General physics formula sheet, Cheat Sheet of Physics

Download General physics formula sheet and more Cheat Sheet Physics in PDF only on Docsity! Page # 1 PHYSICS FORMULA BOOKLET - GYAAN SUTRA INDEX S.No. Topic Page No. 1. Unit and Dimension 2 2. Rectilinear Motion 3 – 4 3. Projectile Motion & Vector 5 – 5 4. Relavitve Motion 5 – 7 5. Newton’s Laws of Motion 7 – 9 6. Friction 9 – 9 7. Work, Power & Energy 10 – 11 8. Circular Motion 11 – 14 9. Centre of Mass 14 – 18 10. Rigid Body Dynamics 18 – 25 11. Simple Harmonic Motion 26 – 28 12. Sting Wave 29 – 31 13. Heat & Thermodynamics 31 – 37 14. Electrostatics 37 – 40 15. Current Electricity 41 – 47 16. Capacitance 47 – 51 17. Alternating Current 52 – 54 18. Magnetic Effect of Current & Magnetic force on charge 54 – 56 19. Electromagnetic Induction 56 – 59 20. Geometrical Optics 59 – 66 21. Modern Physics 67 – 70 22. Wave Optics 70 – 73 23. Gravitation 73 – 75 24. Fluid Mechanics & Properties of Matter 75 – 77 25. Sound Wave 77 – 79 26. Electro Magnetic Waves 79 – 80 27. Error and Measurement 80 – 81 28. Principle of Communication 82 – 83 29. Semiconductor 84 – 85 Page # 2 PHYSICS FORMULA BOOKLET - GYAAN SUTRAA UNIT AND DIMENSIONS Unit : Measurement of any physical quantity is expressed in terms of an internationally accepted certain basic standard called unit. * Fundamental Units. S.No. Physical Quantity SI Unit Symbol 1 Length Metre m 2 Mass Kilogram Kg 3 Time Second S 4 Electric Current Ampere A 5 Temperature Kelvin K 6 Luminous Intensity Candela Cd 7 Amount of Substance Mole mol * Supplementary Units : S.No. Physical Quantity SI Unit Symbol 1 Plane Angle radian r 2 Solid Angle Steradian Sr * Metric Prefixes : S.No. Prefix Sym bol Value 1 Centi c 10–2 2 M ili m 10–3 3 M icro µ 10–6 4 Nano n 10–9 5 Pico p 10–12 6 K ilo K 103 7 M ega M 106 Page # 5 PROJECTILE MOTION & VECTORS Time of flight : T = g sinu2  Horizontal range : R = g 2sinu2  Maximum height : H = g2 sinu 22  Trajectory equation (equation of path) : y = x tan  – 22 2 cosu2 gx = x tan  (1 – R x ) Projection on an inclined plane y x   Up the Incline Down the Incline Range   2 2 cosg )cos(sinu2   2 2 cosg )cos(sinu2 Time of flight   cosg sinu2   cosg sinu2 Angle of projection with incline plane for maximum range 24    24    Maximum Range )sin1(g u2  )sin1(g u2  RELATIVE MOTION BAAB vv)BtorespectwithAofvelocity(v   BAAB aa)BtorespectwithAofonaccelerati(a   Relative motion along straight line - ABBA xxx   Page # 6 CROSSING RIVER A boat or man in a river always moves in the direction of resultant velocity of velocity of boat (or man) and velocity of river flow. 1. Shortest Time : Velocity along the river, vx = vR. Velocity perpendicular to the river, vf = vmR The net speed is given by vm = 2 R 2 mR vv  2. Shortest Path : velocity along the river, vx = 0 and velocity perpendicular to river vy = 2 R 2 mR vv  The net speed is given by vm = 2 R 2 mR vv  at an angle of 90º with the river direction. velocity vy is used only to cross the river, Page # 7 therefore time to cross the river, t = yv d = 2 R 2 mR vv d  and velocity vx is zero, therefore, in this case the drift should be zero.  vR – vmR sin  = 0 or vR = vmR sin  or  = sin–1       mR R v v RAIN PROBLEMS Rmv  = Rv  – mv  or vRm = 2 m 2 R vv  NEWTON'S LAWS OF MOTION 1. From third law of motion BAAB FF  ABF = Force on A due to B BAF = Force on B due to A 2. From second law of motion Fx = dt dPx = max Fy = dt dPy = may Fz = dt dPz = maz 5. WEIGHING MACHINE : A weighing machine does not measure the weight but measures the force exerted by object on its upper surface. 6. SPRING FORCE xkF   x is displacement of the free end from its natural length or deformation of the spring where K = spring constant. 7. SPRING PROPERTY K ×  = constant = Natural length of spring. 8. If spring is cut into two in the ratio m : n then spring constant is given by 1 = nm m   ; 2 = nm .n   k = k11 = k22 Page # 10 WORK, POWER & ENERGY WORK DONE BY CONSTANT FORCE : W = F  . S  WORK DONE BY MULTIPLE FORCES F  = F  1 + F  2 + F  3 + ..... W = [F  ] . S  ...(i) W = F  1 . S  + F  2 . S  + F  3 . S  + ..... or W = W1 + W2 + W3 + .......... WORK DONE BY A VARIABLE FORCE dW =   F.ds RELATION BETWEEN MOMENTUM AND KINETIC ENERGY K = m2 p2 and P = Km2 ; P = linear momentum POTENTIAL ENERGY   2 1 2 1 r r U U rdFdU  i.e., 2 1 r 2 1 r U U F dr W        WrdFU r    CONSERVATIVE FORCES F= – r U   WORK-ENERGY THEOREM WC + WNC + WPS = K Modified Form of Work-Energy Theorem WC = U WNC + WPS = K + U WNC + WPS = E Page # 11 POWER The average power ( P or pav) delivered by an agent is given by P or pav = t W dt SdFP    = dt SdF    = F  . v  CIRCULAR MOTION 1. Average angular velocity  av = 12 12 tt   = t  2. Instantaneous angular velocity   = dt d   3. Average angular acceleration  av = 12 12 tt   = t   4. Instantaneous angular acceleration   = dt d =    d d 5. Relation between speed and angular velocity  v = r and rv   7. Tangential acceleration (rate of change of speed)  at = dt dV = r dt d =  dt dr 8. Radial or normal or centripetal acceleration  ar = r v2 = 2r 9. Total acceleration  rt aaa    a = (at 2 + ar 2)1/2 ra ca a ta v or  P O Where ra t   and var   Page # 12 10. Angular acceleration    = dt d  (Non-uniform circular motion)  ACW Rotation 12. Radius of curvature R = a v2 = F mv2 If y is a function of x. i.e. y = f(x)  R = 2 2 2/32 dx yd dx dy1               13. Normal reaction of road on a concave bridge  N = mg cos  + r mv2   N V mg mgcos concave bridge O 14. Normal reaction on a convex bridge  N = mg cos  – r mv2  N V mg mgcos convex bridge O  15. Skidding of vehicle on a level road  vsafe  gr 16. Skidding of an object on a rotating platform  max = r/g Page # 15 =     n 1i i n 1i ii m rm  cmr  = M 1   n 1i ii rm  CENTRE OF MASS OF A CONTINUOUS MASS DISTRIBUTION xcm =   dm dmx , ycm =   dm dmy , zcm =   dm dmz  dm = M (mass of the body) CENTRE OF MASS OF SOME COMMON SYSTEMS  A system of two point masses m1 r1 = m2 r2 The centre of mass lies closer to the heavier mass.  Rectangular plate (By symmetry) xc = 2 b yc = 2 L Page # 16  A triangular plate (By qualitative argument) at the centroid : yc = 3 h  A semi-circular ring yc =  R2 xc = O  A semi-circular disc yc = 3 R4 xc = O  A hemispherical shell yc = 2 R xc = O  A solid hemisphere yc = 8 R3 xc = O  A circular cone (solid) yc = 4 h  A circular cone (hollow) yc = 3 h Page # 17 MOTION OF CENTRE OF MASS AND CONSERVATION OF MOMENTUM: Velocity of centre of mass of system cmv  = M dt drm.............. dt drm dt drm dt drm n n 3 3 2 2 1 1  = M vm..........vmvmvm nn332211   SystemP = M cmv Acceleration of centre of mass of system cma  = M dt dvm.............. dt dvm dt dvm dt dvm n n 3 3 2 2 1 1  = M am..........amamam nn332211   = M systemonforceNet = M ForceernalintNetForceExternalNet  = M ForceExternalNet extF  = M cma  IMPULSE Impulse of a force F action on a body is defined as :- J  =  f i t t Fdt PΔJ   (impulse - momentum theorem) Important points : 1. Gravitational force and spring force are always non-impulsive. 2. An impulsive force can only be balanced by another impulsive force. COEFFICIENT OF RESTITUTION (e) e = ndeformatioofpulseIm nreformatioofpulseIm =   dtF dtF d r = s impactoflinealongapproachofVelocity impactoflinealongseparationofVelocity Page # 20 2.4 For a larger object :  =  elementd where d = moment of inertia of a small element 3. TWO IMPORTANT THEOREMS ON MOMENT OF INERTIA : 3.1 Perpendicular Axis Theorem [Only applicable to plane lamina (that means for 2-D objects only)]. z = x + y (when object is in x-y plane). 3.2 Parallel Axis Theorem (Applicable to any type of object):  = cm + Md2 List of some useful formula : Object Moment of Inertia 2MR 5 2 (Uniform) Solid Sphere 2MR 3 2 (Uniform) Hollow Sphere MR2 (Uniform or Non Uniform) Page # 21 Ring. 2 MR2 (Uniform) Disc MR2 (Uniform or Non Uniform) Hollow cylinder 2 MR2 (Uniform) Solid cylinder 3 ML2 (Uniform) 12 ML2 (Uniform) Page # 22 3 m2 2 (Uniform) AB = CD = EF = 12 Ma2 (Uniform) Square Plate 6 Ma2 (Uniform) Square Plate  = 12 )ba(M 22  (Uniform) Rectangular Plate 12 )ba(M 22  (Uniform) Cuboid Page # 25 7.6 Impulse of Torque :   Jdt J  Change in angular momentum. For a rigid body, the distance between the particles remain unchanged during its motion i.e. rP/Q = constant For velocities Q P r r  with respect to Q Q P r wr  with respect to ground VQ VQ     cosrV2rVV Q 22 QP For acceleration : , ,  are same about every point of the body (or any other point outside which is rigidly attached to the body). Dynamics :   cmcm , cmext aMF   cmsystem vMP   , Total K.E. = 2cmMv 2 1 + 2 cm2 1  Angular momentum axis AB = L  about C.M. + L  of C.M. about AB cmcmcmAB vMrL   Page # 26 SIMPLE HARMONIC MOTION S.H.M. F = – kx General equation of S.H.M. is x = A sin (t + ); (t + ) is phase of the motion and  is initial phase of the motion. Angular Frequency () :  = T 2 = 2f Time period (T) : T =  2 = k m2 m k Speed : 22 xAv  Acceleration : a = 2x Kinetic Energy (KE) : 2 1 mv2 = 2 1 m2 (A2 – x2) = 2 1 k (A2 – x2) Potential Energy (PE) : 2 1 Kx2 Total Mechanical Energy (TME) = K.E. + P.E. = 2 1 k (A2 – x2) + 2 1 Kx2 = 2 1 KA2 (which is constant) SPRING-MASS SYSTEM (1) smooth surface  k m  T = 2 k m (2) T = K 2   where  = )m(m mm 21 21  known as reduced mass Page # 27 COMBINATION OF SPRINGS Series Combination : 1/keq = 1/k1 + 1/k2 Parallel combination : keq = k1 + k2 SIMPLE PENDULUM T = 2 g  = 2 .effg  (in accelerating Refer- ence Frame); geff is net acceleration due to pseudo force and gravitational force. COMPOUND PENDULUM / PHYSICAL PENDULUM Time period (T) : T = 2 mg  where,  = CM + m2 ;  is distance between point of suspension and centre of mass. TORSIONAL PENDULUM Time period (T) : T = 2 C  where, C = Torsional constant Superposition of SHM’s along the same direction x 1 = A 1 sin t & x 2 = A 2 sin (t + ) A2 A A1 If equation of resultant SHM is taken as x = A sin (t + ) A =  cosAA2AA 21 2 2 2 1 & tan  =   cosAA sinA 21 2 1. Damped Oscillation  Damping force vb–F    equation of motion is dt mdv = –kx – bv  b2 - 4mK > 0 over damping Page # 30      x)k t( sin A y x)k – t( sin A y 1rr 2tt if incident from denser to rarer medium. (v2 > v1) (d) Amplitude of reflected & transmitted waves. Ar = i 21 21 A kk kk   & At = i 21 1 A kk k2  STANDING/STATIONARY WAVES :- (b) y1 = A sin (t – kx + 1) y2 = A sin (t + kx + 2) y1 + y2 =               2 kxcosA2 12 sin         2 t 21 The quantity 2A cos         2 kx 12 represents resultant amplitude at x. At some position resultant amplitude is zero these are called nodes. At some positions resultant amplitude is 2A, these are called antin- odes. (c) Distance between successive nodes or antinodes = 2  . (d) Distance between successive nodes and antinodes = /4. (e) All the particles in same segment (portion between two successive nodes) vibrate in same phase. (f) The particles in two consecutive segments vibrate in opposite phase. (g) Since nodes are permanently at rest so energy can not be trans- mitted across these. VIBRATIONS OF STRINGS ( STANDING WAVE) (a) Fixed at both ends : 1. Fixed ends will be nodes. So waves for which L = 2  L = 2 2 L = 2 3 are possible giving L = 2 n or  = n L2 where n = 1, 2, 3, .... as v =  T fn =  T L2 n , n = no. of loops Page # 31 (b) String free at one end : 1. for fundamental mode L = 4  = or  = 4L fundamental mode First overtone L = 4 3 Hence  = 3 L4  first overtone so f1 =  T L4 3 (First overtone) Second overtone f2 =  T L4 5 so fn =            T L4 )1n2(T L2 2 1n HEAT & THERMODYNAMICS Total translational K.E. of gas = 2 1 M < V2 > = 2 3 PV = 2 3 nRT < V2 > =  P3 Vrms =  P3 = molM RT3 = m KT3 Important Points : – Vrms  T m KT8V   = 1.59 m KT Vrms = 1.73 m KT Most probable speed Vp = m KT2 = 1.41 m KT  Vrms > V > Vmp Degree of freedom : Mono atomic f = 3 Diatomic f = 5 polyatomic f = 6 Page # 32 Maxwell’s law of equipartition of energy : Total K.E. of the molecule = 1/2 f KT For an ideal gas : Internal energy U = 2 f nRT Workdone in isothermal process : W = [2.303 nRT log10 i f V V ] Internal energy in isothermal process : U = 0 Work done in isochoric process : dW = 0 Change in int. energy in isochoric process : U = n 2 f R T = heat given Isobaric process : Work done W = nR(Tf – Ti) change in int. energy U = nCv T heat given Q = U + W Specific heat : Cv = 2 f R Cp =       1 2 f R Molar heat capacity of ideal gas in terms of R : (i) for monoatomic gas : v p C C = 1.67 (ii) for diatomic gas : v p C C = 1.4 (iii) for triatomic gas : v p C C = 1.33 In general :  = v p C C =      f 21 Mayer’s eq.  Cp – Cv = R for ideal gas only Adiabatic process : Work done W = 1 )TT(nR fi   Page # 35 Refrigerator (Heat Pump) Refrigerator Hot (T1) Hot (T2) Q1 Q2 W  Coefficient of performance, W Q2 = 1– T T 1 2 1  = 1– T T 1 2 1  Calorimetry and thermal expansion Types of thermometers : (a) Liquid Thermometer : T =         0100 0   × 100 (b) Gas Thermometer : Constant volume : T =         0100 0 PP PP × 100 ; P = P0 + g h Constant Pressure : T =      VV V T0 (c) Electrical Resistance Thermometer : T =         0100 0t RR RR × 100 Thermal Expansion : (a) Linear :  = TL L 0  or L = L0 (1 + T) Page # 36 (b) Area/superficial :  = TA A 0  or A = A0 (1 + T) (c) volume/ cubical : r = TV V 0  or V = V0 (1 +  T) 32     Thermal stress of a material :    Y A F Energy stored per unit volume : E = 2 1 K(L)2 or 2)L( L AY 2 1E  Variation of time period of pendulum clocks : T = 2 1 T T’ < T - clock-fast : time-gain T’ > T - clock slow : time-loss CALORIMETRY : Specific heat S = T.m Q  Molar specific heat C = T.n Q   Water equivalent = mWSW HEAT TRANSFER Thermal Conduction : dt dQ = – KA dx dT Thermal Resistance : R = KA  Page # 37 Series and parallel combination of rod : (i) Series : eq eq K  = ....... KK 2 2 1 1   (when A1 = A2 = A3 = .........) (ii) Parallel : Keq Aeq = K1 A1 + K2 A2 + ...... (when  1 = 2 = 3 = .........) for absorption, reflection and transmission r + t + a = 1 Emissive power : E = tA U   Spectral emissive power : E = d dE Emissivity : e = temp. T atbody black a of E temp. T atbody a of E Kirchoff’s law : )body(a )body(E = E (black body) Wein’s Displacement law : m . T = b. b = 0.282 cm-k Stefan Boltzmann law : u =  T4 s = 5.67 × 10–8 W/m2 k4 u = u – u0 = e A (T4 – T0 4) Newton’s law of cooling : dt d = k ( – 0) ;  = 0 + (i – 0) e –k t ELECTROSTATICS Coulomb force between two point charges r |r| qq 4 1F 3 21 r0      = r̂ |r| qq 4 1 2 21 r0     The electric field intensity at any point is the force experienced by unit positive charge, given by 0q FE     Electric force on a charge 'q' at the position of electric field intensity E  produced by some source charges is EqF    Electric Potential Page # 40  Potential Energy of an Electric Dipole in External Electric Field: U = -   p E.  Electric Dipole in Uniform Electric Field : torque      p x E ;  F = 0  Electric Dipole in Nonuniform Electric Field: torque      p x E ; U =   Ep , Net force |F| = r Ep    Electric Potential Due to Dipole at General Point (r, ) : V = P r p r r cos .  4 40 2 0 3     The electric flux over the whole area is given by E = S dS.E  = S ndSE  Flux using Gauss's law, Flux through a closed surface E = dSE   = 0 inq  .  Electric field intensity near the conducting surface = 0  n̂  Electric pressure : Electric pressure at the surface of a conductor is given by formula P = 0 2 2  where  is the local surface charge density.  Potential difference between points A and B VB – VA = –  B A rd.E  E  =                V z k̂V x ĵV x î = – V z k̂ x ĵ x î                = – V = –grad V Page # 41 CURRENT ELECTRICITY 1. ELECTRIC CURRENT Iav = t q   and instantaneous current i =. dt dq t qLim 0t     2. ELECTRIC CURRENT IN A CONDUCTOR I = nAeV.   dv ,         2 d m eE 2 1 v =  m eE 2 1 , I = neAVd 3. CURRENT DENSITY n ds dIJ   4. ELECTRICAL RESISTANCE I = neAVd = neA       m2 eE  =          m2 ne2 AE E =  V so I =                 A m2 ne2 V =        A V = V/R  V = IR  is called resistiv ity (it is also called specific resistance) and = 2ne m2 =  1 ,  is called conductivity. Therefore current in conductors is proportional to potential difference applied across its ends. This is Ohm's Law. Units: )m(meterohm),(ohmR  also called siemens, 11m . Page # 42 Dependence of Resistance on Temperature : R = Ro (1 + ). Electric current in resistance I = R VV 12  5. ELECTRICAL POWER P = V Energy = pdt P = I2R = V = R V2 . H = Vt = 2 Rt = t R V 2 H = 2 RT Joule = 2.4 RT2 Calorie 9. KIRCHHOFF'S LAWS 9.1 Kirchhoff’s Current Law (Junction law)  in = out 9.2 Kirchhoff’s Voltage Law (Loop law) IR + EMF =0”. 10. COMBINATION OF RESISTANCES : Resistances in Series: R = R1 + R2 + R3 +................ + Rn (this means Req is greater then any resistor) ) and V = V1 + V2 + V3 +................ + Vn . V1 = V R.........RR R n21 1  ; V2 = V R.........RR R n21 2  ; 2. Resistances in Parallel : Page # 45 Application of potentiometer (a) To find emf of unknown cell and compare emf of two cells. In case , In figure (1) is joint to (2) then balance length = 1 1 = x1 ....(1) in case , In figure (3) is joint to (2) then balance length = 2 2 = x2 ....(2) 2 1 2 1      If any one of 1 or 2 is known the other can be found. If x is known then both 1 and 2 can be found (b) To find current if resistance is known VA – VC = x1 IR1 = x1  = 1 1 R x Similarly, we can find the value of R2 also. Potentiometer is ideal voltmeter because it does not draw any current from circuit, at the balance point. (c) To find the internal resistance of cell. Ist arrangement 2nd arrangement Page # 46 by first arrangement ’ = x1 ...(1) by second arrangement IR = x2  = R x 2 , also  = R'r '    R'r '   = R x 2  R'r x 1   = R x 2 r’ = R 2 21          (d)Ammeter and voltmeter can be graduated by potentiometer. (e)Ammeter and voltmeter can be calibrated by potentiometer. 18. METRE BRIDGE (USE TO MEASURE UNKNOWN RESISTANCE) If AB =  cm, then BC = (100 – ) cm. Resistance of the wire between A and B , R   [  Specific resistance  and cross-sectional area A are same for whole of the wire ] or R =  ...(1) where  is resistance per cm of wire. If P is the resistance of wire between A and B then P    P = () Similarly, if Q is resistance of the wire between B and C, then Q  100 –   Q = (100 – ) ....(2) Dividing (1) by (2), Q P =   100 Page # 47 Applying the condition for balanced Wheatstone bridge, we get R Q = P X  x = R P Q or X =  100 R Since R and  are known, therefore, the value of X can be calculated. CAPACITANCE 1. (i) q  V  q = CV q : Charge on positive plate of the capacitor C : Capacitance of capacitor. V : Potential difference between positive and negative plates. (ii) Representation of capacitor : , ( (iii) Energy stored in the capacitor : U = 2 1 CV2 = C2 Q2 = 2 QV (iv) Energy density = 2 1 r E 2 = 2 1  K E2 r = Relative permittivity of the medium. K= r : Dielectric Constant For vacuum, energy density = 2 1 E 2 (v) Types of Capacitors : (a) Parallel plate capacitor C = d Ar0 = K d A0 A : Area of plates d : distance between the plates( << size of plate ) (b) Spherical Capacitor :  Capacitance of an isolated spherical Conductor (hollow or solid ) C= 4 r R R = Radius of the spherical conductor  Capacitance of spherical capacitor C= 4 )ab( ab  1 2b a  C = )ab( abK4 20   K1 K2 K3 b a Page # 50 Time constant = CReq. I =  0q e – t /  R V e– t /  (ii) Discharging of Capacitor : q = q0 e – t /  q0 = Initial charge on the capacitor I =  0q e – t /  R C q0 0.37v0  t q 5. Capacitor with dielectric : (i) Capacitance in the presence of dielectric : C = d AK 0 = KC0 + + + + + + + + + + + + + + 0 + +  – –  V b+ – b– – – – – – – – – – –  0b C0 = Capacitance in the absence of dielectric. Page # 51 (ii) Ein = E – Eind = 0  – 0 b   = 0K  = d V E : 0  Electric field in the absence of dielectric Eind : Induced (bound) charge density. (iii) b = (1 – K 1 ). 6. Force on dielectric (i) When battery is connected d2 V)1K(bF 2 0   + – b b   d    F x (ii) When battery is not connected F = 2 2 C2 Q dx dC * Force on the dielectric will be zero when the dielectric is fully inside. Page # 52 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). 3. ROOT MEAN SQUARE VALUE: Root Mean Square Value of a function, from t1 to t2, is defined as frms = 12 2 2 1 tt dtf t t   . 4. POWER CONSUMED OR SUPPLIED IN AN AC CIRCUIT: Average power consumed in a cycle =      2 2 o Pdt = 2 1 Vm m cos  = 2 Vm . 2 m . cos  = Vrms rms cos . Here cos  is called power factor. Page # 55 6. Magnetic field on the axis of the solenoid 2 1 B = 2 nI0 (cos 1 – cos 2) 7. Ampere's Law   Id.B 0  8. Magnetic field due to long cylinderical shell B = 0, r < R = Rr, r I 2 0    9. Magnetic force acting on a moving point charge a. )B(qF   (i) B   qB mr   × × × × × × × × × × × × × × × × B r T = qB m2 (ii)   B qB sinmr   T = qB m2 Pitch = qB cosm2  b.  E)B(qF   10. Magnetic force acting on a current carrying wire  BIF     11. Magnetic Moment of a current carrying loop M = N · I · A 12. Torque acting on a loop BM   Page # 56 13. Magnetic field due to a single pole B = 2 0 r m· 4  14. Magnetic field on the axis of magnet B = 3 0 r M2· 4  15. Magnetic field on the equatorial axis of the magnet B = 3 0 r M· 4  16. Magnetic field at point P due to magnet B = 3 0 r M 4   2cos31  S P r N ELECTROMAGNETIC INDUCTION 1. Magnetic flux is mathematically defined as  =  sd.B  2. Faraday’s laws of electromagnetic induction E = – dt d 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. Page # 57 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 = 2 rB 2 5. Fixed loop in a varying magnetic field If magnetic field changes with the rate dt dB , electric field is generated whose average tangential value along a circle is given by E= dt dB 2 r This electric field is non conservative in nature. The lines of force associ- ated with this electric field are closed curves. 6. Self induction  = t IL t )LI( t )N(       . The instantaneous emf is given as  = dt LdI dt )LI(d dt )N(d    Self inductance of solenoid = µ0 n 2 r2. 6.1 Inductor It is represent by electrical equivalence of loop  BA V dt dILV  Energy stored in an inductor = 2 1 L 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 = )e1( R L Rt   Page # 60 Differentiating w.r.t time , we get v(im)x = -v(om)x ; v(im)y = v(om)y ; v(im)z = v(om)z , 3. Spherical Mirror 1 v + 1 u = 2 R = 1 f ..... Mirror formula 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= h h 2 1 m =  v u . (d) On differentiating (a) we get dv du =  v u 2 2 . (e) On dif ferentiating (a) with respect to time we get dv dt v u du dt   2 2 ,where dv dt is the velocity of image along Principal axis and du 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. (g) Optical power of a mirror (in Diopters) = f 1 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 = 12 12 uu vv   (it will always be inverted) Page # 61 4. Refraction of Light vacuum.   speed of light in vacuum speed of light in medium c v . 4.1 Laws of Refraction (at any Refracting Surface) (b) rSin iSin = Constant for any pair of media and for light of a given wave length. This is known as Snell's Law. More precisely, Sin i Sin r = n n 2 1 = v v 1 2 =   1 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= relativen d  nrelative = )refractionofmediumof.I.R(n )incidenceofmediumof.I.R(n r i Apparent shift = d        reln 11 Refraction through a Composite Slab (or Refraction through a number of parallel media, as seen from a medium of R. I. n0) Apparent depth (distance of final image from final surface) = t n rel 1 1 + t n rel 2 2 + t n rel 3 3 +......... + reln n n t Page # 62 Apparent shift = t1          rel1n 11 + t2          rel2n 11 +........+          relnn n1 8. Critical Angle and Total Internal Reflection ( T. I. R.) C = sin 1 n n r d (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)  (r1 + r2) and r1 + r2 = A  = i + e  A. 9.2 Variation of  versus i Page # 65 1 f = (nrel  1) 1 1 1 2R R        1 v  1 u = 1 f  Lens Maker's Formula m = v u Combination Of Lenses: 1 1 1 1 1 2 3F f f f    ... OPTICAL INSTRUMENT SIMPLE MICROSCOPE  Magnifying power : 0U D  when image is formed at infinity f DM   When change is formed at near print D. f D1MD  COMPOUND MICROSCOPE Magnifying power Length of Microscope e0 00 UU DV M  L = V0 + Ue e0 0 fU DV M  L = V0 + fe        e0 0 D f D1 U V M LD = e e 0 fD f.D V   Page # 66 Astronomical Telescope Magnifying power Length of Microscope M = e 0f  L = f + ue. e 0 f f M  L = f0 + fe        D f1 f fM e e 0 D LD= f0 + e e fD Df  Terrestrial Telescope Magnifying power Length of Microscope e 0 U f M  L= f0 + 4f + Ue. e 0 f f M  L = f0 + 4f + fe.        D f 1 f f M e e 0 D LD = f0 + 4f + e e fD Df  Galilean Telescope Magnifying power Length of Microscope e 0 U f M  L = f0 - Ue. e 0 f f M  L = f0 - fe.       d f–1 f fM e e 0 D LD = f0 – e e f–D Df Resolving Power Microscope      sin2 d 1R Telescope.     22.1 a1R Page # 67 MODERN PHYSICS  Work function is minimum for cesium (1.9 eV)  work function W = h0 = 0 hc   Photoelectric current is directly proportional to intensity of incident radiation. ( – constant)  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 = 2 1 0 E 2.c  Momentum of one photon is  h .  Einstein equation for photoelectric effect is h = w0 + kmax   hc = 0 hc  + eVs  Energy E = )A( 12400 0 eV  Force due to radiation (Photon) (no transmission) When light is incident perpendicularly (a) a = 1 r = 0 F = c A , Pressure = c  (b) r = 1, a = 0 F = c A2 , P = c 2 (c) when 0 < r < 1 and a + r = 1 F = c A (1 + r), P = c  (1 + r) Page # 70  A radioactive nucleus can decay by two different processes having half lives t1 and t2 respectively. Effective half-life of nucleus is given by 21 t 1 t 1 t 1  . WAVE OPTICS Interference of waves of intensity 1 and 2 : resultant intensity,  = 1 + 2 + 212  cos () where,  = phase difference. For Constructive Interference : max =  221  For Destructive interference : min =  221  If sources are incoherent  = 1 + 2 , at each point. YDSE : Path difference, p = S2P – S1P = d sin  if d < < D = D dy if y << D for maxima, p = n  y = n n = 0, ±1, ±2 ....... for minima p = p =             3........- 2,- -1,n 2 )1n2( ....3......... 2, 1, n 2 )1n2(  y =             3.......- 2,- -1,n 2 )1n2( ....3......... 2, 1, n 2 )1n2( where, fringe width  = d D Here,  = wavelength in medium. Highest order maxima : nmax =      d total number of maxima = 2nmax + 1 Highest order minima : nmax =       2 1d total number of minima = 2nmax. Page # 71 Intensity on screen :  = 1 + 2 + 212  cos () where,  = p2    If 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, y = (n1 – 2 1 ) 1 = n2 – 2 1 ) 2 Optical path difference popt = p  =  2 p = vacuum 2   popt.  = ( – 1) t. d D = ( – 1)t  B . 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 1 P1 P2 B0O' S2 dsin0 S1 O0 2 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. p =         O'below points for)sind(sin O'&O between points for)sin(sind O above points for)sin(sind 0 0 0 ... (8.1) Page # 72 THIN-FILM INTERFERENCE for interference in reflected light 2d =       ceinterferen veconstructifor) 2 1n( ceinterferen edestructivforn for interference in transmitted light 2d =       ceinterferen edestructivfor) 2 1n( ceinterferen veconstructiforn Polarisation    = tan .(brewster's angle)  + r = 90°(reflected and refracted rays are mutually perpendicular.)  Law of Malus. I = I0 cos2 I = KA2 cos2  Optical activity   CLCt       = rotation in length L at concentration C. Diffraction  a sin  = (2m + 1) /2 for maxima. where m = 1, 2, 3 ......  sin  = a m , m =  1,  2,  3......... for minima.  Linear width of central maxima = a d2   Angular width of central maxima = a 2 Page # 75 When h << Re then v0 = eRg  v0 = 6104.68.9  = 7.92 × 103 ms–1 = 7.92 km s1 Time period of Satellite T =     2 1 e 2 e e hR Rg hR2           =   2 1 3 e e g hR R 2          Energy of a Satellite U = r mGMe K.E. = r2 mGMe ; then total energy E = – e e R2 mGM Kepler's Laws Law of area : The line joining the sun and a planet sweeps out equal areas in equal intervals of time. Areal velocity = time sweptarea = dt )rd(r 2 1  =7 2 1 r2 dt d = constant . Hence 2 1 r2  = constant. Law of periods : 3 2 R T = constant FLUID MECHANICS & PROPERTIES OF MATTER FLUIDS, SURFACE TENSION, VISCOSITY & ELASTICITY : 1. Hydraulic press. p = f a AFor A F a f  . Hydrostatic Paradox PA = PB = PC (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  = g a0 Page # 76 p1 – p2 =  a0 where p1 and p2 are pressures at points 1 & 2. Then h1 – h2 = g a0 (iii) Free surface of liquid in case of rotating cylinder. h = g2 v2 = g2 r22 Equation of Continuity a1v1 = a2v2 In general av = constant . Bernoulli’s Theorem i.e.  P + 2 1 v2 + gh = constant. (vi) Torricelli’s theorem – (speed of efflux) v= 2 1 2 2 A A1 gh2  ,A2 = area of hole A1 = area of vessel. ELASTICITY & VISCOSITY : stress = A F bodytheofarea forcerestoring  Strain,  = ionconfiguratoriginal ionconfiguratinchange (i) Longitudinal strain = L L (ii) v = volume strain = V V (iii) Shear Strain : tan  or  =  x 1. Young's modulus of elasticity Y = LA FL L/L A/F    Potential Energy per unit volume = 2 1 (stress × strain) = 2 1 (Y × strain2 ) Inter-Atomic Force-Constant k = Yr0. Page # 77 Newton’s Law of viscosity, F  A dx dv or F = – A dx dv Stoke’s Law F = 6 r v. Terminal velocity = 9 2   g)(r2 SURFACE TENSION Surface tension(T) = )(linetheofLength )F(lineimaginarytheofeitheronforceTotal  ; T = S = A W Thus, surface tension is numerically equal to surface energy or work done per unit increase surface area. Inside a bubble : (p – pa) = r T4 = pexcess ; Inside the drop : (p – pa) = r T2 = pexcess Inside air bubble in a liquid :(p – pa) = r T2 = pexcess Capillary Rise h = gr cosT2   SOUND WAVES (i) Longitudinal displacement of sound wave  = A sin (t – kx) (ii) Pressure excess during travelling sound wave Pex = x B    (it is true for travelling = (BAk) cos(t – kx) wave as well as standing waves) Amplitude of pressure excess = BAk (iii) Speed of sound C =  E Where E = Ellastic modulus for the medium  = density of medium – for solid C =  Y Page # 80 c Up  energy transferred to a surface in time t is U, the magnitude of the total momentum delivered to this surface (for complete absorption) is p Electromagnetic spectrum Type Wavelength range Production Detection Radio > 0.1m Rapid acceleration and decelerations of electrons in aerials Receiver's aerials Microwave 0.1m to 1mm Klystron value or magnetron value Point contact diodes Infra-red 1mm to 700nm Vibration of atoms and molecules Thermopiles Bolometer, Infrared photographic film Light 700nm to 400nm Electrons in atoms emit light when they move from one energy level to a lower energy The eye, photocells, Photographic film Ultraviolet 400nm to 1nm Inner shell electrons in atoms moving from one energy level to a lower level photocells photographic film X-rays 1nm to 10–3 nm X-ray tubes or inner shell electrons Photograpic film, Geiger tubes, lonisation chamber Gamma rays < 10–3nm Radioactive decay of the nucleus do ERROR AND MEASUREMENT 1. Least Count mm.scale L.C =1mm Vernier L.C=0.1mm Screw gauge L.C=0.1mm Stop Watch L.C=0.1Sec Temp thermometer 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. Page # 81 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) f (x,y,z) = (constant) xa yb zc then maxf f        = max of             z zc y yb x xa 4. Errors in averaging  Absolute Error an = |amean -an|  Mean Absolute Error amean = n|a| n 1i i            Relative error = mean mean a a  Percentage error = mean mean a a ×100 5. Experiments  Reading of screw gauge                              count Least reading scale circular reading scale main gaugescrewofadingReobjectofThicknes least count of screw gauge = divisionscalecircularof.No pitch  Vernier callipers                              count Least reading scale vernier reading scale main callipervernierofadingReobjectofThicknes Least count of vernier calliper = 1 MSD –1 VSD Page # 82 PRINCIPLE OF COMMUNICATION Transmission from tower of height h  the distance to the horizon dT = T2Rh  dM = T R2Rh 2Rh Amplitude Modulation  The modulated signal cm (t) can be written as cm(t) = Ac sin ct + cA 2  cos (C - m) t – cA 2  cos (C + m)  Modulation index m a c kAChange in amplitude of carrier wavem Amplitude of original carrier wave A   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 = max minm c max min A – AA A A – A   If a carrier wave is modulated by several sine waves the total modulated index mt is given by mt = 2 2 2 1 2 3m m m .........    Side band frequencies (fc + fm) = Upper side band (USB) frequency (fc - fm) = Lower side band (LBS) frequency  Band width = (fc + fm) - (fc - fm) = 2fm  Power in AM waves : 2 rmsV P R  (i) carrier power 2 c 2 c c A A2P R 2R        Page # 85 CE Amplifier (i) ac current gain ac =         b c i i VCE = constant (ii) dc current gain dc = b c i i (iii) Voltage gain : AV = i 0 V V   = ac × Resistance gain (iv) Power gain = i 0 P P   = 2ac × Resistance (v) Transconductance (gm) : The ratio of the change in collector in collector current to the change in emitter base voltage is called trans conductance i.e. gm = EB c V i   . Also gm = L V R A RL = Load resistance.  Relation between  and  :    –1 or  =   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. ROUGH WORK Wessocense” Page # 86