Physics Handbook PDF

C HAP TE R Physics HandBook...

C HAP TE R Physics HandBook ALLEN DIMENSIONS OF IMPORTANT PHYSICAL QUANTITIES Physical quantity Dimensions Physical quantity Dimensions Momentum M1 L 1 T –1 Capacitance M–1 L–2 T4 A 2 Calorie M1 L 2 T –2 Modulus of rigidity M1 L– 1 T –2 Latent heat capacity M L T– Magnetic permeability M L T – A– 0 2 2 1 1 2 2 Self inductance M L T – A– Pressure M L– T – 1 2 2 2 1 1 2 Coefficient of thermal conductivity M L T– K – Planck's constant M L T– 1 1 3 1 1 2 1 Power M L T– Solar constant M L T– 1 2 3 1 0 3 Impulse M1 L 1 T –1 Magnetic flux M1 L2 T –2 A– 1 Hole mobility in a semi conductor M– L T A Current density M L– T A 1 0 2 1 0 2 0 1 M L– T– M L– T – N Bulk modulus of elasticity 1 1 2 Young modulus 1 1 2 Potential energy M L T– Magnetic field intensity M L– T A 1 2 2 0 1 0 1 Gravitational constant M – L T– Magnetic Induction M T– A – 1 3 2 1 2 1 0 Light year M0 L1 T0 Electric Permittivity M– L– T A 1 3 4 2 -2 Thermal resistance E M–1 L–2 T3 K Electric Field M1L 1T–3A- 1 Coefficient of viscosity M1 L– 1 T–1 ML T – A– 19 Resistance 2 3 2 SETS OF QUANTITIES HAVING SAME DIMENSIONS 20 LL S.N. Quantities Dimensions n 1. Strain, refractive index, relative density, angle, solid angle, phase, distance gradient, relative permeability, relative permittivity, angle of contact, Reynolds [M 0 L 0 T 0] io number, coefficient of friction, mechanical equivalent of heat, electric susceptibility, etc. 2. Mass or inertial mass [M 1 L 0 T 0 ] ss 3. Mom entum and impulse. [M 1 L 1 T – 1] 4. Thrust, force, weight, tension, energy gradient. [M 1 L 1 T – 2] Se 5. Pressure, stress, Young's modulus, bulk modulus, shear modulus, modulus of rigidity, energy density. [M 1 L – 1 T – 2 ] A 6. Angular momentum and Planck's constant (h). [ M 1 L2 T –1] 7. Acceleration, g and gravitational field intensity. [ M 0 L1 T –2] node06\B0AI-B0\Kota\JEE(Advanced)\Leader\Phy\Sheet\Hand book (E+L)\Eng\01_Unit & Dimension.p65 8. Surface tension, free surface energy (energy per unit area), force gradient, spring constant. [ M 1 L0 T –2] 9. Latent heat capacity and gravitational potential. [ M 0 L2 T –2] 10. Thermal capacity, Boltzmann constant, entropy. [ ML 2 T – 2K – 1 ] 11. Work, torque, internal energy, potential energy, kinetic energy, mom ent of force, (q2 /C), (LI2 ), (qV), (V 2 C), (I 2 Rt), V2 t , (VIt), (PV), (RT), (mL), (mc DT) [M 1 L 2 T – 2] R 12. Frequency, angular frequency, angular velocity, velocity gradient, radioactivity R 1 , , 1 [M 0 L 0 T – 1] L RC LC 13. ælö 12 æmö 1 2 æL ö ç ÷ ,ç ÷ ,ç ÷ , (RC), ( LC ) , time [ M 0 L 0 T 1] ègø è k ø èR ø 14. (VI), (I2 R), (V 2/R), Power [ M L 2 T – 3] E 3 Physics HandBook CH APTER ALLEN Gravitational constant (G) 6.67 × 10 –11 N m 2 kg –2 KEY POINTS Speed of light in vacuum (c) 3 × 10 8 ms –1 Trigonometric functions SOME FUNDAMENTAL CONSTANTS Permeability of vacuum (m 0 ) 4p × 10 –7 H m –1 sinq, cosq, tanq etc and their Permittivity of vacuum (e 0 ) 8.85 × 10 –12 F m –1 arrangement s q are Planck constant (h) dimensionless. 6.63 × 10 –34 Js Atom ic mass unit (am u) 1.66 × 10 –27 kg Dimensions of differential Energy equivalent of 1 amu 931.5 MeV é dny ù éyù coefficients ê n ú = ê n ú 9.1 × 10 –31 kg º 0.511 ë dx û ë x û Electron rest mass (m e ) MeV Dimensions of integrals Avogadro constant (N A ) 6.02 × 10 23 mol–1 é ydx ù = [ yx ] ëê ò úû Faraday constant (F) 9.648 × 10 4 C mol –1 Stefan–Boltzmann constant (s) 5.67× 10 –8 W m –2 K –4 We can't add or subtract two physical quantities of Wien constant (b) 2.89× 10 –3 mK different dimensions. N Rydberg constant (R ¥ ) 1.097× 10 7 m –1 Independent quantities may Triple point for water 273.16 K (0.01°C) be taken as fundamental quantities in a new system of 0 22.4 L = 22.4× 10 –3 m 3 Molar volum e of ideal gas (NTP) units. mol –1 -2 E PRACTICAL PHYSICS 19 Rules for Counting Significant Figures For example : 3.0 × 800.0 = 2.4 × 103 For a number greater than 1 All non-zero digits are significant. 20 The sum or difference can be no more precise than the least precise number involved in the mathematical LL All zeros between two non-zero digits are operation. Precision has to do with the number of significant. Location of decimal does not matter. positions to the RIGHT of the decimal. The more n If the numbe is without decimal part, then the position to the right of the decimal, the more precise io terminal or trailing zeros are not significant. the number. So a sum or difference can have no Trailing zeros in the decimal part are significant. more indicated positions to the right of the decimal ss For a Number Less than 1 as the number involved in the operation with the Any zero to the right of a non-zero digit is significant. LEAST indicated positions to the right of its decimal. All zeros between decimal point and first non-zero For example : 160.45 + 6.732 = 167.18 (after Se digit are not significant. rounding off) A Significant Figures Another example : 45.621 + 4.3 – 6.41 = 43.5 All accurately known digits in measurement plus (after rounding off) node06\B0AI-B0\Kota\JEE(Advanced)\Leader\Phy\Sheet\Hand book (E+L)\Eng\01_Unit & Dimension.p65 the first uncertain digit together form significant Rules for rounding off digits : figure. 1. If the digit to the right of the last reported digit is Ex. 0.108 ® 3SF, 40.000 ® 5SF, less than 5 round it and all digits to its right off. 1.23 × 10 ® 3SF, 0.0018 ® 2SF -19 2. If the digit to the right of the last reported digit is Significant Digits greater than 5 round it and all digits to its right off The product or quotient will be reported as having and increased the last reported digit by one. as many significant digits as the number involved 3. If the digit to the right of the last reported digit is a in the operation with the least number of significant 5 followed by either no other digits or all zeros, digits. round it and all digits to its right off and if the last For example : 0.000170 × 100.40 = 0.017068 reported digit is odd round up to the next even Another example : 2.000 × 104 / 6.0 × 10–3 = digit. If the last reported digit is even then leave it 0.33 × 107 as is. 4 E  C HAP TE R Physics HandBook ALLEN For example if we wish to round off the following Error in Product and Division number to 3 significant digits : 18.3682 A physical quantity X depend upon Y & Z as X = Ya Zb The answer is : 18.4. Another example : Round then maximum possible fractional error in X. off 4.565 to three significant digits. DX DY DZ = a + b The answer would be 4.56. X Y Z Rounding off Error in Power of a Quantity 6.87® 6.9, 6.84 ® 6.8, 6.85 ® 6.8, am Dx é æ Da ö æ Db ö ù 6.75 ® 6.8, 6.65 ® 6.6, 6.95 ® 7.0 x= n then = ± êm ç ÷ + n ç ÷ ú è ø è b øû b x ë a Order of magnitude The quotient rule is not applicable if the numerator Power of 10 required to represent a quantity and denominator are dependent on each other. 49 = 4.9 × 101 » 101 Þ order of magnitude =1 XY 51 = 5.1 × 101 » 102 Þ order of magnitude = 2 e.g if R =. We cannot apply quotient rule X+Y 0.051 =5.1 × 10-2 » 10-1order of magnitude = -1 N to find the error in R. Instead we write the equation Errors Whenever an experiment is performed, two kinds 1 1 1 as follows = +. Differentiating both 0 of errors can appear in the measured quantity. R X Y (1) random and (2) systematic errors. -2 1. Random errors appear randomly because of dR dX dY E the sides, we get - 2 =- 2 -. operator, fluctuations in external conditions and R X Y2 19 variability of measuring instruments. The effect of random error can be some what reduced by taking r x y Thus 2 = 2 + the average of measured values. Random errors R 20 X Y2 have no fixed sign or size. LL Least count 2. Systematic errors occur due to error in the The smallest value of a physical quantity which can procedure, or miscalibration of the instrument etc. n be measured accurately with an instrument is called Such errors have same size and sign for all the the least count of the measuring instrument. io measurements. Such errors can be determined. Vernier Callipers A measurement with relatively small random error ss Least count = 1MSD – 1 VSD is said to have high precision. A measurement with (MSD ® main scale division, VSD ® Vernier scale division) small random error and small systematic error is Se said to have high accuracy. Least Count Error :– If the instrument has known A least count, the absolute error is taken to be equal node06\B0AI-B0\Kota\JEE(Advanced)\Leader\Phy\Sheet\Hand book (E+L)\Eng\01_Unit & Dimension.p65 to the least count unless otherwise stated. 0 1 2 3 4 5 6 14 15 absolute error in a measurement Relative error = size of the measurement A. Systematic errors : They have a known sign. The systematic error is removed before beginning calculations. Bench error Ex. A vernier scale has 10 parts, which are equal to 9 and zero error are examples of systematic error. parts of main scale having each path equal to 1 Propagation of combination of errors 9 mm then least count = 1 mm – mm = 0.1 mm Error in Summation and Difference : 10 x = a + b then Dx = ± (Da+Db) [Q 9 MSD = 10 VSD] E 5 Physics HandBook CH APTER ALLEN Zero Error Main scale Main scale Main scale 0 1 0 1 0 1 0 5 10 0 5 10 0 5 10 Vernier scale Vernier scale Vernier scale without zero error with positive zero error with negative zero error (i) (ii) The zero error is always subtracted from the reading to get the corrected value. If the zero error is positive, its value is calculated as we take any normal reading. N Negative zero error = – [Total no. of vsd – vsd coinciding] ×L.C. Screw Gauge 0 pitch Least count = total no. of divisions -2 on circular scale E 19 Spindle Circular (Head) scale Ratchet 05 10 Linear (Pitch) Scale Sleeve Thimble 20 LL Ex. The distance moved by spindle of a screw gauge for each turn of head is 1mm. The edge of the humble is n 1mm io provided with a angular scale carrying 100 equal divisions. The least count = = 0.01 mm 100 ss Zero Error Positive Zero Error If there is no object between the jaws (i.e. jaws are in (2 division error) i.e., +0.002 cm contact), the screwgauge should give zero reading. But Se due to extra material on jaws, even if there is no object, it A gives some excess reading. This excess reading is called Zero error. Circular scale node06\B0AI-B0\Kota\JEE(Advanced)\Leader\Phy\Sheet\Hand book (E+L)\Eng\01_Unit & Dimension.p65 Negative Zero Error (3 division error) i.e., –0.003 cm 15 10 Circular scale 5 Zero of the circular 0 scale is below the 0 zero of main scale Main scale 95 10 reference line zero of the circular 90 5 scale is above the 0 0 zero of main scale 95 Main scale 90 reference line 85 6 E  C HAP TE R Physics HandBook ALLEN Basic Mathematics used in Physics Quadratic Equation Binomial Theorem - b ± b2 - 4ac n(n - 1) 2 n(n - 1)(n - 2) 3 Roots of ax2 + bx + c=0 are x = (1+x)n = 1 + nx + x + x +.... 2a 2 6 b Sum of roots x1 + x2 = – ; n ( n - 1) 2 n(n - 1)(n - 2) 3 a (1–x)n = 1 – nx + x - x +..... 2 6 c Product of roots x1x2 = If x