Physics Formula Handbook by Competishun

 b  V (ii) Ein = E – Eind = 0 – 0 = K0 = d w.jee books  E : Electric field in the absence of dielectric 0 Eind : Induced (bound) charge density. 1 (iii) b = (1 – K ). 6. Force on dielectric (i) When battery is connected F  0b(K 1)V2 2d  bb + Fd x– Q2 dC (ii) When battery is not connected F = 2C2 dx * Force on the dielectric will be zero when the dielectric is fully inside. WWW.JEEBOOKS.IN Page # 51

ALTERNATING CURRENT 1. AC AND DC CURRENT : A current that changes its direction periodically is called alternating cur- rent (AC). If a current maintains its direction constant it is called direct current (DC). w.jeebooks 3. ROOT MEAN SQUARE VALUE: Root Mean Square Value of a function, from t1 to t2, is defined as frms = t2 .  f 2dt t1 t2  t1 4. POWER CONSUMED OR SUPPLIED IN AN AC CIRCUIT: 2   Pdt 1 o Average power consumed in a cycle = = V m cos  2 2 m  Vm m Page # 52 = 2 . 2 . cos = Vrms rms cos. Here cos is called power factor. WWW.JEEBOOKS.IN

5. SOME DEFINITIONS: The factor cos  is called Power factor. m sin  is called wattless current. Impedance Z is defined as Z = Vm = Vrms m rms w.jeebooks L is called inductive reactance and is denoted by X L. 1 is called capacitive reactance and is denoted by X C. C 6. PURELY RESISTIVE CIRCUIT: I = vs = Vm sin t = m sin t R R m = Vm R rms = Vrms R <P> = Vrmsrmscos  Vrms2 R 7. PURELY CAPACITIVE CIRCUIT: I= = Vm cos t 1 C = Vm cos t = m cos t. XC 1 XC = C and is called capacitive reactance. v VT t i I t WWW.JEEBOOKS.IN Page # 53

I leads by v by /2 Diagrammatically C C (phasor diagram) it is represented as m . w.jeebooks Vm Since º, <P> = Vrms rmscos  MAGNETIC EFFECT OF CURRENT & MAGNETIC FORCE ON CHARGE/CURRENT 1. Magnetic field due to a moving point charge    0  q(v  r ) B 4 r3 vr 2. Biot-savart's Law dB  0I   d    4  r r   3  3. Magnetic field due to a straight wire  1 P r 2 B= 0 I 4 r (sin 1 + sin 2)  rP 4. Magnetic field due to infinite straight wire  B= 0 I 2 r 5. Magnetic field due to circular loop (i) At centre B = 0NI (ii) At Axis 2r B= 0  NR2  2  (R2  x2 )3 / 2  WWW.JEEBOOKS.IN Page # 54

6. Magnetic field on the axis of the solenoid 2 1 B = 0nI (cos 1 – cos 2) 2 7. Ampere's Law w.jeebooks  B.d  0I 8. Magnetic field due to long cylinderical shell B = 0, r < R = 0 I ,r  R 2 r 9. Magnetic force acting on a moving point charge   a. F  q( B) (i)   ××××  B  ×× × r  m × × B× × qB r× × × × × 2m T = qB (ii)  r  m sin B qB 2m Pitch = 2mcos  T = qB qB    F  q ( B)  E  b. 10. Magnetic force acting on a current carrying wire     F I B 11. Magnetic Moment of a current carrying loop M=N·I·A 12. Torque acting on a loop     M B WWW.JEEBOOKS.IN Page # 55

13. Magnetic field due to a single pole B = 40 ·rm2 14. Magnetic field on the axis of magnet w.jeebooks B= 0 · 2M 4 r3 15. Magnetic field on the equatorial axis of the magnet B= 0 · M 4 r3 16. Magnetic field at point P due to magnet B= 0 M 1 3 cos2  4 r3 P r  SN ELECTROMAGNETIC INDUCTION  1. Magnetic flux is mathematically defined as  = B.ds 2. Faraday’s laws of electromagnetic induction d E = – dt 3. Lenz’s Law (conservation of energy principle) According to this law, emf will be induced in such a way that it will oppose the cause which has produced it. Motional emf 4. Induced emf due to rotation Emf induced in a conducting rod of length l rotating with angular speed  about its one end, in a uniform perpendicular magnetic field B is 1/2 B  2. WWW.JEEBOOKS.IN Page # 56

1. EMF Induced in a rotating disc : Emf between the centre and the edge of disc of radius r rotating in a magnetic field B = Br2 2 5. Fixed loop in a varying magnetic field dB If magnetic field changes with the rate dt , electric field is generated w.jeebooks r dB whose average tangential value along a circle is given by E= 2 dt This electric field is non conservative in nature. The lines of force associ- ated with this electric field are closed curves. 6. Self induction  =  ( N)   (LI )   LI . t t t The instantaneous emf is given as  =  d(N)   d(LI)   LdI dt dt dt Self inductance of solenoid = µ0 n2 r2. 6.1 Inductor It is represent by electrical equivalence of loop  VA L dI  VB dt 1 Energy stored in an inductor = L 2 2 7. Growth Of Current in Series R–L Circuit If a circuit consists of a cell, an inductor L and a resistor R and a switch S ,connected in series and the switch is closed at t = 0, the current in the circuit I will increase as I =  (1 e Rt ) L R WWW.JEEBOOKS.IN Page # 57

The quantity L/R is called time constant of the circuit and is denoted by . The variation of current with time is as shown. 1. Final current in the circuit =  , which is independent of L. R w.jeebooks 2. After one time constant , current in the circuit =63% of the final current. 3. More time constant in the circuit implies slower rate of change of current. 8 Decay of current in the circuit containing resistor and inductor: Let the initial current in a circuit containing inductor and resistor be 0. Rt Current at a time t is given as I = 0 e L Current after one time constant : I = 0 e1=0.37% of initial current. 9. Mutual inductance is induction of EMF in a coil (secondary) due to change in current in another coil (primary). If current in primary coil is I, total flux in secondary is proportional to I, i.e. N  (in secondary)  I. or N  (in secondary) = M I. The emf generated around the secondary due to the current flowing around the primary is directly proportional to the rate at which that current changes. 10. Equivalent self inductance : L  VA  VB ..(1) dI / dt 1. Series combination : L = L1 + L2 ( neglecting mutual inductance) L = L + L + 2M (if coils are mutually coupled and they have 12 winding in same direction) L = L1 + L2 – 2M (if coils are mutually coupled and they have winding in opposite direction) 2. Parallel Combination : 11 1 ( neglecting mutual inductance)  L L1 L2 WWW.JEEBOOKS.IN Page # 58

For two coils which are mutually coupled it has been found that M  L1L2 or M =k L1L2 where k is called coupling constant and its value is less w.jeebooksthan or equal to 1. Magnetic Core Es  Ns  p , where denota- S Ep Np s tions have their usual mean- EP ES ings. NS > NP Primary Secondary  ES > EP  coil coil for step up transformer. 12. LC Oscillations 2  1 LC GEOMETRICAL OPTICS 1. Reflection of Light (b) i = r 1.3 Characteristics of image due to Reflection by a Plane Mirror: (a) Distance of object from mirror = Distance of image from the mirror. (b) The line joining a point object and its image is normal to the reflecting surface. (c) The size of the image is the same as that of the object. (d) For a real object the image is virtual and for a virtual object the image is real 2. Relation between velocity of object and image : From mirror property : x = - x , y = y and z = z im om im om im om Here xim means ‘x’ coordinate of image with respect to mirror. Similarly others have meaning. WWW.JEEBOOKS.IN Page # 59

Differentiating w.r.t time , we get v(im)x = -v(om)x ; v(im)y = v(om)y ; v = v(im)z (om)z , 3. Spherical Mirror w.jeebooks11 2 1 ..... Mirror formula += = vuRf x co–ordinate of centre of Curvature and focus of Concave mirror are negative and those for Convex mirror are positive. In case of mirrors since light rays reflect back in - X direction, therefore -ve sign of v indicates real image and +ve sign of v indicates virtual image (b) Lateral magnification (or transverse magnification) m= h2 v h1 m = u. (d) dv v2 . On differentiating (a) we get du =  u2 (e) On dif f erentiating (a) with respect to time we get dv v2 du dv   u2 ,where is the velocity of image along Principal dt dt dt du axis and dt is the velocity of object along Principal axis. Negative sign implies that the image , in case of mirror, always moves in the direction opposite to that of object.This discussion is for velocity with respect to mirror and along the x axis. (f) Newton's Formula: XY = f 2 X and Y are the distances ( along the principal axis ) of the object and image respectively from the principal focus. This formula can be used when the distances are mentioned or asked from the focus. 1 (g) Optical power of a mirror (in Diopters) = f f = focal length with sign and in meters. (h) If object lying along the principal axis is not of very small size, the longitudinal magnification = v2  v1 (it will always be inverted) u2  u1 WWW.JEEBOOKS.IN Page # 60

4. Refraction of Light vacuum.   speed of light in vacuum  c . speed of light in medium v 4.1 Laws of Refraction (at any Refracting Surface) Sini (b) = Constant for any pair of media and for light of a given Sinr wave length. This is known as Snell's Law. More precisely, w.jeebooks Sin i = n2 = v1 = 1 Sin r n1 v2 2 4.2 Deviation of a Ray Due to Refraction Deviation () of ray incident at i and refracted at r is given by  = |i r|. 5. Principle of Reversibility of Light Rays A ray travelling along the path of the reflected ray is reflected along the path of the incident ray. A refracted ray reversed to travel back along its path will get refracted along the path of the incident ray. Thus the incident and refracted rays are mutually reversible. 7. Apparent Depth and shift of Submerged Object At near normal incidence (small angle of incidence i) apparent depth (d) is given by: d= d  n = ni(R.I.of medium of incidence ) nrelative relative nr (R.I.of medium of refraction ) Apparent shift = d  1  1 nrel  Refraction through a Composite Slab (or Refraction through a number of parallel media, as seen from a medium of R. I. n ) 0 Apparent depth (distance of final image from final surface) = t1 + t2 + t3 +......... + tn n1 rel n2 rel n3 rel n n rel WWW.JEEBOOKS.IN Page # 61

Apparent shift = t  1  + t  1  +........+  n  1 1  1  1 nn rel   n 1rel  2  n 2rel    8. Critical Angle and Total Internal Reflection ( T. I. R.)w.jeebooks C = sin 1 nr nd (i) Conditions of T. I. R. (a) light is incident on the interface from denser medium. (b) Angle of incidence should be greater than the critical angle (i > c). 9. Refraction Through Prism 9.1 Characteristics of a prism  = (i + e)  (r + r) and r +r =A 1 2 12  = i + e  A. 9.2 Variation of  versus i WWW.JEEBOOKS.IN Page # 62

(1) There is one and only one angle of incidence for which the angle of deviation is minimum. (2) When  = min , the angle of minimum deviation, then i = e and r1 = r2, the ray passes symmetrically w.r.t. the refracting surfaces. w.jeebooks We can show by simple calculation that min = 2imin – A where i = angle of incidence for minimum deviation and r = A/2. min nrel   =sin A m , where nrel = nprism sin 2 nsurroundings A 2 Alsomin = (n  1) A (for small values of  A) (3) For a thin prism ( A 10o) and for small value of i, all values of  = ( nrel  1 ) A where nrel = nprism nsurrounding 10. Dispersion Of Light The angular splitting of a ray of white light into a number of components and spreading in different directions is called Dispersion of Light. This phenomenon is because waves of different wavelength move with same speed in vacuum but with different speeds in a medium. The refractive index of a medium depends slightly on wavelength also. This variation of refractive index with wavelength is given by Cauchy’s formula. Cauchy's formula n () = a  b where a and b are positive constants 2 of a medium. Anglebetween theraysof theextreme coloursin therefracted (dispersed) light is called angle of dispersion. For prism of small ‘A’ and with small ‘i’ :  = (n – n )A v r Deviation of beam(also called mean deviation)  = y = (ny – 1)A Dispersive power () of the medium of the material of prism is given by: nv  nr  = ny 1 For small angled prism ( A 10o ) with light incident at small angle i : nv  nr = v  r  ny 1 y = y angular dispersion = deviation of mean ray (yellow) WWW.JEEBOOKS.IN Page # 63

[ n = nv  nr if n is not given in the problem ]y y2  = v  r nv  nr nv  nr y ny  1 2 w.jeebooks = [take n= if value of n is not given in y y the problem] nv, nr and ny are R. I. of material for violet, red and yellow colours respectively. 11. Combination of Two Prisms Two or more prisms can be combined in various ways to get different combination of angular dispersion and deviation. (a) Direct Vision Combination (dispersion without deviation) The condition for direct vision combination is : n v nr   n v  n r  2 1  2 1         A A  ny 1 A = ny 1 A (b) Achromatic Combination (deviation without dispersion.) Condition for achromatic combination is: (nv nr) A = (nv  nr) A 12. Refraction at Spherical Surfaces For paraxial rays incident on a spherical surface separating two media: n2  n1 = n 2  n 1 vu R where light moves from the medium of refractive index n1 to the medium of refractive index n2. Transverse magnification (m) (of dimension perpendicular to principal axis) due to refraction at spherical surface is given by m= v R =  v /n2  uR u / n1 13. Refraction at Spherical Thin Lens A thin lens is called convex if it is thicker at the middle and it is called concave if it is thicker at the ends. For a spherical, thin lens having the same medium on both sides: 11  1 1 nlens vu = (nrel  1)    where nrel = nmedium  R1 R2  WWW.JEEBOOKS.IN Page # 64

1 = (nrel  1) 1  1 f    R1 R2  w.jeebooks11 1 v  u = f  Lens Maker's Formula v m= u Combination Of Lenses: 1  1  1  1 ... F f1 f2 f3 OPTICAL INSTRUMENT SIMPLE MICROSCOPE D  Magnifying power : U0  when image is formed at infinity M  D f  When change is formed at near print D. MD  1 D f COMPOUND MICROSCOPE Length of Microscope Magnifying power L = V0 + Ue M  V0D0 U0Ue M  V0D L = V0 + fe U0fe MD  V0  D  LD = V0  D.fe U0 1 fe  D  fe   WWW.JEEBOOKS.IN Page # 65

Astronomical Telescope Length of Microscope Magnifying power L = f + ue. M= f0 w.jeebookse M  f0 L = f0 + fe fe MD  f0 1 fe  L= f+ Dfe fe D D 0 D  fe Terrestrial Telescope Length of Microscope Magnifying power L= f + 4f + U . M  f0 0e Ue M  f0 L = f + 4f + f . fe 0e MD  f0 1 fe  LD = f0 + 4f + Dfe fe D D  fe Galilean Telescope Magnifying power Length of Microscope M  f0 L=f -U. Ue 0e M  f0 L =f -f . fe 0e MD  f0 1 – fe  LD = f0 – feD fe d D – fe Resolving Power Microscope R 1 2 sin   d  Telescope. R 1a   1.22 WWW.JEEBOOKS.IN Page # 66

MODERN PHYSICS  Work function is minimum for cesium (1.9 eV) hc  work function W = h0 = 0  Photoelectric current is directly proportional to intensity of incident radiation. ( – constant) w.jeebooks  Photoelectrons ejected from metal have kinetic energies ranging from 0 to KEmax Here KEmax = eVs Vs - stopping potential  Stopping potential is independent of intensity of light used (-constant)  Intensity in the terms of electric field is I= 1 0 E2.c 2 h  Momentum of one photon is  .  Einstein equation for photoelectric effect is h = w0 + kmax  hc hc  = 0 + eVs 12400  Energy E = (A 0 ) eV  Force due to radiation (Photon) (no transmission) When light is incident perpendicularly (a) a = 1 r = 0 A  F = c , Pressure = c (b) r = 1, a = 0 2A 2 F= c , P= c (c) when 0 < r < 1 and a + r = 1 A  F = c (1 + r), P = c (1 + r) WWW.JEEBOOKS.IN Page # 67

When light is incident at an angle  with vertical. (a) a = 1, r = 0 A cos  Fcos   F= c , P = A = c cos2  (b) r = 1, a = 0 w.jeebooks 2A cos2  2 cos2  F= , P= c c (c) 0 < r < 1, a+r=1  cos2  P = (1 + r) c  De Broglie wavelength = hh h = = mv P 2mKE  Radius and speed of electron in hydrogen like atoms. rn = n2 a0 a0 = 0.529 Å Z Z v0 = 2.19 x 106 m/s vn = n v0  Energy in nth orbit Z2 E1 = – 13.6 eV En = E1 . n2  Wavelength corresponding to spectral lines 1 1  1    =R  n12 n22  for Lyman series n1 = 1 n2 = 2, 3, 4........... Balmer n1 = 2 n2 = 3, 4, 5........... Paschen n1 = 3 n2 = 4, 5, 6...........  The lyman series is an ultraviolet and Paschen, Brackett and Pfund series are in the infrared region.  Total number of possible transitions, is n(n  1) , (from nth state) 2  If effect of nucleus motion is considered, n2 m rn = (0.529 Å) Z .  Z2  En = (–13.6 eV) . n2 m WWW.JEEBOOKS.IN Page # 68

Here µ - reduced mass Mm µ = (M  m) , M - mass of nucleus  Minimum wavelength for x-rays w.jeebooks min = hc = 12400 Å eV0 V0 (v olt)  Moseley’s Law v = a(z – b) a and b are positive constants for one type of x-rays (independent of Z)  Average radius of nucleus may be written as R = R0A1/3, R0 = 1.1 x 10–15 M A - mass number  Binding energy of nucleus of mass M, is given by B = (ZMp + NMN – M)C2  Alpha - decay process A X  A4 Y  4 He Z z2 2 Q-value is       Q = A A4 4 C2 m Z X m z2 Y m 2 He  Beta- minus decay A X  A Y     Z z1 Q- value = [ m ( A X)  m ( A Y )] c 2 z Z1  Beta plus-decay A X  ZA1 Y + + +  z Q- value = [ m ( A X)  m ( A Y )  2me]c 2 z Z1  Electron capture : when atomic electron is captured, X-rays are emitted. A X + e  ZA1Y +  z Q - value = [ m ( A X)  m ( ZA1 Y )] c 2 z  In radioactive decay, number of nuclei at instant t is given by N = N0 e–t , -decay constant.  Activity of sample : A = A0 e–t  Activity per unit mass is called specific activity. 0.693  Half life : T1/2 =   Average life : Tav = T1/ 2 0.693 WWW.JEEBOOKS.IN Page # 69

 A radioactive nucleus can decay by two different processes having half lives t1 and t2 respectively. Effective half-life of nucleus is given by 1 1  1. t t1 t2 w.jeebooks WAVE OPTICS Interference of waves of intensity 1 and 2 : resultant intensity,  = 1 + 2 + 2 12 cos () where,  = phase difference.  max = For Constructive Interference : 2 1  2  min = For Destructive interference : 2 1  2 If sources are incoherent  = 1 + 2 , at each point. YDSE : Path difference, p = SP – SP = d sin  2 1 if d < < D = dy D if y << D for maxima, p = n  y = n n = 0, ±1, ±2 ....... for minima p = p = (2n  1)  n  1, 2, 3.............    1) 2 n  -1, - 2, - 3........ (2n  n  1, 2, 3............. 2 n  -1, - 2, - 3....... (2n  1)  2 y =  (2n   1) 2 D where, fringe width  = d Here,  = wavelength in medium. Highest order maxima : d nmax =    total number of maxima = 2nmax + 1 Highest order minima : nmax =  d  1   2  total number of minima = 2nmax. WWW.JEEBOOKS.IN Page # 70

Intensity on screen :  = 1 + 2 + 2 1 2 cos () where,  = 2 p     Ifw.jeebooks1 = 2, = 41 cos2    2  YDSE with two wavelengths 1 & 2 : The nearest point to central maxima where the bright fringes coincide: y = n11 = n22 = Lcm of 1 and 2 The nearest point to central maxima where the two dark fringes coincide, 11 y = (n1 – 2 ) 1 = n2 – 2 ) 2 Optical path difference popt = p 2 2  =  p =  vacuum popt. DB  = ( – 1) t. d = ( – 1)t  . YDSE WITH OBLIQUE INCIDENCE In YDSE, ray is incident on the slit at an inclination of 0 to the axis of symmetry of the experimental set-up S1 2 P1 1 dsin0 O 0 P2 S2 O' B0 We obtain central maxima at a point where, p = 0. or 2 = 0. This corresponds to the point O’ in the diagram. Hence we have path difference. d(sin 0  sin )  for points above O d(sin p = d(sin 0  sin )  for points between O & O' ... (8.1)   sin 0 )  for points below O' Page # 71 WWW.JEEBOOKS.IN

THIN-FILM INTERFERENCE 2d for interference in reflected light n for destructive interference w.jeebooks = (n  1) 2 for constructive interference for interference in transmitted light 2d n for constructive interference  for destructive interference = (n  1 ) 2 Polarisation    = tan .(brewster's angle)  + r = 90°(reflected and refracted rays are mutually perpendicular.)  Law of Malus. I = I cos2 0 I = KA2 cos2  Optical activity      tC LC  = rotation in length L at concentration C. Diffraction where m = 1, 2, 3 ......  a sin  = (2m + 1) /2 for maxima.  sin  = m , m =  1,  2,  3......... for minima. a  2d Linear width of central maxima = a 2  Angular width of central maxima = a WWW.JEEBOOKS.IN Page # 72

   0  sin  / 22 where  = a sin     /2   w.jeebooks Resolving power . R=   2 – 1  where ,   1  2 ,  = 2 - 1 2 GRAVITATION GRAVITATION : Universal Law of Gravitation F m1 m2 or F = G m1 m2 r2 r2 where G = 6.67 × 10–11 Nm2 kg–2 is the universal gravitational constant. Newton's Law of Gravitation in vector form :  = Gm1m2 rˆ12 &  = Gm1m2 F12 r2 F2 1 r2 Now rˆ12  rˆ21 , Thus  G m1 m2 rˆ12 . F21  r2  Comparing above, we get F12  F21 F GM Gravitational Field E = m = r2 Gravitational potential : gravitational potential, GM dV V=– r . E = – dr . 1.  GM & E= GMr rˆ Ring. V = x or (a2  r 2 )1/ 2 (a2  r 2 )3 / 2 GM cos  or E = – x2 WWW.JEEBOOKS.IN Page # 73

Gravitational field is maximum at a distance, r = ± a 2 and it is – 2GM 3 3 a2 2. Thin Circular Disc. w.jeebooks  V =  1   2GM 2  r 2GM r 2GM a2  a2  r2 a2 1 r2  a2  a2 1  cos     & E=–  =– 1 3. Non conducting solid sphere 2  (a) Point P inside the sphere. r < a, then V=  GM (3a2  r 2 ) &E=– GMr 3GM and E = 0 2a3 a3 , and at the centre V = – 2a (b) Point P outside the sphere . r > a, then V =  GM GM r & E = – r2 4. Uniform Thin Spherical Shell / Conducting solid sphere (a) Point P Inside the shell.  GM E=0 r < a , then V = a & (b) Point P outside shell.  GM GM r > a, then V = r & E = – r2 VARIATION OF ACCELERATION DUE TO GRAVITY : 1. Effect of Altitude GMe = g 1 h 2 ~ g 1 2h  when h << R. Re Re gh = Re  h2 2. Effect of depth gd = g 1  d  Re 3. Effect of the surface of Earth The equatorial radius is about 21 km longer than its polar radius. We know, g = GMe Hence gpole > g .equator R 2 e SATELLITE VELOCITY (OR ORBITAL VELOCITY) 11 2  GMe  2  g R e  2   v0 =  Re  h =  Re  h   WWW.JEEBOOKS.IN Page # 74

When h << Re then v0 = gRe  v0 = 9.8  6.4  106 = 7.92 × 103 ms–1 = 7.92 km s1 Time period of Satellite w.jeebooks 1 T= 2Re  h = 2  Re  h3 2   1 Re  g   RgeRe2h 2   Energy of a Satellite U = GMem K.E. = GMem ; then total energy E = – GM em r 2r 2R e Kepler's Laws Law of area : The line joining the sun and a planet sweeps out equal areas in equal intervals of time. = 1 r (rd) =7 1 r2 d = constant . area swept 2 2 dt Areal velocity = dt time 1 T2 Hence r2  = constant. Law of periods : R3 = constant 2 FLUID MECHANICS & PROPERTIES OF MATTER FLUIDS, SURFACE TENSION, VISCOSITY & ELASTICITY : 1. Hydraulic press. p = f  F or F  A  f . aA a Hydrostatic Paradox P = P = P ABC (i) Liquid placed in elevator : When elevator accelerates upward with acceleration a0 then pressure in the fluid, at depth ‘h’ may be given by, p = h [g + a0] and force of buoyancy, B = m (g + a0) (ii) Free surface of liquid in horizontal acceleration : tan  = a0 g WWW.JEEBOOKS.IN Page # 75

p1 – p2 =  a0 where p1 and p2 are pressures at points 1 & 2. Then h1 – h2 = a0 g w.jeebooks (iii) Free surface of liquid in case of rotating cylinder. v2 2r2 h = 2g = 2g Equation of Continuity a1v1 = a2v2 In general av = constant . Bernoulli’s Theorem P1 i.e.  + 2 v2 + gh = constant. 2gh (vi) Torricelli’s theorem – (speed of efflux) v= 1 A 2 ,A2 = area of hole 2 A12 A1 = area of vessel. ELASTICITY & VISCOSITY : stress = restoringforce  F area of the body A change in configuration Strain,  = original configuration L (i) Longitudinal strain = L V (ii) v = volume strain = V x (iii) Shear Strain : tan  or  =  F / A FL 1. Young's modulus of elasticity Y=  L / L AL 11 Potential Energy per unit volume = (stress × strain) = (Y × strain2 ) 22 Inter-Atomic Force-Constant k = Yr0. WWW.JEEBOOKS.IN Page # 76

Newton’s Law of viscosity, dv dv F  A dx or F = – A dx 2 r 2(  )g w.jeebooksStoke’s Law F = 6 r v.Terminal velocity = 9 SURFACE TENSION  Surface tension(T) = Total force on either of the imaginary line (F) ; Length of the line () W T=S= A Thus, surface tension is numerically equal to surface energy or work done per unit increase surface area. Inside a bubble : 4T (p – pa) = r = pexcess ; Inside the drop : 2T (p – pa) = r = pexcess 2T Inside air bubble in a liquid :(p – p ) = = p ra excess Capillary Rise 2T cos  h= rg SOUND WAVES (i) Longitudinal displacement of sound wave  = A sin (t – kx) (ii) Pressure excess during travelling sound wave P=  B  (it is true for travelling ex x = (BAk) cos(t – kx) wave as well as standing waves) Amplitude of pressure excess = BAk E (iii) Speed of sound C =  Where E = Ellastic modulus for the medium  = density of medium – for solid Y C=  WWW.JEEBOOKS.IN Page # 77

where Y = young's modulus for the solid – for liquid B C=  w.jeebooks where B = Bulk modulus for the liquid – for gases C= B P   RT   M0 where M0 is molecular wt. of the gas in (kg/mole) Intensity of sound wave : <> = 22f2A2v = Pm2 <>  P2 2v m (iv) Loudness of sound : L = 10 log10    dB 0 where I0 = 10–12 W/m2 (This the minimum intensity human ears can listen) P Intensity at a distance r from a point source =   4r2 Interference of Sound Wave if P = p sin (t – kx + 1) 1 m1 1 P2 = pm2 sin (t – kx2 + 2) resultant excess pressure at point O is p=P +P 12 p = p0 sin (t – kx + ) p0 = p 2  pm2 2  2pm1pm2 cos  m1 where  = [k (x2 – x1) + (1 – 2)] and I = I + I + 2 1 2 1 2 (i) For constructive interference  = 2n and  p0 = pm1 + pm2 (constructive interference) (ii) For destructive interfrence  = (2n+ 1) and  p0 = | pm1 – pm2 | (destructive interference) 2 If  is due to path difference only then  =  x. Condition for constructive interference : x = n Condition for destructive interference : x = (2n + 1)   2 WWW.JEEBOOKS.IN Page # 78

(a) If p = p and  m1 m2 resultant p = 0 i.e. no sound (b) If pp0m1==2ppmm2&anI0d=4=I1 0 , 2, 4, ... w.jeebooks p0 = 2pm1 Close organ pipe : f= v , 3v , 5v ,.......... (2n  1)v n = overtone 4 4 4 4 Open organ pipe : f= v , 2v , 3v ,.......... nV 2 2 2 2 Beats : Beatsfrequency = |f – f |. 12 Doppler’s Effect The observed frequency, f = f  v  v 0  v  v s and Apparent wavelength =  v  vs   v    ELECTRO MAGNETIC WAVES Maxwell's equations E  dA  Q / 0 (Gauss's Law for electricity) B  dA  0 (Gauss's Law for magnetism) (Faraday's Law) E  d  –dB (Ampere-Maxwell Law) dt  B  d  0ic  0 0 d E dt Oscillating electric and magnetic fields E= Ex(t) = E0 sin (kz - t) = E0 sin 2 z – v t  = E0 sin 2 z – t       T  E0/B0 = c c = 1/ 00 c is speed of light in vaccum v  1/  v is speed of light in medium WWW.JEEBOOKS.IN Page # 79

p  U energy transferred to a surface in time t is U, the magnitude of the tcotal momentum delivered to this surface (for complete absorption) is p Electromagnetic spectrum w.jeebooks Type Wavelength Production Detection Radio range > 0.1m Rapid acceleration and Receiver's aerials Microwave decelerations of electrons in Infra-red 0.1m to 1mm aerials Point contact diodes 1mm to 700nm Klystron value or magnetron Light value Thermopiles Bolometer, 700nm to Vibration of atoms and Infrared photographic Ultraviolet 400nm molecules film The eye, photocells, X-rays 400nm to 1nm Electrons in atoms emit light Photographic film Gamma when they move from one 1nm to 10–3 nm energy level to a lower photocells photographic rays < 10–3nm energy film Inner shell electrons in atoms moving from one Photograpic film, Geiger energy level to a lower level tubes, lonisation chamber X-ray tubes or inner shell do electrons Radioactive decay of the nucleus ERROR AND MEASUREMENT 1. Least Count mm.scale Vernier Screw gauge Stop Watch Temp thermometer L.C =1mm L.C=0.1mm L.C=0.1mm L.C=0.1Sec L.C=0.1°C 2. Significant Figures  Non-zero digits are significant  Zeros occurring between two non-zeros digits are significant.  Change of units cannot change S.F.  In the number less than one, all zeros after decimal point and to the left of first non-zero digit are insignificant  The terminal or trailing zeros in a number without a decimal point are not significant. WWW.JEEBOOKS.IN Page # 80

3. Permissible Error  Max permissible error in a measured quantity = least count of the measuring instrument and if nothing is given about least count then Max permissible error = place value of the last number  f (x,y) = x + y then (f)max = max of (  X  Y) w.jeebooks f (x,y,z) = (constant) xa yb zc then  f   f max = max of   a x  b y  c z  x y z 4. Errors in averaging  Absolute Error an = |a -a | mean n  Mean Absolute Error amean =  n | ai |  n  i1  amean  Relative error = amean amean  Percentage error = amean ×100 5. Experiments  Reading of screw gauge Thicknes of object  Readingof screw gauge  main    crseciraacdlueilnagr Lcoeuanstt  scale   reading   pitch least count of screw gauge = No.of circular scale division  Vernier callipers Thicknes of object  Re adingof vernier calliper  main   rsveceaardnleiinegr Lcoeuanstt  scale  reading Least count of vernier calliper = 1 MSD –1 VSD WWW.JEEBOOKS.IN Page # 81

PRINCIPLE OF COMMUNICATION Transmission from tower of height h w.jeebooks the distance to the horizon dT = 2RhT  dM = 2RhT  2RhR Amplitude Modulation  The modulated signal cm (t) can be written as cm(t) = Ac sin ct + A c cos (C - m) t – A c cos (C + m) 2 2  Modulation index ma  Change in amplitude of carrier wave  kAm Amplitude of original carrier wave Ac where k = A factor which determines the maximum change in the amplitude for a given amplitude Em of the modulating. If k = 1 then ma = Am  Amax – Amin Ac Amax – Amin  If a carrier wave is modulated by several sine waves the total modulated index mt is given by mt = m12  m22  m32  .........  Side band frequencies (f + f ) = Upper side band (USB) frequency cm (fc - fm) = Lower side band (LBS) frequency  Band width = (fc + f) - (f - f) = 2f m c m m  Power in AM waves : P  Vr2ms R (i) carrier power Pc   Ac 2  Ac2  2  2R   R WWW.JEEBOOKS.IN Page # 82

 maAc 2  maAc   (ii) Total power of side bands P =  2 2    2 2   ma2 A 2 sb c w.jeebooks R 2R 4R (iii) Total power of AM wave PTotal = Pc + Pab = Ac2  1  ma2  2R  2  (iv) Pt   ma2  and Psb  ma2 / 2 Pc 1 2  Pt  ma2  1 2  (v) Maximum power in the AM (without distortion) will occur when ma = 1 i.e., Pt = 1.5 P = 3Pab (vi) If I = Unmodulated current and I = total or modulated current ct Pt  2t t   ma2   Pc c2  c 1 2  Frequency Modulation  Frequency deviation  = = (fmax - f) = f - f = k . Em c c min f 2  Carrier swing (CS) = CS = 2 × f  Frequency modulation index (m ) f =. m =   fmax – fc  fc – fmin  kf Em f fm fm fm fm  Frequency spectrum = FM side band modulated signal consist of infi- nite number of side bands whose frequencies are (fc ± fm), (fc ± 2fm), (f ± 3f )......... cm  Deviation ratio = (f )max (fm )max  Percent modulation , m = (f )actual (f )max WWW.JEEBOOKS.IN Page # 83

SEMICONDUCTOR Conductivity and resistivity w.jeebooks  P (– m) (–1m–1) 102 – 108 Metals 10–2 -10–6 semiconductors 10–5 -10–6 105 – 10–6 Insulators 1011 –1019 10–11 – 10–19 Charge concentration and current [ n = e] In case of intrinsic semiconductors  P type n >> e  i = ie + ih  e n = i2  Number of electrons reaching from valence bond to conduction bond. = A T3/ 2e–Eg/2kT (A is positive constant)   = e ( e m + n n) e for  hype n = Na >> e. for  – type e = Na >> h V  Dynamic Resistance of P-N junction in forward biasing =  Transistor  CB amplifier Samll change incollector current (ic ) (i) ac current gain c = Samll change incollector current (ie ) (ii) dc current gain dc = Collector current (ic ) value of dc lies Emitter current (ie ) between 0.95 to 0.99 Change inoutput voltage(V0 ) (iii) Voltage gain AV = Change in input voltage(Vf ) AV = aac × Resistance gain Change inoutput power (P0 ) (iv) Power gain = Change in input voltage(PC )  Power gain = a2ac × Resistance gain (v) Phase difference (between output and input) : same phase (vi) Application : For High frequency WWW.JEEBOOKS.IN Page # 84

CE Amplifier (i) ac current gain ac =  ic  VCE = constant ib w.jeebooks (ii) dc current gain dc = ic ib (iii) Voltage gain : AV = V0 = ac × Resistance gain Vi (iv) Power gain = P0 = 2ac × Resistance Pi (v) Transconductance (g ) : The ratio of the change in collector in m collector current to the change in emitter base voltage is called trans ic A V conductance i.e. gm = VEB . Also gm = RL RL = Load resistance.  Relation between  and  :   or   1–  = 1  (v) Transconductance (gm) : The ratio of the change in collector in collec- tor current to the change in emitter base voltage is called trans conductance i.e. gm = . Also gm = RL = Load resistance. WWW.JEEBOOKS.IN Page # 85