1

FIVE IMPORTANT CONCEPTS OF PHYSICS FOR JEE MAIN

1. DIMENSIONS AND ERRORS IN MEASUREMENT

DIMENSIONS OF A PHYSICAL QUANTITYThe dimensions of a physical quantity are the powers to which the unit of fundamental quantities are raised to represent that quantity.

Dimensions of fundamental quantitiesS.No. Quantity Dimensions

1. Length [L]

2. Mass [M]

3. Time [T]

4. Electric current [A]

5. Temperature [K]

6. Luminous intensity [Cd]

7. Amount of substance [Mol]

Note:Two supplementary fundamental quantities that is plane angle and solid angle have no dimensions.

Dimensional equation : The equation obtained by equating a physical quantity with its dimensions formula is called dimensionalequation of the given physical quantity. Example : The dimensional equation of momentum is

[Momentum] = [MLT1]

Dimensions of some physical quantitiesS.No. Physical Relation with Unit Dimensional

quantity other quantities formulae

1. Force Mass Acceleration N [MLT2]2. Work Force Displacement J [ML2T2]

3. PressureForceArea N/m

2 [ML1T2]

4. Force constantForce

Distance N/m [ML0T2]

5. Gravitational constant G2

2Force distance

MassNm2/kg2 [M1L3T2]

6. Impulse of force Force Time Ns [MLT1]

7. StressForceArea N/m

2 [ML1T2]

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8. StrainChange in dimensionOriginal dimension [M

0L0T0]

Principle of homogeneity of dimensionsAccording to this principle, the dimensions of all the terms occurring on both sides of the equation must be same.Uses of dimensions

Conversion of unit of one system to another : It is based on the fact that product of numerical value contained in and the unit ofphysical quantity remains constant, that is, larger unit has smaller magnitude or n [u] = constant.If a physical quantity has dimensional formula [ MaLbTc] and units of that quantity in two systems are[M1

a L1b T1

c] and [M2a L2

b T2c] respectively, then

n1 [u1] = n2 [u2]

n2 = n1 1

2

[ ][ ]uu

or 2n =1 1 1

12 2 2

M L TM L T

a b c

n

Where n1 and n2 are numerical values in first and second system of units.

To check the accuracy of a formula : It is based on homogeneity principle of dimension. According to it, formula is correctwhen L.H.S. = R.H.S. dimensionally.To derive the formula by dimensional analysis methodLet a physical quantity x depends on the another quantities P, Q and R. Then

x (P)a (Q)b (R)c

or, x = k (P)a (Q)b (R)c .....(1)Now consider dimensional formula of each quantity in both sides.

MxLyTz = 3 3 31 1 1 2 2 2 x y zx y z x y za b c[M L T ] [M L T ] [M L T ]

MxLyTz = 3 3 31 1 1 2 2 2 cx cy czax ay az bx by bzM L T M L T M L T

MxLyTz = 1 2 3 1 2 3 1 2 3ax bx cx ay by cy az bz czM L T

Now comparing the powers of both sides ax1 + bx2 + cx3 = x ....(2)ay1 + by2 + cy3 = y ....(3)az1 + bz2 + cz3 = z ....(4)

After solving equations (2), (3) and (4) value of a, b and c will be m, n and o may be find outNow substitute the values of x, y and z in equation (1)Then obtained formula will be

x = (P)m (Q)n (R)o

ERRORS IN MEASUREMENTEvery measurement is limited by the reliability of the measuring instrument and skill of the person making the measurement. If werepeat a particular measurement, we usually do not get the same result every time. This imperfection in measurement can be expressedin two ways :Accuracy and precisionAccuracy refers to the closeness of observed values to its true value of the quantity while precision refers to closeness between thedifferent observed values of the same quantity. High precision does not mean high accuracy. The difference between accuracy andprecision can be understand by the following example : Suppose three students are asked to find the length of a rod whose length isknown to be 2.250 cm. The observations are given in the table.

3

Student Measurement-1

Measurement-2

Measurement-3

Average length

A. 2.25 cm 2.27 cm 2.26 cm 2.26 cm B. 2.252 cm 2.250 cm 2.251 cm 2.251 cm C. 2.250 cm 2.250 cm 2.251 cm 2.250 cm

It is clear from the above table, that the observations taken by student A are neither precise nor accurate. The observations of studentB are more precise. The observations of student C are precise as well as accurate.Error : Each instrument has its limitation of measurement. While taking the observation, some uncertainty gets introduced in theobservation. As a result, the observed value is somewhat different from true value. Therefore,

Error = True value Observed valueSystematic errors : The errors which tend to occur of one sign, either positive or negative, are called systematic errors. Systematicerrors are due to some known cause which follow some specified rule. We can eliminate such errors if we know their causes.Systematic errors may occur due to zero error of an instrument, imperfection in experimental techniques, change in weather conditionslike temperature, pressure etc.Random errors : The errors which occur randomly and irregularly in magnitude and sign are called random errors. The cause ofrandom errors are not known. If a person repeat the observations number of times, he may get different readings every time. Randomerrors have almost equal chances for positive and negative sign. Hence the arithmetic mean of large number of observations can betaken to minimize the random error.

Note:

1. Least count error always associated with the observation. Therefore they occur with both random and systematic errors.2. The accuracy of measurement is related to the systematic errors but its precision is related to the random errors, which include

least count also.

Mean value of a quantity : Since the probability of occurrence of positive and negative errors are same, so the arithmetic mean of allobservations can be taken as the true value of a observed quantity.If a1, a2, ................an are the observed values of a quantity, then its true value a can be given by

a = 1 2mean................ na a aa

n

=1

1 ni

ia

n

The absolute errors in individual observations are:

1 1a a a 2 2a a a ........................... n na a a

The mean absolute error is defined as

a = 1 2| | | | ................. | |na a a

n

= `1| |

n

ii

an

Thus the final result of the observed quantity can be expressed as a = a a .It is clear from above that any observed value can by ( ) ( )a a a a a .Relative or fractional error : The ratio of the mean absolute error to the true value of the quantity is called relative error.

4

Thus relative error = a

aPercentage error : If relative error is expressed in percentage is called percentage error.

Thus percentage error 100a

a

Note: Absolute error has the unit of quantity. But relative error has no unit.2. ROTATIONAL MOTION AND MOMENT OF INERTIA

MOMENT OF INERTIA AND RADIUS OF GYRATIONA rigid body having constituent particles of masses m1, m2, ....mn and r1, r2 ... rn be their respective distances from the axis of rotationthen moment of inertia is given by,

I = m1 r12 + m2 r2

2 + ... + mn rn2 2

1

ni i

im r

The moment of inertia of continuous mass distribution is given by

2I r dmwhere r is the perpendicular distance of the small mass dm from the axis of rotation.Its SI unit is kgm2. It is a tensor.Radius of gyration :The radius of gyration of a body about its axis of rotation may be defined as the distance from the axis of rotation at which, if theentire mass of the body were concentrated, its moment of inertia about the given axis would be same as with its actual distributionof mass.Radius of gyration k is given by,

I = MK2

or

121 2

2In

i ii

i

m rK

M m

where, M = m i

Also,2 2 2 2

1 2 3 ..... nr r r rkn

r1m1

r2m2

r3 m3k m

r4 m4

X

YTherefore, radius of gyration (k) equals the root mean square of the distances of particles from the axis of rotation.GENERAL THEOREMS ON MOMENT OF INERTIA

Theorem of perpendicular axis :According to this theorem the moment of inertia of a plane lamina (a plane lamina is a 2-dimensional body. Its third dimensionis so small that it can be neglected.) about an axis, perpendicular to the plane of lamina is equal to the sum of the moment ofinertia of the lamina about two axes perpendicular to each other, in its own plane and intersecting each other at the point,where the perpendicular axes passes through it.If Ix and Iy be the moment of inertia of a plane lamina (or 2D rigid body) about the perpendicular axis OX and OY respectively,which lie in plane of lamina and intersect each other at O, then moment of inertia (Iz) about an axis passing through (OZ) and

5perpendicular to its plane is given by

Xp(x, y)

x

Y

yo

Z

r

Ix + Iy = Iz

Let us consider a particle of mass m at point P distance r from origin O, where 22 yxrso Ix + Iy = my

2 + mx2 = mr2

i.e., Iz = Ix + IyTheorem of parallel axes :

(Derived by Steiner) This theorem is true for both plane laminar body and thin 3D body. It states that the moment of inertia ofa body about any axis is equal to its moment of inertia about a parallel axis through its centre of mass plus the product of themass of the body and the square of the distance between two axes.

P A

Q B

Mc.m

O

r

Let AB be the axis in plane of paper about which, the moment of inertia (I) of plane lamina is to be determined and PQ an axisparallel to AB, passing through centre of mass O of lamina is at a distance r from AB.Consider a mass element m of lamina at point P distant x from PQ. Now the moment of inertia of the element about AB = m (x + r)2so moment of inertia of whole lamina about AB isI = m(x+r)2 = mx2 + mr2 +2 mxrWhere first term on R.H.S is mx2 = Ic.m. moment of inertia of lamina about PQ through its centre of mass, second term on R.H.S.is mr2 = r2 m = Mr2, M is whole mass of lamina, third term on R.H.S is ( mx) r = 0, because mx is equal to moments of allparticles of lamina about an axis PQ, passing through its centre of mass. Hence

I = Ic.m. + M.r2

i.e., the moment of inertia of lamina about AB = its moment of inertia about a parallel axis PQ passing through its centre of mass+ mass of lamina(distance between two axes)2

6

Moment of inertia and radius of gyration of different objects

Shape of body Rotational axis Figure Moment of Radius ofinertia gyration

(1) Ring (a) Perpendicular to plane M = mass passing through centre R = radius of mass

MR2 R

(b) Diameter in the plane

12

MR2R2

(c) Tangent perpendicularto plane

2MR2 2 R

(d) Tangent in the plane

32

MR23 R2

(2) Disc (a) Perpendicular to plane passing through centre of mass

21 MR2

R2

(b) Diameter in the plane

2MR4

R2

(c) Tangent in the plane

54

MR25

2 R

M

R cm

CI

M

R cm

BI

M

R cm

CM'dI I

MR cm

DI

Ic

RM

cm

Id

Y

Z

IcR

M

cmId

7

(d) Tangent perpendicular to plane

32

MR2 32

R

(3) Thin walled (a) Geometrical axis MR2 R

cylinder

(b) Perpendicular to length 2 2R LM2 12

2 2R L2 12passing through centre of mass

(c) Perpendicular to length 2 2R LM2 3

2 2R L2 3passing through one end

(4) Solid cylinder (a) Geometrical axis2MR

2R2

(b) Perpendicular to length2 2R LM

4 12

2 2R L4 12

passing through centre of mass

(c) Perpendicular to length 2 2R LM4 3

2 2R L4 3

passing through one end

IcId

RM

cm

Ic

M

cmL

R

IcM

cm

L

R

IcM

cm

L

R

Id

Ic

M

cm

R

L

M

L

Ic

cm

R

I

M

L

Ic

cm

Id

R

8

3. THERMODYNAMICSFIRST LAW OF THERMODYNAMICSThe first law of thermodynamics is based on conservation of energy. According to this law heat Q supplied to a system is equal to thesum of the change in internal energy ( U) and work done by the system (W). Thus we can write

Q = U + WMore about first law of thermodynamics :1. Heat supplied to the system taken as positive and heat given by the system taken as negative.2. It makes no different between heat and work. It does not indicate that why the whole of heat energy cannot be converted into

work.3. Heat and work depend on the initial and final states but on the path also. The change in internal energy depends only on initial

and final states of the system.4. The work done by the system against constant pressure P is W = P V. So the first law of thermodynamics can be written as

Q U P V .5. Differential form of the first law;

dQ = dU + dWor dQ = dU + PdV.

THERMODYNAMICAL PROCESSESAny process may have own equation of state, but each thermodynamical process must obey PV = nRT.1. Isobaric process

If a thermodynamic system undergoes physical change at constant pressure, then the process is called isobaric.(i) Isobaric process obeys Charles law, V T

(ii) Slope of ~P V curve, dPdV = 0.

(ii) Specific heat at constant pressure

CP = 52R

for monoatomic and CP = 72R

for diatomic

(iv) Bulk modulus of elasticity: As P is constant, P = 0

and B =PV

V

= 0

(v) Work done: W = P V nR T(vi) First law of thermodynamics in isobaric process

Q = U W = U P V = U nR T= VnC T nR T = ( )Vn C R T

= PnC T(vii) Examples: Boiling of water and freezing of water at constant pressure etc.

9

2. Isochoric or isometric processA thermodynamical process in which volume of the system remain constant, is called isochoric process.

(i) An isochoric process obeys Gay - Lussacs Law, P T

(ii) Slope of P V curve, dPdV =

(ii) Specific heat at constant volume

VC = 32R

for monoatomic and VC = 52R

for diatomic

(iv) Bulk modulus of elasticity : As V is constant, V = 0

B =PV

V

= `

(v) Work done : W = P V = 0(vi) First law of thermodynamics in ischoric process

Q = U + W = U+ 0or Q = U

= nCV T

3. Isothermal processA thermodynamical process in which pressure and volume of the system change at constant temperature, is called isothermalprocess.(i) An isothermal process obeys Boyles law PV = Constant.(ii) The wall of the container must be perfectly conducting so that free exchange of heat between the system and surroundings

can take place.(iii) The process must be very slow, so as to provide sufficient time for the exchange of heat.(iv) Slope of P V curve:

For isothermal processPV = Constant

After differentiating w.r.t. volume, we getdPP VdV = 0

10

ordPdV =

PV or tan =

PV

(v) Specific heat at constant temperature:As T = 0,

C =Q

n T =

(vi) Isothermal elasticity: Bulk modulus at constant temperature is called isothermal elasticity. It can bedefined as

isoE = B = PV

V

= dPdVV

From abovedPdVV

= P

Eiso = P

(vii) Work done : W =

f

i

V

VPdV

By PV = nRT P = nRTV

W =

f

i

V

V

dVnRTV

fi

VVnRT nV

or W =f

i

VnRT n

V

Here f

i

VV is called expansion ratio.

Also PiVi = PfVf,f

i

VV

i

f

PP

W =f

i

VnRT n

V = i

f

PnRT n

P

(viii) First law of thermodynamics in isothermal process.As T = 0, U = 0

Q = U + W = 0 + Wor Q = W

114. Adiabatic process

An adiabatic process is one in which pressure, volume and temperature of the system change but heat will not exchangebetween system and surroundings.(i) Adiabatic process must be sudden, so that heat does not get time to exchange between system and surroundings.(ii) The walls of the container must be perfectly insulated.(iii) Adiabatic relation between P and V

According to first law of thermodynamicsdQ = dU + dW

For adiabatic process, dQ = 0, dU + dW = 0 ...(1)For one mole of gas

dU = CVdT and dW = PdVSubstituting these values in equation (1), we have

CVdT + PdV = 0 ...(2)For one mole of an idea gas,

PV = RT ...(3)After differentiating equation (3), we get

PdV + VdP = RdT

or dT =PdV +VdP

RFrom equation (2)

VPdV VdPC PdV

R = 0

or CV PdV + CVVdP + RPdV = 0or (CV + R) PdV + CVVdP = 0or CPPdV + CVVdP = 0After rearranging, we get

P

V

C dV dPC V P = 0

Substituting P

V

CC = , we have

ordV dPV P = 0 ...(4)

Integrating equation (4), we get

dV dPV P = C

or nV n P = C

or nV nP = C

or ( )n PV = C

or PV = Ce

or PV = k

Adiabatic relation between V and T & P and TFor one mole of gas

PV = RT, or P = RTV

Substituting in PV = k, we get

RT VV = k

12

or 1V T =kR

= new constant

Also V =RTP

RTPP

= k

or 1P T =k

R= another constant

(iv) Slope of P V curve : We have PV k . On differentiating,

we have 1dPP V VdV = 0

ordPdV =

PV

or tan =PV

As slope of isothermal curve = PV

Slope of adiabatic curve = slope of isothermal curve.Since > 1, so slope of adiabatic always be greater than slope of isothermal curve.

(v) Specific heat : C =Q

n T = 0

n T = 0

(vi) Adiabatic elasticity: Bulk modulus of gas at constant heat is called adiabatic elasticity. If can be defined as

Ead = B = PV

V

= dPdVV

AsdpdVV

= P

Ead = P

As Eiso = P, Ead = Eiso

(vii) Work done : W =

f

i

V

VPdV

For adiabatic process PV = i iPV = f fP V = k

or P = kV

W =

f

i

V

VkV dV

= 1

(1 )

f

i

V

V

Vk

13

=1 11 [ ]

(1 ) f ikV kV

=1 11 [( ) ( ) ]

(1 ) f f f i i iP V V PV V

=1

(1 ) f f i iP V PV

or W =( )

( 1)i i f fPV P V

Also PiVi = nRTi and PfVf = nRTf

W = [ ]1 i fnR T T

(viii) First law of thermodynamics in adiabatic process

Q = U + W

As Q = 0, U = W

or Uf Ui = WUf = Ui W

Summary of four gas processes :Fig. shows four different processes: Isobaric, isothermal, adiabatic and isochoric.

P

1

1 Isobaric = ; =2 Isothermal ( / ); 0

3 , Adiabatic 0;4 Isochoric ; 0

V

f i

V

U Q W U nC TP Q nC T W P VT Q W nRT n V V U

PV TV Q W UV Q U nC T W

Some resultsConstant Process Path

= and = for all pathsquantity type

P V diagram representing four different processes for an ideal gas4 : CAPACITORS AND GROUPING OF CAPACITORS.

CAPACITOR OR CONDENSERA capacitor is an electrical device which is used to store electrical energy in the same way that a bucket is a container for storing wateror a tank is a container for storing gas. Each of these devices has fixed capacity which does not depend on the quantity to be stored.A system consisting of two conductors of any shape, called plates placed at some separation constitutes a capacitor. When acapacitor is charged, its plates have equal and opposite charges of +Q and Q. The charge of capacitor means the charge on positiveplate. (Note that Q is not the net charge on the capacitor, which is zero). The symbol that we use to represent a capacitor is .

CAPACITANCE OF PARALLEL PLATE CAPACITORIt consists of two metallic plates of any shape but of equal size and placed at some separation. This constitutes a parallel plate

14capacitor. Suppose a charge Q is given to plate A; it will induce a charge Q to the other plate B, which is earthed. If d is the separationbetween the plate, then potential difference between the plates

V = E d.

Here E = 0

. If A is the area of each plate, then = QA

.

V =0

( / )Q A d

Capacitance C =QV

or C = 0A

d. (1)

Effect of dielectricAbove obtained expression for the capacitance holds only when the plates are in vacuum or air. Michael Faraday, first investigatedand found that, when a conductor is charged in the presence of dielectric, the charge on the conductor

[Q]in dielectric medium = k[Q]air, keeping V constant.

Thus [C]in dielectric medium = k[C]air (2)

Note:If there are n identical plates at equal distances from each other and the alternate plates are connected together, then the capacitance

of the arrangement will be C = (n 1) 0A

d.

Capacitance of parallel plate capacitor with a compound dielectricOn introduction of dielectric in the space between the plates of a capacitor, there is decrease in the potential difference between theplates. The potential difference between the plates

V = Vair + Vmed= Eair (d t) + E med t

=0 0

( )d t tk

Capacity C =QV

=

0 0( )

Atd t

k

15

or C = 0

( )

Atd tk

FORCE BETWEEN THE PLATES OF A CAPACITORMethod I :Consider a parallel plate capacitor with plate area A. Suppose a charge Q is given to one plate and Q to the other plate. Theforce on plate B due to plate A (or vice-versa)

FBA = EA QB.The electric field at the position of plate B due to the plate A is

EA =0 02 2

QA

FBA =02

Q QA

or FBA = FAB =2

02Q

FA

...(1)

If E is the electric field between the plates, then

E =0 0

QA

F = 2012

E A ...(2)

Method II :The expression for the force can also be derived by energy method. Suppose F is the force of attraction between the plates.If x is the separation between the plates, then

16

U = 201

( )2

E Ax

anddUdx

= 2012

E A

By the definition F =dUdx

= 2012

E A

CAPACITORS IN SERIES AND PARALLELIn some circuits, there are many capacitors connected together, we can sometimes replace that combination with an equivalentcapacitor. Here we are discussing two basic combinations of capacitors.

Capacitors in seriesFig. above shows three capacitors connected in series with a battery. In series means that the capacitors are connected by a wire andthere is no other way of flowing of charge. When a potential V is applied across them, the same amount Q of charge will appear on eachof the capacitor. The sum of the potential differences across all the capacitors is equal to the applied potential difference V. Thus, if V1,V2 and V3 are the potential differences across the capacitors C1, C2 and C3 respectively, then

V = V1 + V2 + V3.

Here 1 21 2

,Q QV VC C

and 33

QVC

. If Ceq is the equivalent capacitance, then

V = Q/C.

QC

=1 2 3

Q Q QC C C

oreq

1C = 1 2 3

1 1 1C C C ...(1)

For n-capacitors, we can write

17

eq

1C

=1

1n

iiC ...(2)

Capacitors in parallelFig. shows three capacitors are connected in parallel with a battery. In parallel connection, one plate of all the capacitors areconnected to the wire through the positive terminal of the battery and similarly second plates are connected together throughnegative terminal of the battery. Thus, each capacitor in parallel has the same potential difference V, which produces charge on thecapacitor. If Q1, Q2 and Q3 are the charges on the capacitors C1, C2 and C2 respectively, then

Q = Q1 + Q2 + Q3.Here Q1 = C1V, Q2 = C2V and Q3 = C3V. If Ceq is the equivalent capacitance, then total charge Q = Ceq V.

or Ceq V = C1V + C2V + C3V

Ceq = C1 + C2 + C3 ...(3)

For n-capacitors, we can write Ceq =1

n

ii

C ...(4)

5. SEMICONDUCTOR DIODE & LOGICAL GATESp-n JUNCTION DIODEWhen a p-type semiconductor is joined with a n-type semiconductor by appropriate method, the resulting device is called p-njunction diode.

Symbol of p-n junction diode is .

Diffusion and drift currentIn p-type crystal, the majority carriers are the holes and in n-type crystal, the majority carriers are the electrons. Due to difference inconcentration of charge carriers, hole / electron start diffusing from their side to other side. Only those holes / electrons cross thejunction which have high kinetic energy. Due to which the diffusion current takes place from the p-side to n-side of the crystal.

18Because of the thermal energy, the collision occur in each part of the material of the diode. Because of these collisions a covalent bondbreaks and the electron jumps to the conduction band. An electron-hole pair is created. Also a conduction electron fills up a vacantbond and so destroying an electron-hole pair. If an electron-hole pair is created in the depletion layer, they are send to their regionsby the electric field inside depletion layer. This constitutes drift current from the n-side to p-side of the crystal. Thus drift current andthe diffusion current are in opposite directions. In steady state both the currents are equal in magnitude and so there is no net transferof charge across the junction diode.

Depletion layerInitially, due to diffusion of charge carriers, a thin layer is formed on both sides of the junction. In this layer the charge carriers becomestatic due to mutual attraction force between the opposite charges. This layer is called depletion layer. The thickness of depletionlayer is of the order of 106 m.

Potential barrierIn equilibrium, the opposite charge carriers accumulate on both sides of the junction. This produces an electric field Ei from hole sideto electron side. The corresponding potential difference across the junction is called potential barrier. It can be denoted by V0 andequal to Ed. It is 0.3 V for Ge and 0.7 V for Si.

BIASING OF JUNCTION DIODEThe junction diode can be connected across the battery in following two ways :

(i) Forward biasIn forward bias (FB), the positive terminal of the battery is connected to p-side and negative terminal to n-side of the diode. Ifexternal electric field produced by the battery exceeds the internal electric field Ei, then hole move from p-region to n-region andelectrons from n-region to p-region. A current is thus set-up across the junction diode due to the diffusion of charge carriers.This results decrease in the thickness of depletion layer, and so potential barrier decreases. If d1 is the thickness of depletionlayer, then potential barrier V = Eid1. The following are the important points regarding with the forward bias:1. The forward current is of the order of mA.

192. Within the junction diode the current flows due to both types of charge carriers, while in external circuit (connecting

metallic wires) the current flows due to free electrons.3. The variation of current is not proportional to V; the graph between i and V is non linear.

(ii) Reverse biasIn reverse bias (RB), the positive terminal of the battery is connected to n-side and negative terminal of the p-side of the diode.

The external electric field produced by the battery is established to help the internal electric field iE . Because of this, majority

charge carriers are prevented to cross the junction while minority carriers are pushed to cross the junction, which constitutes avery small reverse current. The thickness of the depletion layer increases and so potential barrier increases. If d2 is the thicknessof the depletion layer, then potential barrier V = Eid2. The following are the important points regarding with reverse bias :1. The reverse current is order of A and due to drift of minority charge carriers.2. Within the junction diode the current is due to the both types of minority charge carriers but in external circuit it is due to

electrons only.3. The graph between i and V is non-linear.

Avalanche breakdownIf the reverse bias voltage is made sufficiently high, the covalent bonds near the junction break down and so large number of chargecarriers are produced. Which then increases the reverse current abruptly. This is called avalanche breakdown which was explained byZener.Forward and reverse resistanceThe resistance offered by diode in forward bias is called forward resistance Rf. The resistance offered by diode in reverse bias is called

reverse resistance Rr. For an ideal diode Rf = 0 and Rr = also V0 = 0. But for practical diode the ratio of 40000rf

RR

for Ge and

100000 for Si.

20

LOGIC GATESAn electronic circuit based on some logic and connected between input and output, is called a logic gate. The logic gates are basedon Boolean algebra, which has two digits; 0 and 1, and three operators; OR, AND, and NOT. The meaning of 0 (zero) is; OFF, false, no,open, low and the meaning of 1 (one) is : ON, true, yes, close high.OR operatorIn Boolean algebra, OR operator is used for addition symbolically, it is represented by sign +. For addition of A and B, one can writeA + B = Y, and reads as; A OR B is equal to Y. The expression like A + B = Y is called Boolean expression. The electrical circuit for ORoperator is as follows :

Truth tableTruth table represents the all possible inputs and outputs. For the OR operator it is :

Inputs Outputs Switch S1 Switch S2 Bulb glows

Open Open No Open Close Yes Close Open Yes Close Close Yes

In Boolean digits, it can be represented as follows :

A B A + B = Y 0 0 0 1 0 1 0 1 1 1 1 1

AND operatorIn Boolean algebra, AND operator is used for multiplication. Symbolically it is represented by sign dot (.). For the multiplication of Aand B, one can write, A.B = Y, and reads as; A AND B is equal to Y. The electrical circuit for AND operator is as follows :

Truth table Inputs Outputs Switch S1 Switch S2 Bulb glows

Open Open No Open Close No Close Open No Close Close Yes

In Boolean digits, it can be represented as follows :

21

A B A B = Y 0 0 0 0 1 0 1 0 0 1 1 1

NOT operatorIn Boolean algebra, NOT operator is used for inversion. Symbolically, it is represented by a sign bar over the input (). For inversionof A, one can write A Y, and reads as; A NOT is equal to Y. The electrical circuit for NOT operator is as follows :

Truth tableSwitch S Bulb glows

Open Yes Close No

In Boolean digits, it can be represented as follows :

A A Y 0 1 1 0

Some basic Boolean laws1. Boolean postulates :

A + 0 = A, 1.A = A1 + A = 1, 0.A = 0

A + A = 12. Identity law : A + A = A, A. A = A

3. Negation law : A = A4. Absorption laws :

A + A. B = A,A. (A + B) = A

A . (A + B) = A . B

5. Boolean identities : A + A B = A + B,

A ( A +B) = ABA + BC = (A + B) (A + C),

.A B A C = AC AB .

COMBINATION OF GATESBy combing any two or all the three basic gates, we can get new logic gates.NAND gateWhen output of AND gate becomes the input of NOT gate, this results in a new gate, which is called NAND gate. Therefore we canhave;

The Boolean expression of NAND gate is :

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A.B = Y.Truth table A B A.B A.B Y

0 0 0 1 0 1 0 1 1 0 0 1 1 1 1 0

NOR gateWhen output of OR gate becomes the input of NOT gate, this results in a new gate, which is called NOR gate. Therefore we can have;

The Boolean expression of NOR gate is :

A B = Y.Truth table

A B A + B A B 0 0 0 1 0 1 1 0 1 0 1 0 1 1 1 0

Exclusive OR or XOR gateIt consists of two NOT gates, two AND gates and one OR gate. The output signal is high if either input A or input Bbut not when both are high. The XOR gate may be expressed as :

The Boolean expression of XOR gate is :

AB AB = Y.

The symbol of XOR gate is shown in figure :

Truth table

A B A B AB AB AB AB Y 0 0 1 1 0 0 0 0 1 1 0 1 0 1 1 0 0 1 0 1 1 1 1 0 0 0 0 0

Exclusive NOR or XNOR gateWhen output of XOR becomes the input of NOT gate, this results in a new gate, which is called XNOR gate. Thus

XOR NOT XNOR.The Boolean expression of XNOR gate is :

A B AB = Y.The symbol of X NOR is :

Building blocksThe NAND gate is known as universal building block; with the help of it, we can get basic gates. The other universal building blockis NOR gate.NOT from NAND gate

23When both the inputs A and B of the NAND gate are joined together then it works as the NOT gate. Thus we have;

A.A = ATruth table

Input OutputA B Y

0 11 0

AND from NAND gateWhen two NAND gates are connected in series, the output is equivalent to AND gate. That is;

NAND + NAND= (AND + NOT) + (AND + NOT) AND

Truth table A B A.B A.B Y

0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 1

NOT

OR from NAND gateWhen the outputs of two NOT gates (which are obtained from NAND gates) is given to the input of the another NANDgate, the resultant gate works as the OR gate.Truth table

A B A B A B Y

0 0 1 1 0 0 1 1 0 1 1 0 0 1 1 1 1 0 0 1

OR