Physics Handbook By Disha

UNITS AND MEASUREMENTS The SI system : It is the international system of units. At present internationally accepted for measurement. In this system there are seven fundamental and two supplementary quantities and their corresponding units are: Quantity 1. Length 2. Mass 3. Time 4. Electric current 5. Temperature 6. Luminous intensity 7. Amount of substance Supplementary 1. Plane angle 2. Solid angle

Unit metre kilogram second ampere kelvin candela mole

Symbol m kg s A K cd mol

radian steradian

rad sr

Dimensions : These are the powers to which the fundamental units are raised to get the unit of a physical quantity. Uses of dimensions (i) To check the correctness of a physical relation. (ii) To derive relationship between different physical quantities. (iii) To convert one system of unit into another. n 1u 1 = n 2 u 2 a b c n1[M1a Lb1T1c ] = n2 [M 2 L 2T2 ]

Significant figures : In any measurement, the reliable digits plus the first uncertain digit are known as significant figures. Error : It is the difference between the measured value and true value of a physical quantity or the uncertainty in the measurements. Absolute error : The magnitude of the difference between the true value and the measured value is called absolute error. Da1 = a - a1 , Da2 = a - a2 , Dan = a - an Mean absolute error n

Da =

1 | Da1 | + | Da2 | +.....+ | Dan | | Dai | = n i =1 n

å

Relative error : It is the ratio of the mean absolute error to its true value or relative error =

Da a

Percentage error : It is the relative error in per cent.

æ Da ö Percentage error = ç ´ 100% è amean ÷ø

MOTION IN A STRAIGHT LINE Average speed, Vav =

s1 + s2 + s3

t1 + t 2 + t 3 a t +a t Average acceleration, aav = 1 1 2 2 t1 + t 2

The area under the velocity-time curve is equal to the displacement and slope gives acceleration. If a body falls freely, the distance covered by it in each subsequent second starting from first second will be in the ratio 1 : 3 : 5 : 7 etc. 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. If air resistance acting on a body is considered, the time taken by the body to reach maximum height is less than the time to fall back the same height. a For a particle having zero initial velocity if s µ t , where a > 2 , then particle’s acceleration increases with time. a For a particle having zero initial velocity if s µ t , where a < 0 , then particle’s acceleration decreases with time. Kinematic equations : v = u + at (t) ; v2 = u2 + 2at (s) a 1 S = ut + at (t)2; Sn = u + (2 n - 1) 2 r 2 applicable only when | a t | = a t is constant. at = magnitude of tangential acceleration, S = distance If acceleration is variable use calculus approach. r r r Relative velocity : v BA = vB - vA

MOTION IN A PLANE If T is the time of flight, h maximum height, R horizontal range of a projectile, a its angle of projection, then the relations among these quantities.

gT 2 8 gT2 = 2R tan a R tan a = 4h h=

...... (1); ....... (2); ....... (3)

2

PHYSICS T=

On a banked road, the maximum permissible speed

2u sin q u 2 sin 2 q ;h = g 2g

u2 u 2 sin 2q when q = 45° ; Rmax = g g For a given initial velocity, to get the same horizontal range, there are two angles of projection a and 90° – a. The equation to the parabola traced by a body projected horizontally from 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. R=

Equation of trajectory is y = x tan q -

gx 2 2

2u cos 2 q

, which

is parabola. Equation of trajectory of an oblique projectile in terms

xö æ of range (R) is y = x tan q ç1 - ÷ è Rø Maximum height is equal to n times the range when the projectile is launched at an angle q = tan–1(4n). In a uniform circular motion, velocity and acceleration are constants only in magnitude. Their directions change. In a uniform circular motion, the kinetic energy of the r r r body is a constant. W = 0, a ¹ 0, P ¹ constant, L=constant

v2 = wv (always Centripetal acceleration, a r = w 2 r = r r r r applicable) a r = w ´ v

LAWS OF MOTION Newton’s first law of motion or law of inertia : It is resistance to change. r r r r Newton’s second law : F = ma, F = dp / dt 2 r r Impulse : Dp = FDt, p2 - p1 = ò F dt r r1 Newton’s third law : F12 = - F21 Frictional force fs £ (fs ) max = ms R ;

æ ö1/2 çç R u s ∗ tan π ÷÷ Vmax = çç g ÷ è 1, u tan π ø÷ s

WORK, ENERGY AND POWER Work done W = FS cosq Relation between kinetic energy E and momentum, P = 2mE K.E. = 1/2 mV2; P.E. = mgh If a body moves with constant power, its velocity (v) is related to distance travelled (x) by the formula v µ x3/2. W Power P = = F.V t Work due to kinetic force of friction between two contact surfaces is always negative. It depends on relative displacement between contact surfaces. WFK= -FK (Srel ) .

S W= S DK , S W Þ total work due to all kinds of forces, S DK Þ total change in kinetic energy. SWconservative = -S DU ; SWconservative Þ Total work due to all kinds of conservative forces. SDu Þ Total change in all kinds of potential energy. velocity of separation Coefficient of restitution e = velocity of approach The total momentum of a system of particles is a constant in the absence of external forces.

SYSTEM OF PARTICLES & ROTATIOAL MOTION The centre of mass of a system of particles is defined as mr the point whose position vector is R = å i i M The angular momentum of a system of n particles about n

fk = m k R

the origin is L = å ri ´ p i ; L = mvr = Iw i =1

Circular motion with variable speed. For complete circles, the string must be taut in the highest position, u 2 ³ 5ga . 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 2ga < u < 5ga .

The torque or moment of force on a system of n particles

Conical pendulum : w = g / h where h is height of a point of suspension from the centre of circular motion. The acceleration of a lift actual weight - apparent weight a= mass If ‘a’ is positive lift is moving down, and if it is negative the lift is moving up.

1 The kinetic energy of rotation is K = Iw 2 2

about the origin is t = å ri ´ Fi i

The moment of inertia of a rigid body about an axis is defined by the formula I = å mi ri 2

The theorem of parallel axes : I'z = Iz + Ma2 Theorem of perpendicular axes : Iz = Ix + Iy For rolling motion without slipping vcm = Rw. The kinetic energy of such a rolling body is the sum of kinetic energies of translation and rotation :

3

FORMULAE BOOK 1 1 2 mvcm + Iw 2 2 2 A rigid body is in mechanical equilibrium if (a) It is translational equilibrium i.e., the total external force on it is zero : S Fi = 0. (b) It is rotational equilibrium i.e., the total external torque on it is zero : S ti = Sri × Fi = 0. If a body is released from rest on rough inclined plane, n tan q (Ic = nmr2) then for pure rolling m r ³ n +1 æ n ö Rolling with sliding 0 < ms < çè ÷ tan q ; n + 1ø K=

g sin q < a < g sin q n +1

GRAVITATION Newton’s universal law of gravitation Gm1m 2 Gravitational force F = r2 G = 6.67 × 10 –11

Nm 2 kg 2

The acceleration due to gravity. (a) at a height h above the Earth’s surface g(h) =

æ 2h ö = g ç1 ÷ for h << RE (R E + h) è RE ø GM E

2

æ 2h ö GM E g(h) = g(0) ç 1 where g(0) = è R E ÷ø R 2E (b) at depth d below the Earth’s surface is æ GM E æ d ö d ö 1= g (0) ç1 ÷ 2 ç è R E ÷ø RE è RE ø (c) with latitude l g1 = g – Rw2 cos2l GM Gravitational potential Vg = – r GM Intensity of gravitational field I = 2 r The gravitational potential energy g(d) =

Gm1m 2 V=+ constant r The escape speed from the surface of the Earth is ve =

2GM E = RE

2gR E and has a value of 11.2 km s–1.

Orbital velocity, vorbi =

GM E = RE

gR E

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.

Kepler’s 3rd law of planetary motion. T2 µ a3 ;

T12 T22

=

a13 a 32

MECHANICAL PROPERTIES OF SOLIDS Hooke’s law : stress µ strain Young’s modulus of elasticity Y = Compressibility =

1 Bulk modulus

F Dl Al

Y = 3k (1 – 2s) Y = 2n (1 + s) If S is the stress and Y is Young’s modulus, the energy density of the wire E is equal to S2/2Y. If a is the longitudinal strain and E is the energy density of a stretched wire, Y Young’s modulus of wire, then E 1 2 is equal to Ya 2 F Thermal stress = = Y a Dq A

MECHANICAL PROPERTIES OF FLUIDS 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. Pressure exerted by a liquid column P = hrg Bernoulli’s principle P + rv2/2 + rgh = constant Surface tension is a force per unit length (or surface energy per unit area) acting in the plane of interface. Stokes’ law states that the viscous drag force F on a sphere of radius a moving with velocity v through a fluid of viscosity h F = – 6phav. Terminal velocity VT =

2 r 2 (r – σ)g 9 h

The surface tension of a liquid is zero at boiling point. The surface tension is zero at critical temperature. If a drop of water of radius R is broken into n identical drops, the work done in the process is 4pR2S(n1/3 – 1) 3T 1 1 , J r R Two capillary tubes each of radius r are joined in parallel. The rate of flow is Q. If they are replaced by single capillary tube of radius R for the same rate of flow, then R = 21/4 r. Ascent of a liquid column in a capillary tube 2s cos ε h= rθg

and fall in temperature Dq =

4

PHYSICS Coefficient of viscosity, n = -

F æ dv ö Aç ÷ è dx ø

Velocity of efflux V = 2 gh

THERMAL PROPERTIES OF MATTER Relation between different temperature scales : C F , 32 K , 273 < = 100 180 100 The coefficient of linear expansion (al), superficial (b) and volume expansion (av) are defined by the relations :

DA DV Dl = a l DT ; = bDT ; = a V DT l A V av = 3al ; b = 2a l In conduction, heat is transferred between neighbouring parts of a body through molecular collisions, without any

TC - TD flow of matter. The rate of flow of heat H = KA , L where K is the thermal conductivity of the material of the bar. Convection involves flow of matter within a fluid due to unequal temperatures of its parts. Radiation is the transmission of heat as electromagnetic waves. Stefan’s law of radiation : E = sT4, where the constant s is known as Stefan’s constant = 5.67 × 10–8 wm–2 k–4. Wein’s displacement law : lmT = constant, where constant is known as Wein’s constant = 2.898 × 10 –3 mk.

dQ = - k (T2 - T1 ) ; where T1 is dt the temperature of the surrounding medium and T2 is the temperature of the body. Heat required to change the temperature of the substance, Q = mcDq c = specific heat of the substance Heat absorbed or released during state change Q = mL L = latent heat of the substance Mayer’s formula cp – cv = R

Newton’s law of cooling:

First law of thermodynamics: DQ = DU + DW, where DQ is the heat supplied to the system, DW is the work done by the system and DU is the change in internal energy of the system. 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 æV ö Q = W = nRT ln ç 2 ÷ è V1 ø In an adiabatic process of an ideal gas PVg = TVg–1

=

P g -1

KINETIC THEORY Ideal gas equation PV = nRT 1 Kinetic theory of an ideal gas gives the relation P = nmv 2 , 3 Combined with the ideal gas equation it yields a kinetic interpretation of temperature. 1 3 nmv 2 = k B T , v rms = (v 2 )1/2 = 2 2

3k B T m

The law of equipartition of energy is stated thus: the energy for each degree of freedom in thermal equilibrium is 1/2 (kBT) The translational kinetic energy E =

3 k B NT . This leads 2

2 E. 3 Degree of freedom : Number of directions in which it can move freely. Root mean square (rms) velocity of the gas

to a relation PV =

C=

3RT = M

3P r

Most probable speed Vmp = Mean free path l =

KT

2RT = M

2KT m

2 pd 2 P

OSCILLATIONS Displacement in SHM : Y = a sin wt or, y = a cos wt

THERMODYNAMICS

Tg

Work done by an ideal gas in an adiabatic change of state nR (T1 - T2 ) from (P1, V1, T1) to (P2, V2, T2) is W = g -1 T The efficiency of a Carnot engine is given by h = 1 - 2 T1 Second law of thermodynamics: No engine operating between two temperatures can have efficiency greater than that of the Carnot engine. dQ Entropy or disorder S = T

Cp

= constant, where g = C . v

The particle velocity and acceleration during SHM as functions of time are given by, v (t) = – wA sin (w + f ) (velocity), a (t) = – w2A cos (wt + f) = – w2x (t) (acceleration) Velocity amplitude vm = w A and acceleration amplitude am =w2A. 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), m

m (period) k Such a system is also called a linear oscillator. T = 2p

5

FORMULAE BOOK æ l cosq ö Time period for conical pendulum T = 2p where çè g ÷ø q angle between string & vertical. 1 Energy of the particle E = k + u = mω2 A 2 2

WAVES The displacement in a sinusoidal wave y (x, t) = a sin (kx – wt + f) where f is the phase constant or phase angle. t x Equation of plane progressive wave : = a sin 2p æç - ö÷ èT Vø 2p t 2 px cos Equation of stationary wave : Y = 2a sin T l The speed of a transverse wave on a stretched string v = T/m. 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 µ is v = B/r. The speed of longitudinal waves in a metallic bar is v=

Y/r

For gases, since B = g P, the speed of sound is v = gP / r The interference of two identical waves moving in opposite directions produces standing waves. For a string with fixed ends, standing wave y (x, t) = [2a sin kx ] cos wt The separation between two consecutive nodes or antinodes is l/2. A stretched string of length L fixed at both the ends vibrates 1 v with frequencies f = . 2 2L The oscillation mode with lowest frequency is called the fundamental mode or the first harmonic. 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, .... The lowest frequency è 2 ø 2L given by v/4L is the fundamental mode or the first harmonic. V Open organ pipe n1 : n2 : n3 ....... 1, 2, 3......., n = 2l Beats arise when two waves having slightly different frequencies, f1 and f2 and comparable amplitudes, are superposed. The beat frequency fbeat = f1 – f2 The Doppler effect is a change in the observed frequency of a wave when the source S and the observer O moves relative æ v ± v0 ö to the medium. f = f0 çè v ± v ÷ø s

ELECTROSTATICS

k (q1q 2 ) r ˆr21 Coulomb’s Law : F21 = force on q2 due to q1 = 2 r21

where k =

1 = 9 × 109 Nm2 C–2 4pe 0

Electric field due to a point charge q has a magnitude | q |/4pe0r2 Field of an electric dipole in its equatorial plane r r -p 1 -p @ E= , for r >> a 4pe 0 (a 2 + r 2 )3/2 4pe 0 r 3 Dipole electric field on the axis at a distance r from the centre: r r r 2pr 2p E= @ for r >> a 4pe 0 (r 2 - a 2 ) 2 4pe 0 r 3 r Dipole moment p = q 2a r In a uniform electric field E , a dipole experiences a torque r r r r t given by t = p ´ E but experiences no net force. r The flux Df of electric field E through a small area element r r r DS is given by Df = E.DS Gauss’s law: The flux of electric field through any closed surface S is 1/e0 times the total charge enclosed i.e., Q Thin infinitely long straight wire of uniform linear charge r l density l : E = nˆ 2pe 0 r Infinite thin plane sheet of uniform surface charge density s r s E= nˆ 2e 0 Thin spherical shell of uniform surface charge density s : r s r E= rˆ (r ³ R) ; E = 0 (r < R) 4pe r 2 0

r Electric Potential : V(r) =

1 Q . 4pe 0 r An equipotential surface is a surface over which potential has a constant value. r r Potential energy of two charges q1, q2 at r1, r2 is given by 1 q1q 2 , where r12 is distance between q1 and q2. 4pe 0 r12 Capacitance C = Q/V, where Q = charge and V = potential difference For a parallel plate capacitor (with vacuum between the A plates), C = e 0 . d The energy U stored in a capacitor of capacitance C, with U=

1 1 1 Q2 charge Q and voltage V is U = QV = CV 2 = 2 2 2 C For capacitors in the series combination, 1 1 1 1 = + + + ......... Ceq C1 C2 C3

In the parallel combination, Ceq = C1 + C2 + C3 + ... where C1, C2, C3... are individual capacitances.

CURRENT ELECTRICITY q t Current density j gives the amount of charge flowing per r second per unit area normal to the flow, J = nqvd

Electric current, I =

6

PHYSICS Mobility, µ =

Vd I and Vd = E Ane

l Resistance R = r , r = resistivity of the material A r r Equation E = rJ another statement of Ohm’s law,, r= resistivity of the material. Ohm’s law I µ V or V = RI (a) Total resistance R of n resistors connected in series R = R1 + R2 +..... + Rn (b) Total resistance R of n resistors connected in parallel 1 1 1 1 = + + ...... + R R1 R 2 Rn . Kirchhoff’s Rules – (a) Junction rule: At any junction of circuit elements, the sum of currents entering the junction must equal the sum of currents leaving it. (b) Loop rule: The algebraic sum of changes in potential around any closed loop must be zero. The Wheatstone bridge is an arrangement of four resistances R1 , R2 , R3, R4 . The null-point condition is given by R1 R 3 = R2 R4 The potentiometer is a device to compare potential differences. The device can be used to measure potential difference; internal resistance of a cell and compare emf’s of æl ö two sources. Internal resistance r = R ç 1 - 1÷ è l2 ø

RC circuit : During charging : q = CE (1 – e–t/RC) During discharging : q = q0e–t/RC According ‘to Joule’s Heating law, H = I2Rt

MAGNETISM The total force on a charge q moving with velocity v i.e., r r r r Lorentz force. F = q (v ´ B + E) . A straight conductor of length l and carrying a steady current r I experiences a force F in a uniform external magnetic field r r r r B , F = Il ´ B , the direction of l is given by the direction of the current. r r m 0 d l ´ rr I 3 . Biot-Savart law dB = 4p r 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 =

m 0 IR 2

. 2 (x 2 + R 2 )3/2 The magnitude of the field B inside a long solenoid carrying m 0 NI a current I is : B = µ0nI. For a toroid one obtains, B = . 2pr r r Ampere’s Circuital Law : Ñò B.d l = m 0 I , where I refers to the C

current passing through S.

m 0 I1I2 Nm -1 . 2pa The force is attractive if currents are in the same direction and repulsive currents are in the opposite direction. r r r r For current carrying coil M = NIA ; torque = rt = M ´ B

Force between two long parallel wires F =

æ Ig Conversion of (i) galvanometer into ammeter, S = ç ç I - Ig è

(ii) galvanometer into voltmeter, S =

ö ÷G ÷ ø

V -G Ig

r r B0 The magnetic intensity, H = . m0 r The magnetisation M of the material is its dipole moment per unit volume. The magnetic field B in the material is, r r r B = m 0 (H + M) r r r r For a linear material M = cH . So that B = mH and c is called the magnetic susceptibility of the material. m = m 0m r ; m r = 1 + c .

ELECTROMAGNETIC INDUCTION The magnetic flux

r r r r fB = B.A = BA cos q , where q is the angle between B & A .

df B dt 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 induced emf (motional emf) across ends of a rod e = Blv dI The self-induced emf is given by, e = - L dt L is the self-inductance of the coil.

Faraday’s laws of induction : e = - N

m0 N2 A l A changing current in a coil (coil 2) can induce an emf in a nearby coil (coil 1). dI e1 = - M12 2 , M12 = mutual inductance of coil 1 w.r.t coil dt 2. m NN A M= 0 1 2 l L=

Growth of current in an inductor, i = i 0 [1 - e - Rt /L ] For decay of current, i = i 0 e - Rt /L

ALTERNATING CURRENT For an alternating current i = im sin wt passing through a resistor R, the average power loss P (averaged over a cycle) due to joule heating is (1/2)i2mR. E.m.f, E = E0 sin wt

7

FORMULAE BOOK Root mean square (rms) current I =

im 2

= 0.707 i m .

E0 2 The average power loss over a complete cycle P = V I cosf. The term cos f is called the power factor. An ac voltage v = vm sinwt applied to a pure inductor L, drives a current in the inductor i = im sin (wt – p/2), where im = vm/XL. XL = wL is called inductive reactance. An ac voltage v = vm sinwt applied to a capacitor drives a current in the capacitor: i = im sin (wt + p/2). Here, Erms =

im =

vm 1 , XC = is called capacitive reactance. XC wC

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, w0 = 1 (X L = XC ) . LC

Impedance z = R 2 + (x L – x C ) 2 Transformation ratio, K =

NS ES IP = = N P E P IS

Step up transformer : NS > NP; ES > EP; IP > IS Step down transformer NP > NS; EP > ES and IP < IS w0L 1 = is an R w 0 CR indicator of the sharpness of the resonance, the higher value of Q indicating sharper peak in the current.

The quality factor Q defined by Q =

RAY OPTICS

1 1 1 + = v u f V I Magnification M = = u O

Mirror equation:

n 2 sin [(A + D m ) / 2)] , where Dm is = n1 sin (A / 2)

the angle of minimum deviation. Dispersion is the splitting of light into its constituent colours. The deviation is maximum for violet and minimum for red. Scattering µ

1 l4

d v - dr , where dv, dr are deviation of d violet and red and d the deviation of mean ray (usually yellow).

Dispersive power w =

1 (n 2 - n1 ) æ 1 1 ö Lens maker’s formula : f = ç n1 è R1 R 2 ÷ø 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, f2, f3,.. are in contact, 1 1 1 1 the effective focal f = f + f + f + ..... 1 2 3 The total power of a combination of several lenses P = P1 + P2 + P3 +....... Chromatic aberration if satisfying the equation w1 w 2 + = 0 or in terms of powers w P + w P = 0 . 1 1 2 2 f1 f 2

For compound microscope M =

V0 æ D ö 1+ u 0 çè fe ÷ø

when final image at D V0 D . when final image at infinity.. M= u 0 fe

WAVE OPTICS

Reflection is governed by the equation Ð i = Ð r' and refraction by the Snell’s law, sini/sinr = n, where the incident ray, reflected ray, refracted ray and normal lie in the same plane.

Prism Formula n 21 =

For refraction through a spherical interface (from medium 1 to 2 of refractive index n 1 and n2, respectively) n 2 n1 n 2 - n1 = v v R C Refractive index of a medium m = (C = 3 × 108 m/s) V 1 r= (C = Critical angle) sin C Condition for TIR : 1. Ray of light must travel from denser to rarer medium 2. Angle of incidence in denser medium > critical angle. 1 1 1 Lens formula - = v u f

Wavefront : It is the locus of all the particles vibrating in the same phase. The resultant intensity of two waves of intensity I0/4 of éf ù phase difference f at any points I = I0 cos 2 ê ú , ë2û where I0 is the maximum density. Intensity I µ (amplitude)2 l Condition for dark band : d = (2n - 1) , for bright band : 2 d = ml Dl Fringe width b = d A thin film of thickness t and refractive index µ appears dark by reflection when viewed at an angle of refraction r if 2µt cos r = nl (n = 1, 2, 3, etc.) A single slit of width a gives a diffraction pattern with a central maximum. The intensity falls to zero at angles of l 2l ± ,± , etc. a a

8

PHYSICS 2 2 Amplitude of resultant wave R = a + b + 2ab cos f

Intensity of wave I = I1 + I2 + 2 I1I2 cosf Brewster law : µ = tan ip

MODERN PHYSICS Energy of a photon E = hn =

æ DV ö ri = ç BE ÷ è DI B ø V

hc l

Current amplification factor, b = æ DIC ö çè DI ÷ø B V

Einstein’s photoelectric equation

CE

1 mv 2max = V0 e = hn - f 0 = h ( n - n0 ) 2 Mass defect, DM = (Z mp + (A – Z )mn) – M ; D Eb = DM c2. 1 amu = 931 MeV Z

2

´ 13.6eV (For hydrogen like atom) n2 According to Bohr’s atomic model, angular momentum for the electron revolving in stationary orbit, mvr = nh/2p

Radius of the orbit of electron r =

n 2h2 4π 2 mkze 2

Bragg’s law : 2d sin q = nl. Radius of the nucleus R = R o A1/3 Law of radioactive decay : N = N0e–lt. Activity =

dN = -lN (unit is Becquerel) N

Half life period, T1/2 =

æ DV ö r0 = ç CE ÷ è DI C ø I

CE

h Momentum of a photon P = l

En = -

Diodes can be used for rectifying an ac voltage (restricting the ac voltage to one direction). 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. The important transistor parameters for CE-configuration are: Input resistance Output resistance

0.693 l

Characteristics X-rays : l Ka < l La

Moseley law : n = a (Z – b)2 Pure semiconductors are called ‘intrinsic semiconductors’. The presence of charge carriers (electrons and holes) number of electrons (n e ) is equal to the number of holes (n h). The number of charge carriers can be changed by ‘doping’ of a suitable impurity in pure semiconductors known as extrinsic semiconductors (n-type and p-type). In n-type semiconductors, n e >> n h while in p-type semiconductors nh >> ne. 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. 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.

B

DI

; a = æç C ö÷ è DI E øV

CE

The voltage gain of a transistor amplifier in common emitter configuration is: æv ö R A v = ç 0 ÷ = b C , where R and R are respectively the C B RB è vi ø

resistances in collector and base sides of the circuit. The important digital circuits performing special logic operations are called logic gates. These are: OR, AND, NOT, NAND, and NOR gates. NAND gate is the combination of NOT and AND gate. NOR gate is the combination of NOT and OR gate.

COMMUNICATION SYSTEMS Transmitter, transmission channel and receiver are three basic units of a communication system. 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 quantised levels. Low frequencies cannot be transmitted to long distances. Therefore, they are superimposed on a high frequency carrier signal by a process known as modulation. In the process of modulation, new frequencies called sidebands are generated on either side. If an antenna radiates electromagnetic waves from a height hT, then the range dT

2Rh T R = radius of earth.

Effective range, d = 2Rh T + 2Rh R hT = height of transmitting antenna; h R = height of receiving antenna Critical frequency Vc = 9(Nmax)1/2 where Nmax = no. density of electrons/m3 2

æ Vmax ö -1 è Vc ÷ø

Skip distance, Dskip = 2h ç

h = height of reflecting layer of atmosphere.

1 Power radiated by an antenna µ 2 l