering the size of conductors is called a capacitor (or condenser). A capacitor consists of two conductors carrying equal and opposite charges separated by a distance. The capacitance of a capacitor depends upon (i) Size of conductors (ii) Separation between the conductors (iii) Medium between the conductors but it is independent of metal of conductors.

If $C_{air}$ is the capacitance of the capacitor in air and $C_{medium}$ is the capacitance of capacitor in medium, then dielectric constant K of medium.

$K = \dfrac{C_{medium}}{C_{air}}$

### # Parallel Plate Capacitor

It consists of two parallel metallic plates P1 and P2 (say) of any shape, each of area A and at a distance d apart. The plate P1 is given a positive charge and the plate P2 is earthed. The negative charge is induced on the nearer face of plate P2 and positive charge on the farther face of P2 which is transferred to earth. Thus the plates have charges and If K is dielectric constant of medium between the plates, the electric field strength between plates.

Paralle plate capacitor

$E = \dfrac{\sigma}{\epsilon} = \dfrac{\sigma}{\epsilon ^o K}$

Where $\sigma$ =(q/A) is the surface density of the plates.

The capacitance of the capacitor is given by: $C = \dfrac{K \epsilon _0 A}{d}$

If medium between the plates is air, then (K = 1) so that:

$C_{air} = \dfrac{\epsilon _0 A}{d} \text{equation 1}$

#### # Special case

When the space between the plates is partly filled with dielectric:

If a dielectric slab of dielectric constant K and thickness t is introduced between the plates of an air-capacitor, then the effective distance between the plates is reduced by $t ( 1 - \dfrac{1}{K} )$ and so the capacitance of capacitor, becomes:

$C = \dfrac{\epsilon _0 A}{d-t(1- \dfrac{1}{K}} \cdots \text{equation 2} )$

Capacitor

### #Induced charge

In the case of a conductor the induced charge is equal and (apposite to the inducing charge (q); but in the case of a dielectric, the induced charge is given by:

$\text{qinduced}$ = $- q ( 1 - \dfrac{1}{K} )$

### # Combination of Capacitors:

(a) Series arrangement: In this arrangement the second plate of one is connected with the first plate of the next capacitor and the second plate of last capacitor is earthed. If the first plate of first capacitor is given a charge + Q, then due to induction charge +Q and –Q are induced on the two plates of each capacitor as shown in fig. In this arrangement, (i) the charges of individual capacitors are equal. (ii) The total p.d. i.e. across AB is shared by the capacitors in the inverse ratio of their capacities i.e.

$Q = C_1 V_1 = C_2 V_2 = C_3 V_3$

Such that,

$V = V_1 + V_2 + V_3$

Series Arrangement

The effective capacitance, C is given as:

$\dfrac{1}{C} = \dfrac{1}{C_1} + \dfrac{1}{C_2} + \dfrac{1}{C_3}$

C is less than $C_1 , C_2 \text{and} C_3$

I.e.  to decrease the capacitance, the capacitors are connected in series.

(b) Parallel Arrangement: In this arrangement the first plates of all capacitors are connected to a common point A and the second plates to another common point B. When a charge Q is given to arrangement, it gets shared by all capacitors; the first plate becomes positively charged and second plate negatively charged. The potential difference between A and B being V, remains same for all capacitors.

Parallel Arrangement

(i) The p.d. across the individual capacitors is the same.

(ii) The total charge Q from A to B by individual capacitors in direct ratio of their capacities.

I.e.

$Q = q_1 + q_2 + q_3$

So that $q_1 = C_1 V , q_2 = C_2 V \text{and} q_3 = C_3 V$

So that the equivalent capacitance C is given by:

$CV = C_1 + C_2 V + C_3 V$

The equivalent capacitances is: $C = C_1 + C_2 + C_3$

Energy Stored in a Capacitor: The energy of a charged capacitor of capacitance C is given by:

Where,

Q = charge on each plate of capacitor

V = potential difference between plates

$\epsilon$ = permittivity of medium between the plates

E = electric field strength between the plates

t= volume of space between the plates.

This energy resides in the space between the plates as electric potential energy.

An Important Remark: When the battery across the capacitor remains connected the p.d. remains the same but when the battery is disconnected the charge on the capacitor remains same.

### # Sharing of charges

When two capacitors of capacitances C1 and C2 charged to potentials V1 and V2 (or charges q1 and (q2), are connected together in parallel with positive plates connected together, then charges are redistributed until their potentials are the same.

The common potential,

$V = \dfrac{q_1 + q_2}{C_1 + C_2} = \dfrac{C_1 V_1 + C_2 V_2}{C_1 + C_2}$

Charges after sharing become q’1 and q’2 given by q’1=$C_1 V$ and q’2=$C_2 V$

I.e. Charges are distributed in ratio of their capacitances.

Energy Loss: There is a loss of energy during sharing of charges given by:

$\delta V$ =$U_{initial} - U_{final}$

Remark: If charged capacitors are so connected in parallel that positive plate of one is connected to the negative plate of the other, then net charge on capacitor becomes q1-q2.

Ie.

Common potential, V=$\dfrac{q_1 - q_2}{C_1 + C_2}$

11. Capacitance of a spherical capacitor: If there are two concentric conducting spheres of radii a and b, the outer sphere being earthed and inner sphere carrying a charge + q then capacitance is:

C=$\dfrac{4 \pi \epsilon _0 ab}{b-a}$

If we assume a spherical capacitoe formed by earth’s surface and the top of stratosphere height (10 Km say), then a=6400 Km, b=6410 Km

b-a=10 Km

So,

C=$\dfrac{1}{9 \times 10^9} \times \dfrac{6400 \times 10^3 \times 6410 \times 10^3}{10 \times 1063} = 0.4F$

# Three electric vectors D, P and E: If an electric field E is applied across a parallel plate capacitor filled with a dielectric of dielectric constant K (or permittivity), then polarization P = Induced charge per unit area (opposite to free charge)

### # Atmospheric Electricity

The atmosphere surrounds the earth like a blanket. The atmosphere is a very variable system. Therefore only its average properties may be described. The radius of earth is about 6400 km and the atmosphere extends to about 300 km above the surface of earth which is about 1/20th radius of earth. When we go up, the temperature and density vary. At about 300 km, its density falls to $10^{-10}$ times its ground level value.

### # Electric Properties of Atmosphere:

1. A low altitudes the atmosphere is poor conductor. The conductivity is only due to presence of ions, small nuclei of dirt; water vapor carrying static charges etc. the conductivity in low atmosphere is very variable. It varies from day to day.

2. At the top of stratosphere (ie, about 50 Km) the atmosphere is pretty conducting. The conductivity increases from earth’s surface towards the top of stratosphere.

3. At ground level there is vertical electric field of about 100 V/m all over the earth. The field weakness at higher altitudes and becomes negligible at 50 Km.

4. The potential drop from 50 Km to earth’s surface is nearly 400 Kv. Most of the potential drop occurs at low altitudes.

5. The surface density of earth is $- 10^{-9} C / m^2$ the total charge being $- 0.5 \times 10^6 C$.

6. The number of protons entering the earth’s surface per second is $2 \times 10^7 per m^2$. This is equivalent to positive charge of +1800 C.

7. To maintain the constancy of negative charge of earth and potential difference between earth and potential difference between earth and stratosphere, there are about $4 \times 10^4$ thunder storms off somewhere in every two seconds. The duration of each storm in about 1 hour.

8. Within each thunder cloud positive charge upward to a height of about 6 Km while negative charges collect at about 2 to 3 km above ground, the bottom of the cloud.

9. The amount of negative charge may be $-20^0 to -30^0 C$ . As the end of storm, the negative charge burst along narrow path from cloud to earth to maintain earth at negative potential.

10. In the last stages of a storm, there are about 200 flashes or bolts, each lasting about $2 \times 10^{-3}$ sec. The peak current in each bolt is about $- 10^4 A$ in the downward direction. The calculation shows that each bolt deposits $- 20^0 C$ of charge on earth. After each bolt the thunder cloud gets charged again and gets ready for nest bolt.

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