LECTURE 4

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Introduction

• The branch of engineering which deals with the flow of electrons (i.e. electric current) is called

current electricity and is important in many ways. For example, it is the electric current by

means of which electrical energy can be transferred from one point to another for utilisation.

• There can be another situation where charges (i.e. electrons) do not move but remain static or

stationary on the bodies.

• Such a situation will arise when the charged bodies are separated by some insulating medium,

disallowing the movement of electrons.

• This is called static electricity and the branch of engineering which deals with static electricity

is called electrostatics.

• The most useful outcomes of static electricity are the development of lightning rod and the

capacitor.

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Electrostatics

• The branch of engineering which deals with charges at rest is called electrostatics.

• When a glass rod is rubbed with silk and then separated, the former becomes

positively charged and the latter attains equal negative charge.

• It is because during rubbing, some electrons are transferred from glass to silk.

• Since glass rod and silk are separated by an insulating medium (i.e., air), they

retain the charges.

• In other words, the charges on them are static or stationary.

• Note that the word ‘electrostatic’ means electricity at rest.

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Importance of Electrostatics

• A few important applications of electrostatics are :

a) Electrostatic generators can produce voltages as high as 106 volts. Such high

voltages are required for X-ray work and nuclear bombardment.

b) We use principles of electrostatics for spray of paints, powder, etc.

c) The principles of electrostatics are used to prevent pollution.

d) The problems of preventing sparks and breakdown of insulators in high voltage

engineering are essentially electrostatic.

e) The development of lightning rod and capacitor are the outcomes of electrostatics.

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Methods of charging a conductor

• An uncharged conductor can be charged by the following two methods :

– By conduction

– By induction

• By conduction. In this method, a charged body is brought in contact with the

uncharged conductor

• By Induction. In this method, a charged body is brought close to the uncharged

conductor but does not touch it.

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Coulomb’s Laws of Electrostatics

• First law. This law relates to the nature of force between two charged bodies and

may be stated as under :

Like charges repel each other while unlike charges attract each other.

• In other words, if two charges are of the same nature (i.e. both positive or both

negative), the force between them is repulsion. On the other hand, if one charge is

positive and the other negative, the force between them is an attraction.

• Second law. This law tells about the magnitude of force between two charged

bodies and may be stated as under :

• The force between two point charges is directly proportional to the product of their

magnitudes and inversely proportional to the square of distance between their

centres.

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• where k is a constant whose value depends upon the medium in which the charges

are placed and the system of units employed.

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Capacitance and Capacitors

• It is well known that different bodies hold different charge when given the same

potential.

• This charge holding property of a body is called capacitance or capacity of the

body.

• In order to store sufficient charge, a device called capacitor is purposely

constructed.

• A capacitor essentially consists of two conducting surfaces (say metal plates)

separated by an insulating material (e.g., air, mica, paper etc.).

• It has the property to store electrical energy in the form of electrostatic charge. The

capacitor can be connected in a circuit so that this stored energy can be made to

flow in a desired circuit to perform a useful function.

• In many circuits (e.g., radio and television circuits), capacitors are intentionally

inserted to introduce the desired capacitance.

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Capacitors and Capacitance

 In its simplest form a capacitor consists of two plates which are separated by an insulating

material known as a dielectric.

 A capacitor has the ability to store a quantity of static electrical energy.

 Static electric fields arise from electric charges, electric field lines beginning and ending on

electric charges.

 Thus the presence of the field indicates the presence of equal positive and negative electric

charges on the two plates of Figure below.

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Capacitor

• Any two conducting surfaces separated by an insulating material is called a

capacitor or condenser.

• Its purpose is to store charge in a small space.

• The conducting surfaces are called the plates of the capacitor and the insulating

material is called the dielectric. (air, mica, waxed paper, ceramics etc.)

• Capacitance can be defined as the amount of charge required to create a unit

potential difference between the plates.

Note the following points carefully:

• The ability of a capacitor to store charge (i.e. its capacitance) depends upon the

area of plates, distance between plates and the nature of insulating material (or

dielectric).

• A capacitor is generally named after the dielectric used e.g. air capacitor, paper

capacitor, mica capacitor etc.

• The capacitor may be in the form of parallel plates, concentric cylinder or other

arrangement. 10

How does a Capacitor Store Charge ?

• Fig. shows how a capacitor stores charge when connected to a d.c. supply.

• The parallel plate capacitor having plates A and B is connected across a battery of

V volts as shown in Fig. (i).

• When the switch S is open as shown in Fig. (i), the capacitor plates are neutral i.e.

there is no charge on the plates. When the switch is closed as shown in Fig. (ii), the

electrons from plate A will be attracted by the +ve terminal of the battery and these

electrons start accumulating on plate B.

• The result is that plate A attains more and more positive charge and plate B gets

more and more negative charge.

• This action is referred to as charging a capacitor because the capacitor plates are

becoming charged.

• This process of electron flow or charging (i.e. detaching electrons from plate A and

accumulating on B) continues till p.d. across capacitor plates becomes equal to

battery voltage V. When the capacitor is charged to battery voltage V, the current

flow ceases as shown in Fig.(iii). 11

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• If now the switch is opened

as shown in Fig. (iv), the

capacitor plates will retain

the charges.

• Thus the capacitor plates

which were neutral to start

with now have charges on

them.

• This shows that a capacitor

stores charge.

Points to Note

• When a d.c. potential difference is applied across a capacitor, a charging current

will flow until the capacitor is fully charged when the current will cease. This

whole charging process takes place in a very short time, a fraction of a second.

Thus a capacitor once charged, prevents the flow of direct current.

• The current does not flow through the capacitor i.e. between the plates. There is

only transference of electrons from one plate to the other.

• When a capacitor is charged, the two plates carry equal and opposite charges (say +

Q and –Q). This is expected because one plate loses as many electrons as the other

plate gains. Thus charge on a capacitor means charge on either plate

• The energy required to charge the capacitor (i.e. transfer of electrons from one

plate to the other) is supplied by the battery. 13

Capacitance

• The ability of a capacitor to store charge is known as its capacitance.

• It has been found experimentally that charge Q stored in a capacitor is directly

proportional to the p.d. V across it.

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• The constant C is called the capacitance of the

capacitor. Hence capacitance of a capacitor can

be defined as :

• The ratio of charge on capacitor plates to the p.d.

across the plates is called capacitance of the

capacitor.

• A capacitor is said to have a capacitance of 1 farad if a charge of 1 coulomb

accumulates on each plate when a p.d. of 1 volt is applied across the plates.

Factors Affecting Capacitance

• The ability of a capacitor to store charge (i.e. its capacitance) depends upon the

following factors :

• Area of plate. The greater the area of capacitor plates, the larger is the capacitance

of the capacitor and vice-versa. It is because the larger the plates, the greater the

charge they can hold for a given p.d. and hence greater will be the capacitance.

• Thickness of dielectric. The capacitance of a capacitor is inversely proportional to

the thickness (i.e. distance between plates) of the dielectric. The smaller the

thickness of dielectric, the greater the capacitance and vice-versa. When the plates

are brought closer, the electrostatic field is intensified and hence capacitance

increases.

• Relative permittivity of dielectric. The greater the relative permittivity of the

insulating material (i.e., dielectric), the greater will be the capacitance of the

capacitor and vice-versa.

• It is because the nature of dielectric affects the electrostatic field between the

plates and hence the charge that accumulates on the plates. 15

Capacitance of Parallel-Plate Capacitor

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Capacitor Circuits

Capacitors in Series

• Consider three capacitors, having capacitances C1, C2 and C3 farad respectively,

connected in series across a p.d. of V volts.

• In series connection, charge on each capacitor is the same (i.e. +Q on one plate and

−Q on the other) but p.d. across each is different.

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Capacitors in Parallel

• Consider three capacitors, having capacitances C1, C2 and C3 farad respectively,

connected in parallel across a p.d. of V volts.

• In parallel connection, p.d. across each capacitor is the same but charge on each is

different.

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Practice Questions

• Qn.1: In the circuit shown, the total charge is 750μC. Determine the values of V1,

V and C2.

• Qn.2: Two capacitors A and B are connected in series across a 200 V d.c. supply.

The p.d. across A is 120 V. This p.d. is increased to 140 V when a 3μF capacitor is

connected in parallel with B. Calculate the capacitances of A and B.

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Solution

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Solution

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Practice Questions

• Qn. 3: Obtain the equivalent capacitance for the network shown in Fig (a). For

300V d.c. supply, determine the charge and voltage across each capacitor.

• Qn.4: Find the charge on 5 μF capacitor in the circuit shown in Fig. (b). [An: 9μC]

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Solution

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Solution

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Energy Stored in a Capacitor

• Charging a capacitor means transferring electrons from one plate of the capacitor

to the other.

• This involves expenditure of energy because electrons have to be moved against

the *opposing forces.

• This energy is stored in the electrostatic field set up in the dielectric medium.

• On discharging the capacitor, the field collapses and the stored energy is released.

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Behaviour of Capacitor in a D.C. Circuit

• When d.c. voltage is applied to an uncharged capacitor, there is transfer of

electrons from one plate (connected to +ve terminal of source) to the other plate

(connected to –ve terminal of source).

• This is called charging current because the capacitor is being charged.

• The capacitor is quickly charged to the applied voltage and charging current

becomes zero.

• Under this condition, the capacitor is said to be fully charged.

• When a wire is connected across the charged capacitor, the excess electrons on the

negative plate move through connecting wire to the positive plate.

• The energy stored in the capacitor is dissipated in the resistance of the wire.

• The charge is neutralised when the number of free electrons on both plates are

again equal.

• At this time, the voltage across the capacitor is zero and the capacitor is fully

discharged. 28

Points to Note

• When d.c. voltage is applied to an uncharged capacitor, the capacitor is quickly

(not instantaneously) charged to the applied voltage.

When the capacitor is fully charged, capacitor voltage becomes constant and is equal

to the applied voltage. Therefore, dV/dt = 0 and so is the charging current. Note that

dV/dt is the slope of v–t graph of a capacitor.

• A capacitor can have voltage across it even when there is no current flowing.

• The voltage across a capacitor (Q = CV) is proportional to charge and not the

current.

• There is no current through the dielectric of the capacitor during charging or

discharging because the dielectric is an insulating material. There is merely

transfer of electrons from one plate to the other through the connecting wires. 29

• When the capacitor is fully charged, there is no circuit current. Therefore, a fully

charged capacitor appears as an open to d.c.

• An uncharged capacitor is equivalent to a *short circuit as far as d.c. voltage is

concerned. Therefore, a capacitor must be charged or discharged by connecting a

resistance in series with it to limit the charging or discharging current.

• When the circuit containing capacitor is disconnected from the supply, the

capacitor remains charged for a long period. If the capacitor is charged to a high

value, it can be dangerous to someone working on the circuit.

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Magnetic Circuits

• The space (or field) in which a magnetic pole

experiences a force is called a magnetic field.

Properties of magnetic lines of force.

• (i) Each magnetic line of force forms a closed loop i.e.

outside the magnet, the direction of a magnetic line of force is

from north pole to south pole and it continues through the

body of the magnet to form a closed loop.

• (ii) No two magnetic lines of force intersect each other.

• (iii) Where the magnetic lines of force are close together, the

magnetic field is strong and where they are well spaced out,

the field is weak.

• (iv) Magnetic lines of force contract longitudinally and widen

laterally.

• (v) Magnetic lines of force are always ready to pass through

magnetic materials like iron in preference to pass through

non-magnetic materials like air. 31

Magnetic Flux (Φ)

• The total number of magnetic lines of force produced by a magnetic source is

called magnetic flux. It is denoted by Greek letter Φ(phi)..

• Units are in Weber (Wb)

• The more the magnetic lines of force, the greater the magnetic flux and the

stronger the magnetic field.

• Magnetic Flux Density (B) • The magnetic flux density is defined as the magnetic flux passing normally per

unit area i.e.

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Absolute and Relative Permeability

• Permeability of a material means its conductivity for magnetic flux.

• The greater the permeability of a material, the greater is its conductivity for

magnetic flux and vice-versa.

• The absolute (or actual) permeability *μ0 = of air or vacuum is 4π× 10−7 H/m.

• μr = 1 for air or non-ferrous materials

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Magneto-motive force (mmf) (Fs)

• This is the source of magnetic flux in a magnetic circuit. E.g. Permanent magnet or

a current carrying conductor. Provided there is a coil of N turns with current (I)

passing through it, Fs = IN (AT).

• Magnetising Force (H)

• The magnetising force (H) produced by an electric current is defined as the m.m.f.

set up per unit length of the magnetic circuit

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Relation Between B and H

• The flux density B produced in a material is directly proportional to the applied

magnetising force H.

• The greater the magnetising force, the greater is the flux density and vice-versa

• Hence relative permeability of a material is equal to the ratio of flux density

produced in that material to the flux density produced in air by the same

magnetising force.

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Reluctance

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Comparison Between Magnetic and Electric Circuits

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Series Magnetic Circuits

• In a series magnetic circuit, the same flux (Φ) flows through each part of the

circuit.

• It can just be compared to a series electric circuit which carries the same current

throughout.

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Practice Question

• An iron ring of cross sectional area 6 cm2 is wound with a wire of 100 turns and

has a saw cut of 2 mm.

• Calculate the magnetising current required to produce a flux of 0·1 mWb if mean

length of magnetic path is 30 cm and relative permeability of iron is 470.

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Solution

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Electromagnetic Induction

• The phenomenon of production of e.m.f. and hence current in a conductor or coil

when the magnetic flux linking the conductor or coil changes is called

electromagnetic induction.

• Flux Linkages

• The product of number of turns (N) of the coil and the magnetic flux (Φ) linking the

coil is called flux linkages i.e.

• Flux linkages = N Φ

• Experiments show that the magnitude of e.m.f. induced in a coil is directly

proportional to the rate of change of flux linkages. If N is the number of turns of

the coil and the magnetic flux linking the coil changes (say increases) from Φ1to

Φ2 in t seconds, then,

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Faraday’s Laws of Electromagnetic Induction

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Faraday’s First Law.

•When the magnetic flux linking a conductor or coil changes, an e .m.f. is induced in it.

•Any change in the magnetic field of a conducting coil causes an emf to be induced in the coil.

•If the conductor circuit is closed, the induced emf will cause current to circulate through the circuit and this current is called induced current.

•The induction of emf requires a conductor, a magnetic field and linking or cutting of flux by the conductor. The linking of magnetic field by the conductor can occur in three ways:

1. By moving a conductor in a stationary permanent magnet or dc electromagnet. This configuration is used in all dynamos, generators and motors.

2. By moving an electromagnet with respect to a stationary conductor. This configuration is used in large ac generators (especially synchronous generators)

3. Having a stationary conductor and a stationary electromagnet and variation of flux by feeding an alternating current to the magnet. This is used in transformers.

Faraday’s Second Law

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How to increase emf induced in a coil

1. By increasing the number of turns in the coil i.e N-from the formulae derived

above it is easily seen that if number of turns of coil is increased, the induced emf

also increases.

2. By increasing magnetic field strength surrounding the coil. Mathematically if

magnetic field increases, flux increases and if flux increases emf induced will also

get increased.

– Theoretically, if the coil is passed through a stronger magnetic field, there will

be more lines of force for coil to cut and hence there will be more emf induced.

3. By increasing the speed of the relative motion between the coil and the magnet – If

the relative speed between the coil and magnet is increased from its previous

value, the coil will cut the lines of flux at a faster rate, so more induced emf would

be produced.

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Lenz’s Law

• This law states that the electromagnetically induced current (due to Faraday’s law) always

flows in such direction that the action of the magnetic field set up by it tends to oppose the

very cause which produces it.

• Usually, a negative sign is used in Faraday’s law of electromagnetic induction, to indicate

that the induced emf (e) and the change in magnetic flux (dΦ) have opposite signs.

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Self & Mutual Inductance

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• Inductance is the name given to the property of a circuit whereby there is an e.m.f.

induced into the circuit by the change of flux linkages produced by a current

change.

• Self inductance (L) is when the e.m.f. is induced in the same circuit as that in

which the current is changing.

• Mutual inductance (M) is when an e.m.f. is induced in a circuit by a change of flux

due to current changing in an adjacent circuit.

Coefficient of Coupling

• The coefficient of coupling (k) between two coils is defined as the fraction of

magnetic flux produced by the current in one coil that links the other.

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Practice

• Qn. A solenoid with 900 turns has a total flux of 1.33 × 10–7 Wb through its air

core when the coil current is 100 mA. If the flux takes 75 ms to grow from zero to

its maximum level, calculate the inductance of the coil. Also, calculate the induced

e.m.f. in the coil during the flux growth.

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Practice Question

• Coils A and B in a magnetic circuit have 600 and 500 turns respectively. A current

of 8 A in coil A produces a flux of 0·04 Wb. If the coefficient of coupling is 0

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