Class TABLE OF CONTENT
MOTION IN A STRAIGHT LINE............................................... 1 MOTION IN A PLANE............................................................... 2 LAWS OF MOTION................................................................... 3 WORK, ENERGY AND POWER............................................... 4 SYSTEM OF PARTICLES AND ROTATIONAL MOTION......... 5 GRAVITATION........................................................................... 6 MECHANICAL PROPERTIES OF SOLIDS.............................. 7 MECHANICAL PROPERTIES OF FLUIDS.............................. 7 THERMAL PROPERTIES OF MATTERS................................ 8 THERMODYNAMICS............................................................... 9 KINETIC THEORY.................................................................... 10 OSCILLATIONS........................................................................ 11 WAVES...................................................................................... 12 - 13 ELECTROSTATICS................................................................... 14 - 15 CURRENT ELECTRICITY.......................................................... 16 MAGNETISM.............................................................................. 17 - 18 ELRCTRO MAGNETIC INDUCTION.......................................... 19 ALTERNATING CURRENT......................................................... 20 RAY OPTICS............................................................................... 21 WAVE OPTICS............................................................................ 22 MODERN PHYSICS.................................................................... 23 SEMICONDUCTOR..................................................................... 24 - 25 COMMUNICATION SYSTEM...................................................... 26
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Class
1
MOTION IN A STRAIGHT LINE 1. The area under the velocity-time curve and time axis gives the displacement of the object for given interval of time. 2. If a body falls freely, the distance covered by it in each subsequent second starting from first second will be in the ration 1:3:5:7, etc. 3. If a body is thrown vertically up with an initial velocity u, it takes u/g second to reach maximum height and u/g second to return, if air resistance is negligible. 4. If air resistance acting on a body is considered, the time taken by the body is considered, the time taken by the body to reach maximum height is less than the time to fall back the same height.
s s la
5. For particle having zero initial velocity if sa t a where a > 2 then particle's acceleration increases with time.
C
6. For a particle having zero initial velocity if sa t a , where a < 0 then particle's acceleration decreases with time. 7. Kinematic equations : v = u + at ; v 2 = u 2 + 2as 1 2 s = ut + at 2
Application only when particles move with constant acceleration is variable use calculus approach. 8. If acceleration is variable use calculus approaches. r
r
9. Relative velocity : v BA = v B - vA
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Class
2
MOTION IN A PLANE 1. If T is the time of flight, h maximum height, R horizontal range of a projectile, α its angle of projection, then the relations among these quantities h=
gT 2 ...............(1); 8
gT 2 = 2R tan a ........(2)
R tan a = 4h.............(3)
2. For a given initial velocity, to get the same horizontal range, there are two angles of projection α and 90o – α. 3. The equation to the parabola traced by a body projected horizontally form the top of a tower of height y, with a velocity u is y = gx2/2u2, where x is the horizontal distance covered by it from the foot of the tower.
s s la
4. At any instant if v is the velocity of projectile making angle β with the horizontal, then
C
vy = v sin b = u sinq - gt
vx = v cos b = u cos q
5. Equation of trajectory is
y = x tan q -
gx 2 2u 2 cos2 q
which is parabola.
6. Equation of trajectory of an oblique projectile in terms of range ® is y = x tan q æç1 - x ö÷ R è
ø
7. Maximum height is equal to n times the range when the projectile is launched at an angle q = tan -1 (4 n)
8. In a uniform circular motion, velocity and acceleration are constant only in magnitude. Their directions change. r
ur
9. In a uniform urcircular motion, the kinetic energy of the body is a constant. W = 0, a ¹ 0, p ¹ constant, L = constant 10.
ar = w 2 r =
v2 = wv r
ar = 4p 2 n 2 r =
4p 2 r T2
(Always applicable) (Applicable in uniform circular motion) uur ur r n = frequency of rotation, T = time period of rotation. ar = w ´ v
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Class
3
LAWS OF MOTION ur
r
ur
uur
1. Newton's second law : F = ma, = F = dp / dt ur ur uur uur t 2 2. Impulse: D p = F D t, p2 - p1 = Fdt ò ur
uuur t1
3. Newton's third law : F 12 = -F21
4. Frictional force f s £ ( f s )max = ms R ; f k = mk R 5. Circle motion with variable speed. For complete circles, the string must be taut in the highest position,u 2 ³ 5 gl where l is the length of string.
s s la
Circular motion ceases at the instant when the string becomes slack, i.e. when T =0, range of values of u for which the string does go slack is
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2 gl < u < 5 gl
6. Conical pendulum : w = g / h where h is height of a point of suspension from the centre of circular motion. 7. The acceleration of a lift a = actualweight - apparentweight , where the weight is in N. If 'a' is mass
positive the lift is moving down, and if it is negative the lift is moving up.
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Class
4
WORK, ENERGY AND POWER 1.
If a light body and a heavy body have equal kinetic energy, then heavy body had greater momentum.
2.
Work due to static force of friction on system as whole is always zero.
3.
If a body moves with constant power, its velocity (v) is related to distance travelled (x) by the formula va x3 /2
4.
Work due to kinetic force of friction between two contact surfaces is always negative. It depends on relative displacement between constant surfaces
s s la
wFK = - Fk ( Srel ) 5. 6.
7.
8.
åW = å DK , åW Þ total change in kinetic energy. = - å D U ; åW ÞTotal work due to all kinds of conservative forces. åW å DU Þ Total change in all kinds of potential energy. conservative
e=-
conservative
Velocity of separation of colliding bodies velocity of approach of colliding bodies
C
The total moment of a system of particles is a constant in the absence of external forces.
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Class
5
SYSTEM OF PARTICLES AND ROTATIONAL MOTION 1. The centre of mass of a system of particles is defined as the point whose position vector m .r R = is the centre of gravity of an extended body is that point where the total åM gravitational torque on the body is zero. The centre of gravity of a body coincides with its centre of mass only if the gravitational field does not very from one part of the body to the other. i
i
n
2. The angular momentum of a system of n particles about the origin is L = å ri ´ pi i= 1
The torque or moment of force on a system of n particles about the origin is t = å ri ´ pi i
s s la
2
3. The moment of inertia of a rigid body about an axis is defined by the formula I = å mi .ri where ri is the perpendicular distance of the ith point of the body from the axis. The kinetic energy of rotation is K = 1 Iw 2 2
4. The theorem of parallel axes: I z = I z + Ma 2
C
5. For rolling motion without slipping vcm = Rw is the velocity of translation (i.e., of the centre of mass), R is the radius and m is the mass of the body. The kinetic energies of translation and rotation: K = 1 mv 2 + 1 Iw 2 2
2
6. A rigid body is mechanical equilibrium if (a)It is in translational equilibrium i.e., the total external force on it is zero : å Fi = 0 (b)It is in rotational equilibrium i.e., the total external torque on it is zero å = å ri ´ Fi = 0 : 7. If a body is released form rest on rough inclined plane, then for pure rolling n 2 ur ³ tan q (Ic = nmr ) Rolling with sliding 0 < u < æ n ö tan q ; g sin q < a < g sin q n +1
s
ç n + 1÷ è ø
n +1
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Class
6
GRAVITATION 1.
Angular momentum conservation leads to Kepler's second law. However, it is not special to the inverse square law of gravitation. It holds for any central force.
2.
They acceleration due to gravity (a) At a height h above the Earth's surface g ( h) =
GM E GM æ 2h ö = 2 E ç1 ÷ 2 (RE + h) R E è RE ø
æ 2h ö g ( h) = g(0) ç1 ÷ è RE ø
(b)
GM
E where g (0) = R 2 E
At depth d below the Earth's surface is g(d) =
3.
for h < < RE
s s la
æ GM E æ d ö d ö ç1 ÷ = g (0) ç1 ÷ 2 R E è RE ø è RE ø
C
The gravitational potential energy V = - Gm1 m2 + cons tan t r
4.
The escape speed form the surface of the Earth is Ve = and has a value of 11.2 km s
-1
2GM E = 2 gRE RE
5.
A geostationary (geosynchronous communication) satellite moves in a circular orbit in the equatorial plane at a approximate distance of 4.22 ×104 km from the Earth's centre
6.
Vmax =
7.
GM s æ 1 + e ö GM s æ 1 + e ö ç ÷ ;Vmin = ç ÷ a è 1- e ø a è 1- e ø
Whenever force responsible for orbital motion obeys inverse square law, then only square of time period is directly proportional to cube of average distance 2 3 2 3 T a between planet and sun. T aa ; 1 = 1 T22
a32
Applicable only when both planets revolve around same mass. Length of semi major axis is the average distance between sun and planet during its complete orbital motion. 8.
2 1- n n If F a r then T a (r)
IfU a r m then T 2a (r) 2 -m
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Class
7
MECHANICAL PROPERTIES OF SOLIDS 1. If S is the stress and Y is Younger's modulus, the energy density of the wire E is equal to S2/2Y. 2. If α is the longitudinal strain and E is the energy density of a stretched wire, Y Younger's modulus of wire, then E is equal to 1 Ya 2 2
MECHANICAL PROPERTIES OF FLUIDS
s s la
1. Pascal's law : A change in pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid and the walls of the containing vessel.
C
2. Bernoulli's principle states that as we move along a 2streamline, the sum of the pressure (P), the kinetic energy per unit volume (pv /2) and the potential energy per unit volume (pgy) remains a constant. P + rv 2 / 2 + r gy = cons tan t 3. Surface tension is a force per unit length (or surface energy per unit area) acting in the plane of interface between the liquid and bounding surface. 4. Stokes' law states that the viscous drag force F on a sphere of radius a moving with velocity v through a fluid of viscosity is η, F = 6ph av 5. The surface tension of a liquid is zero at boiling point. The surface tension is zero at critical temperature. 6. If a drop of water of radius R is broken into n identical drops, the work done in the process is 21/34(n1)RSp-. 7.Two capillary tubes each of radius r are joined in parallel. The rate of flow of liquid is Q. If they1/4are replaced by single capillary tube of radius R for the same rate of flow, then R=2 r. 8.If radius of a drop is doubled its terminal velocity increases to four times.
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Class
8
THERMAL PROPERTIES OF MATTERS 1. The coefficient of linear expansion (a l ), superficial expansion (β) and volume expansion (a v ) are defined by the relations: Dl DA DV = al D T; = b DT ; = a v DT l A V
Where Dl and DV denote the change in length l and volume V due to change of temperature DT. The relation between them is: a v = 3a l ; b = 2a l
2. In conduction, heat is transferred between neighboring parts of a body through molecular collisions, without any flow of matter. For a bar of length L and uniform cross section A with its ends maintained at temperatures TC and TD, the rate of flow of heat H is : H = KA TC - TD where K is the thermal conductivity of the material of the L bar.
s s la
3. Convection involves flow of matter within a fluid due to unequal temperatures of its parts.
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4. Radiation is the transmission of heat as electromagnetic waves. Radian heat travels with the speed of light. It requires no medium. Stefan's law of radiation: The energy emitted by a black body per unit area per second is directly proportional to the fourth power of its Kelvin temperature. E = σT4 , where the constant σ is known as Stefan's constant. Wien's displacement law states that lmT = cons tan t where lm is the wavelength corresponding to maximum energy. The constant is known as Wien's constant.5. 5. Newton's Law of Cooling says that the rate of cooling of a body is proportional to the excess temperature of the body over the surroundings: dQ = - k (T2 - T1 ) dt
where T1 is the temperature of the surrounding medium and T2 is the temperature of the body.
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Class
9
THERMODYNAMICS 1. The first law of thermodynamics is the general law of conservation of energy applied to any system in which energy transfer form or to the surroundings (through heat and work) is taken into account. It states that DQ = DU + D W, where is the heat supplied to the system, DQ is the work done by the system and DW is the change in internal energy of the system. If Q > 0 heat is added to the system, if Q < 0heat is removed to the system, if W > 0 work is done by the system, if W < 0 work is done on the system quantity. 2. In isothermal quasi-static process, heat is absorbed or given out by the system even through at every stage the gas has same temperature as that of the surrounding reservoir. This is possible because of the infinitesimal difference in temperature between the system and the reservoir. In an isothermal expansion of an ideal gas from volume V1 to V2 at temperature T the heat absorbed (Q) equals the work done (W) by the gas, each given by Q = W = nRT .In æ V2 ö
s s la
ç ÷ è V1 ø
C
Cp g = PV = cons tan t , where 3. In an adiabatic process of an ideal gas Cv g
Work done by an ideal gas in an adiabatic change of state from (P1, V1, T1) to (P2, V2, T2) is W=
nR(T1 - T2 ) g -1
4. The efficiency of a Carnot engine is given by h = 1-
T1 T2
No engine operating between two temperatures can have efficiency greater than that of the Carnot engine.
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Class
10
KINETIC THEORY 1 3
1. Kinetic theory of an ideal gas gives the relation P = nmv
2
2
where n is number density of molecules, m the mass of the molecule and v is the mean of squared speed. Combined with the ideal gas equation it yields a kinetic interpretation of temperature. 2 2 3k BT 1 3 nmv = k BT , vrms = (v )1/ 2 = 2 2 m
2. The law of equipartition of energy is stated thus: the energy for each degree of freedom in thermal equilibrium is 1/2(kBT).
s s la
2 3
3. The translational kinetic energy E = 3 k B NT . This leads to a relation PV = E . 2
C
g RT vs g 4. Speed of sound in a gas vs = M , v = 3 i .e, vs » vrms rms
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Class
11
OSCILLATIONS 1. The particle velocity and acceleration during SHM as functions of time are given by, v(t ) = - w A sin(w + f ) (velocity), a(t ) = - w 2 A cos(wt + f ) = - w 2 x(t ) (acceleration)
Where x(t) = A cos(wt + f ) Velocity amplitude vm = w A and accelearation amplitude am = w 2 A
s s la
2. A particle of mass m oscillating under the influence of a Hooke's law restoring force given by F = -k x exhibits simple harmonic motion with w=
k (angular frequency),T = 2p m (period) m k
C
Such a system is also called a linear oscillator.
3. A body of mass M is suspended form a spring whose force constant is K and mass is m. The time period of this system will be (M + m / 3) 2p
k
4. Time period for conical pendulum T = 2p æç l cosq ö÷ where θ is angle between string & è g ø vertical.
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Class
12
WAVES 1. The displacement in a sinusoidal wave propagating in the positive x direction is given by y (x, t) = a a sin( kx - w t + f ) where a is the amplitude of the wave, k is the angular wave number, (kx - wt + f ) is the phase, and f is the phase constant or phase angle. 2. The speed of a transverse wave on a stretched string is set by the properties of the string. The speed on a string with tension T and linear mass density μ is v = T . m
3. Sound waves are longitudinal mechanical waves that can travel through solids, liquids, or gases. The speed v of sound wave in a fluid having bulk modulus B and density ρ is B. v=
s s la
r
The speed of longitudinal waves in a metallic bar (stretched wire) is v =
Y r
For gases, since B = g P (Adiabatic bulk modulus of elasticity), the speed of sound is v = gP/ r
C
4. The interference of two identical waves moving in opposite directions produces standings waves. For a string with fixed ends, the standing wave is given by y ( x, t ) = [2a sin kx]cos w t . Standing waves are characterized by fixed locations of zero displacement called nodes and fixed locations of maximum displacements called antinodes. The separation between two consecutive nodes or antinodes is l / 2 A stretched string of length L fixed at both the ends vibrates with frequencies given by 1 v . f = 2 2L
The oscillation mode with lowest frequency is called the fundamental mode or the first harmonic. The second harmonic is the oscillation mode with n = 2 and so on. A pipe of length L with one end closed and other end open (such as air columns) vibrates with frequencies given by 1ö v æ f = çn + ÷ , n = 0,1, 2, 3,...... 2 ø 2L è
The set of frequencies represented by the above relations are the normal modes of oscillation of such a system. The lowest frequency given by v/4L is the fundamental mode or the first harmonic.
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Class
13
5. Beats arise when two waves having slightly different frequencies, f1 and f2 and comparable amplitudes, are superposed. The beat frequency is f beat = f1 - f 2 6. The Doppler effect is a change in the observed frequency of a wave when the source S and the observe O moves relative to the medium. For sound the observed frequency f is æ v ± v0 ö given in terms of the source frequency f0 by f = f0 ç ÷ è v ± vs ø
Here v is the speed of sound through the medium, v0 is the velocity of observer relative to the medium, and vs is the source velocity relative to the medium. In using this formula, velocities in the direction OS should be treated should be treated as positive and those opposite to it should be taken to be negative.
s s la
7. Doppler effect formula in light : d l = v where dl is change in wavelength of a spectral l c line
C
of original wave length l and v, the speed of the source and c is the speed of light.
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Class
14
ELECTROSTATICS ur
1. Coulomb's Law: F 21= force q2 due to q1 = 1
k (q1 q2 ) $ r 21 where 21r$ is a unit vector in the r212
direction from q1 to q2 and k = 4pe is the constant of proportionality. 0
2. Electric field due to a point charge q has a magnitude | q | /4pe 0 r 2 it is radially outwards from q, if q is positive, and radially inwards if q is negative. Like Coulomb force, electric field also satisfies upper position principle. 3. An electric dipole is a pair of equal andur opposite charges q and –q separated by some distance 2a. Its dipole moment vector p has magnitude 2qa and is in the direction of the dipole axis form –q to q.
s s la
Field of an electric dipole in its equatorial plane (i.e. the plane perpendicular to its axis and passing through its centre) at a distance r form the centre: ur ur ur - p 1 -p , E= @ 4pe 0 ( a2 + r 2 )3/ 2 4pe 0 r 3
for r >>a
C
Dipole electric field on the axis at a distance r from the center: ur E=
ur ur 2 pr 2p , @ 4pe 0 (r 2 - a 2 )2 24pe 0 r 3
for r >>a
The 1/r3 dependence of dipole electric fields should be noted in contrasturto the 1/r2 dependence of electricr field due to r aurpoint ur charge. In a uniform electric field E , a dipole experiences a torque t given by t = p ´ E but experiences zero net force. ur
ur
4. The flux Df of electric field E passing through a small area element DS is given by ur ur Df = E .D S
5. Gauss's law : The flux of electric field passing through any closed surface S is1 / e 0 times the total charge enclosed by S. ur
The law is especially useful in determining electric field E , when the source distribution has simple symmetry :
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Class
15 ur
l
(a). Thin infinitely long straight wire of uniform linear charge density l : E = 2pe n$ 0 where r is the perpendicular distance of the point from the wire and n$ is the radial unit vector in the plane normal to the wire passing through the point. ur (b). Infinity thin plane sheet of uniform surface charge density E = s $n
2e 0 ur ur (c). Thin spherical shell of uniform surface charge density E = s r$ (r ³ R); E = 0(r < R ) 4pe 0 r 2 r 6. Potential : V (r ) = 1 Q 4pe 0 r
For a charge configuration q1, q2, ………, qn with position vectors r1, r2, …. ,rn the potential at a point P is given by the superposition principle 1 æ q1 q2 qn ö , where r1p V= + + ..... + is the distance between q1 and P , and so on. ç ÷
s s la
4pe 0 è r1 p
r2 p
rn p ø
7. The electrostatic potential at a point with position vector rr due to a point dipole of dipole ur ur $ moment p placed at the origin is 1 p .r . V (r ) =
C
4pe 0 r 2
8. An equipotential surface is a surface over which potential has a constant value. For a point charge, concentric sphere centered at a location of the charge are equipotential ur surfaces. The electric field at a point is perpendicular to the equipotential surface E ur through the point. E is in the direction of the steepest decrease of potential. 9. Potential energy stored in a system of charges is the work done (by an external agency) in assembling the charges at their locations. Potential energy of two charges 1 q1 q2 , where r is distance between q1 and q2. q1, q2 at r distance is given by , U=
4pe 0 r
10. Capacitance is defined C = Q/V, where Q is the charge on positive plate and V is the potential difference between plates. C is determined purely geometrically, by the shapes, sizes and relative positions of the two plates. The unit of capacitance is farad:, -1 1F = 1 C V . For a parallel plate capacitor (with vacuum between the plates), A, C = e0 where A is the area of each plate and d the separation between them. d 11. The energy U stored in a capacitor of capacitance C, with charge Q and voltage V is 1 1 1 Q2 2 U = QV = CV = 2 2 2 C
12. For capacitors in the series combination, the total capacitance C is given by 1 1 1 1 = + + + ... C C1 C2 C3
In the parallel combination, the total capacitance C is given by C = C1 + C2 + C3 + ….. where C1, C2, C3 …… are individual capacitances.
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Class
16
CURRENT ELECTRICITY 1. Current density j gives the amount of charge flowing per second per unit area normal r to the flow, j = nqv d ur
2. EquationE = r j another statement of Ohm's law, i.e., a conducting material obeys Ohm's law when the resistivity of the material does not depend n the magnitude and direction of applied electric field. 3. (a) Total resistance R of n resistors connected in series is given by R = R1 + R2+….+Rn (b). Total resistance R of n resistors connected in parallel is given by 1 1 1 1 = + + .......... + R R1 R2 Rn
s s la
. Where R1 + R2+….+Rn are individual resistance.
C
4. Kirchhoff's Rules – (a) Junction Rule : At any junction is equal to the sum of currents leaving it. (b) Loop Rule : The algebraic sum of the changes in potential in any closed loop is zero. 5. The Wheatstone is an arrangement of four resistances –R1, R2, R3, R4. The null-point condition is given by R1 = R3 , Using which the value of one resistance can be R2
R4
determined, knowing the other three resistances. 6. The potentiometer is a device to compare potential differences. Since the method involves a condition of no current flow, the device can be used to measure potential difference; internal resistance of a cell and compare emf's of two sources. æl ö r = R ç 1 - 1÷ 7. RC circuit : During charging : q = CE (1- e -t /rc ) è l2 ø During discharging : q = q0 e -t /rc
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Class
17
MAGNETISM 1. The total on a charge q moving with velocity v in the presence of electric and ur ur magnetic fields ur E urandur Bur , respectively is called the Lorentz force. It is given by the expression: F = q éëE + (V ´ B )ùû . ur
2. A straight conductor of length _ andurcarrying current I experiences a force F in a ur ur a usteady r uniform external magnetic field B , F = I E ´ B , the direction of l is given by the direction of the current. uur
uur
3. The Biot-Savart law asserts that the magnetic fielddB due to an element dl uu carrying a r r uur steady current I at a point P a a distance r from the current element is: . m0 dl ´ r
s s la
dB =
4p
I
r3
To obtain the total field at P, we must integrate this vector expression over the entire length of the conductor ur
C
4. Magnetic field due to straight current carrying conductor B = q1andq 2
m 0I (sin q1 + sin q2 ) , where 4p a
are the angles between the line joining the point to the ends of conductor and perpendicular through the point to the conductor. 5. The magnitude of the magnetic field due to a circular coil of radius R carrying a current I at an axial distance x from the Centre is B = m0 IR 2 . 2( x 2 + R 2 )3/ 2
6. The magnitude of the field B inside a long solenoid carrying a current I is : B = m0 NI , where m NI , where N is the total number n is the number ofturns per unit length. For a toroid, B= 0 of turns and r is the mean radius. 2p R 7. Ampere's Circuital Law: Let ur uuran open surface S be bounded by a loop C. Then the Ampere's law states that Ñò B.dl = m0 I , where I refers to the current passing through S. c
8. Force per unit length between 'two long parallel wires carrying currents I1, I2 and separated by distance a in a free space or air F = m0 I1 I 2 Nm-1 . 2p a
The force is attractive if currents are in the same direction and repulsive currents are in the opposite direction. uur
ur
r
uur ur
9. For current carrying coil M = NI A ; torque = t = M ´ B
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Class
18
uur
uur placed in an external magnetic field B0 . The magnetic intensity 10. Consider a material uur B0 is defined as, H = .
m0 uur The magnetization M ofurthe material uur uur is its dipole moment per unit volume. The magnetic field B in the material is, B = m0 ( H + M ) . uur
uur
ur
uur
11. for a linear material and c m is called the magnetic B = mH M = c m H. So that susceptibility of the material. The three quantities, c m , the relative magnetic m permeability m, rand the magnetic permeability are related as follows: m = m 0 mr ;
mr = 1 + c m
s s la
C
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Class
19
ELRCTRO MAGNETIC INDUCTION ur
1. The magnetic flux ur through a surface of area A placed in a uniform magnetic field B ur ur ur Is defined as fB = B.A = BA cosq where q is the angle between B and A . 2. Faraday's laws of induction imply that the emf induced in a coil of N turns is directly related to the rate of change of flux through it e = - N df B dt
3. Lenz's law states that the polarity of the induced emf is such that it tends to produce a current which opposes the change in magnetic flux that produces it. The negative sign in the expression for Faraday's law indicates this fact 4. When a metal rod of length l is placed normal to a uniform magnetic field B and moved with a velocity v perpendicular to the field, the induced emf (called motional emf) across its ends is e = Blv
s s la
C
5. When a current in a coil changes, it induces a back emf in all the same coil. The selfinduced emf is given by e = - L dI L is the self-inductance of the coil. It is a measure of dt
the inertia of the coil against the change of current through it. 6. A changing current in a coil (coil 2) can induce an emf in nearby coil (coil l). This relation is given by, dI 2 dt , The quantity M12 is called mutual inductance of coil l with respect to coil 2. M 12 = K L1 L2 . e 1 = - M 12
7. LR circuit: for growth current, i/ = i0 [1 - e -R t/ L ] for decay of current, i = i0 - e- Rt / L
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Class
20
ALTERNATING CURRENT 1. For an alternating current i = im sin cot passing through a resistor R, the average power loss P (averaged over a cycle) due to joule heating is (l/2) i2mR. To express it in the same form as the dc power (P = I2R), a special value of current is used. lt is called root mean square (rms) current and is denoted by, I=
im 2
= 0.707im
The average power loss over a complete cycle is given by P = VI cos f . The term cosfis called the power factor. When a value is given for ac voltage or current, it is ordinarily the rms value.
s s la
2. An ac voltage v = v m sin wt applied to a pure inductor drives a current in the i = im sin(w t - p / 2 ), where i = vm / X L. X L = w L is called inductive reactance. The current in the inductor lags the voltage by p / 2 . The average power supplied to an inductor over one complete cycle is zero. An v = v m sin wt ac applied to a capacitor drives a current in the capacitor i = im sin(w t + p / 2) . Here,
im =
C
Vm 1 is called capacitive reactance. ,XC = XC wC
3. An interesting characteristic of a series RLC circuit is the phenomenon of resonance. The circuit exhibits resonance, i.e.. The amplitude of the current is maximum at the resonant frequency, w = 1 ( X = X ) . The quality factor Q defined byQ = w0 = 1 0
LC
L
C
LC
w0 CR
is an indicator of the sharpness of the resonance, the higher value of Q indicating sharper peak in the current.
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Class
21
RAY OPTICS 1. Reflection is governed by the equation Ði=Ðr and refraction by the Snell's law, sin i/sin r = n, n is refractive index where the incident ray, reflected ray, refracted ray and normal lie in the same plane. 2. Mirror Equation:
1 1 1 + = v u f
.
3. For a prism of the angle A, of refractive index n2 placed in a medium of refractive index n1, n2 sin[( A + Dm ) / 2], where Dm is the angle minimum deviation. Dispersion is the n21 =
n1
=
sin( A / 2)
splitting of light into its constituent colours. The deviation is maximum for violet and minimum for red. Dispersive power to is the ratio of angular dispersion( d v - d r )to the mean deviation d w = d v - d r , where d v , d r ,, are deviation of violet and red respectively
s s la
d
and 6 the deviation of mean ray (usually yellow).
C
4. For refraction through a spherical interface (from medium 1 to 2 of refractive index n1 and n2 respectively) n2 = n1 = n2 - n1 . Thin lens formula 1 - 1 = 1 , Lens maker's formula: v v 1 (n2 - n1 ) æ 1 1 ö = ç - ÷ f n1 è R1 R2 ø
v
v u
f
The power of a lens P = 1/f. The SI unit for power of a lens is dioptre (D): 1 D = 1 m-1. If several thin lenses of focal length f1 , f 2 ,are f in contact, the effective focal length of their combination, is given by 1 1 1 1 f
=
f1
+
f2
+
f3
+ .......
The total power of a combination of several lenses is P = P1 + P2 + P3 + ... If distance between lens is d then power of combination = P = P1 + P2 - d ´ P1 P2 Chromatic aberration is the colorings of image produced by lenses. This can be avoided by combining a convex and a concave lens of focal lengths f2and f2 and ispersive powers w1 , w2 respectively satisfying the equation w1 w2 =0 or in terms of power w1 P1 + w 2 P = 0 + f1
f2
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Class
22
WAVE OPTICS 1. Young's double slit of separation d gives equally spaced fringes of angular separation l / d . The source, mid-point of the slits, and central bright fringe lie in a straight line. An extended source will destroy the fringes if it subtends angle more than l / d at the slits. The resultant intensity of two waves of intensity I0/4 of phase difference f at any points is given by , where 0I is the maximum intensity. 2 éf ù I = I 0 cos ê ú ë2 û
Condition for dark band: d = (2 n - 1)
l , 2
For bright band: d = nl , Fringe width b =
Dl d
s s la
2. A thin film of thickness I and refractive index m appears dark by reflection when viewed at an angle of refraction r if 2m t cos r = nl (n = 1,2,3, etc). The minimum thickness (n = 1) of a film which appears dark by reflection at normal incidence (r= 0°) is 2m t = l . The minimum thickness of a film, which appears bright under normal incidence of monochromatic light of wavelength, l is 2mt = l 2
C
3. A single slit of width a gives a diffraction pattern with a central maximum. The intensity falls to zero at angles of, ± l , ± l , etc. with successively weaker secondary a 2a maxima in between
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Class
23
MODERN PHYSICS 1. Einstein's photoelectric equation is 1 2 mv max = V0 e = hv - f0 = h(v - v0 ) 2
2. The nuclear mass M is always less than the total mass, åm, of its constituents. The difference in mass of a nucleus and 35 its constituents is called the mass defect, DM = ( Zm p + ( A - Z ) mn ) - M ;
DEb = D Mc 2
1 amu = 931 MeV 3.
En = -
Z2 ´ 13.6ev (for hydrogen like atom) n2
s s la
4. Bragg's law: 2d sin q = nl . 5. Law of radioactive decay: N = N 0 e- lt . Activity =
dN = -l N N
6. Half Time Period, T1/2
(unit is Becquerel) 0.693 = l
C
7. X- Rays: lmin = 12400 A° V
Characteristics X- Rays: lka < l La 2 Moseley law: v = a (Z - b) maxima in between
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Class
24
SEMICONDUCTOR 1. Pure semiconductors are called 'intrinsic semiconductors'. The presence of charge carriers (electrons and holes) is an 'intrinsic' property of the material and these are obtained as a result of thermal excitation. The number of electrons (ne) is equal to the number of holes (nh) in intrinsic conductors. Holes are essentially electron vacancies with an effective positive charge. 2. The number of charge carriers can be changed by 'doping' of a suitable impurity in pure semiconductors. Such, semiconductors are known as extrinsic semiconductors. These are of two types (n-type and p-type). 3. In n-type semiconductors, ne >> nh while in p-type semiconductors nh >> ne.
s s la
4. n-type semiconducting Si or Ge is obtained by doping with pentavalent atoms (donors) like As, Sb, P, etc., while p-type Si or Ge can be obtained by doping with trivalent atom (acceptors) like B,Al, In, etc.
C
5. p-n junction is the 'key' to all semiconductor devices. When such a junction is made, a 'depletion layer' is formed consisting of immobile ion-cores devoid of their electrons or holes. This is responsible for a junction potential barrier 6. In forward bias (n-side is connected to negative terminal of the battery and p—side is connected to the positive), the barrier is decreased while the barrier increases in reverse bias. 7. Diodes can be used for rectifying an ac voltage (restricting the ac voltage to one direction). 8. Zener diode is one such special purpose diode. In reverse bias, after a certain voltage, the current suddenly increases (breakdown voltage) in a Zener diode. This property has been used to obtain voltage regulation.
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Class
25
9. The transistors can be connected in such a manner that either C or E or B is common to both the input and output. This gives the three configurations in which a transistor is used: Common Emitter (CE), Common Collector (CC) and Common Base (CB). The plot between IC and VCE for fixed IB is called output characteristics while the plot between IB and VBE with fixed VCE is called input characteristics. The important transistor parameters for CE-configuration are: æ DV
ö
Input resistance, ri = ç BE ÷ è DI B øVCE æ DVCE ö ÷ è DI C ø I B
Output resistance, r0 = ç
s s la
æ DI ö
Current amplification factor, b = ç C ÷ è DI B ø V
CE
C
The voltage gain of a transistor amplifier in common emitter configuration is: æv ö R Av = ç 0 ÷ = b C ,Where R and R are respectively the resistances in collector and C B RB è vi ø
base sides of the circuit. 10.The important digital circuits performing special logic operations are called logic gates. These are: OR, AND, NOT, NAND, and NOR gates.
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Class
26
COMMUNICATION SYSTEM 1. Transmitter, transmission channel and receiver are three basic units of a communication system. In the process of transmission of message/ information signal, noise gets added to the signal anywhere between the information source and the receiving end. 2. Two important forms of communication system are: Analog and Digital. The information to be transmitted is generally in continuous waveform for the former while for the latter it has only discrete or quantized levels. 3. Low frequencies cannot be transmitted to long distances. Therefore, they are superimposed on a high frequency carrier signal by a process known as modulation.
s s la
4. In modulation, some characteristic of the carrier signal like amplitude, frequency or phase varies in accordance with the modulating or message signal. Correspondingly, they are called Amplitude Modulated (AM), Frequency Modulated (FM) or Phase Modulated (PM) waves. In the process of modulation, new frequencies called sidebands are generated on either side (higher and lower than the earner frequency) of the carrier by an amount equal to the highest modulating frequency.
C
5. If an antenna radiates electromagnetic waves from a height hp then the range dT is given by 2 RhT where R is the radius of the earth.
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Class
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Mumbai Intelivisto Consulting India Private Limited
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