... the total number ** of ** nodes ** and ** the position ** of ** the nodes measured from the left end ** of ** string Sound Waves 15 Sound waves are the most common examples ** of ** longitudinal waves The speed ** of ** sound waves ... power form: P = ** **** 21** ρ ** A ** v **2 ** smax (15 **.23** ) Thus, the power ** of ** ** a ** periodic sound wave is proportional to the square ** of ** the angular frequency ** and ** the square ** of ** the displacement amplitude (as in the case ... sound which has ** a ** pressure-variation amplitude Pmax = ** 28** Pa (the threshold ** of ** pain), ** and ** can detect the faintest sound which has Pmax = ** 2.** 8 × 10−5 Pa (the threshold ** of ** hearing) ** For ** ** a ** sound of...

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... at one end ** A ** person at the other end hears two sounds as ** a ** result ** of ** the propagation ** of ** two longitudinal waves, one that traveled through the rod ** and ** the other that traveled through the air at ... at constant pressure to the heat capacity at constant volume **(a)** Show that Bad = γ P ** for ** an ideal gas (b) Show that the speed ** of ** ** a ** longitudinal wave √ in the adiabatic process ** of ** an ideal gas ... sinusoidal wave ** and ** has the same frequency f ** and ** wavelength λ as any one ** of ** the contributing waves y1 ** and ** y2 , **(2)** It has an amplitude ** of ** **2A ** cos(φ **/2)** , (3) It has ** a ** phase ** of ** φ **/2 ** Now, let us consider...

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... diffraction, ** and ** polarization ** of ** light ** and ** need to get satisfactory descriptions ** of ** these phenomena, we treat light as waves Such ** a ** study is referred to as physical optics H ** A ** Radi ** and ** J ** O ** ** Rasmussen,** ... position ** of ** the nodes ** and ** antinodes (c) What is the maximum amplitude ** of ** an element at an antinode? (9) The two sources shown in the evacuated vessel ** of ** Fig 16 **.22** , are 1 **.2 ** m apart, ** and ** send sound ... wall at this end does not allow longitudinal motion ** of ** the air ** and ** acts like ** a ** pressure antinode (point ** of ** maximum pressure variation) The open end ** of ** ** a ** pipe acts like the end ** of ** ** a ** string that...

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... sin[ **(A ** + δm ) **/2]** sin **(A/** ** 2)** −−−−−−−−−−→ When ** A ** is small n= ** A ** + δm ** A ** (17. **12)** The most charming example ** of ** color dispersion is that ** of ** ** a ** rainbow To understand the formation ** of ** ** a ** rainbow we consider ** a ** ... An anti-shoplifter convex spherical mirror has ** a ** radius ** of ** curvature ** of ** 0.4 m Locate ** and ** describe the image formed by ** a ** man standing 3.8 m away from the mirror Solution: The focal length ** of ** ** a ** ... at ** a ** real focal point after reflecting from ** a ** concave mirror (b) The same rays will diverge from ** a ** convex mirror ** and ** appear to come from ** a ** virtual focal point When parallel rays reach the surface...

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... in air falls on water at an angle ** of ** incidence ** of ** 45◦ ** and ** then passes through ** a ** glass block before it emerges out to air again The surfaces ** of ** water ** and ** glass are parallel ** and ** their indexes ** of ** ... the air-alcohol boundary at an angle ** of ** incidence ** of ** 60◦ ? (6) ** A ** beam ** of ** light in air falls on ** a ** liquid surface at an angle ** of ** incidence ** of ** 55◦ The liquid has an unknown index ** of ** refraction **(a)** ... Eq 18 **.23** Part **(a)** ** of ** Fig 18. **12 ** displays the variation ** of ** I as ** a ** function ** of ** δ **/2 ** Part (b) is ** a ** representation ** of ** the obtained photograph 18.4 Diffraction ** of ** Light Waves 619 Fig 18. **12 ** **(a)** ** A ** sketch...

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... defined as the direction ** of ** polarization ** of ** the wave However, an ordinary beam ** of ** light contains ** a ** large number ** of ** electromagnetic waves emitted by atoms having random vibrational orientations, → ** and ** ... the Coulomb constant Coulomb found that each charged particle (also called ** a ** point charge) exerts ** a ** force ** of ** this magnitude on the other particle, ** and ** the two forces form an action–reaction pair ... materials nonconductors or insulators Spotlight Insulators are materials that contain electrons that are bound to their atoms ** and ** cannot move freely through the material Semiconductors are materials...

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... Gravitational attraction ** A ** classical model ** of ** the hydrogen atom Solution: **(a)** From Coulomb’s law, the magnitude ** of ** the electrostatic force ** of ** → attraction Fe between the electron ** and ** the proton ... ** 20** Electric Fields When x ** a ** we can take out ** a ** factor ** of ** x from each denominator ** of ** the brackets ** of ** the last formula ** for ** x > **+a ** ** and ** then expand each ** of ** these terms by binomial expansion Therefore, ... equal point charges ** of ** magnitude q lie on ** a ** semicircle ** of ** radius R as √ shown in Fig 19 **.21** Show that the net force on q2 has ** a ** magnitude kq2 / 2R2 ** and ** points downward away from the center C of...

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... that: (x dx x = 3 **/2 ** **+a ** ) ** a2** (x + ** a2** ) **(20** .33) Thus: L **/2 ** E = 2kλ ** a ** = 2kλ ** a ** x dx = kλ ** a ** √ ** 2 ** 3 **/2 ** (x + ** a ** ) ** a2** x + ** a2** L **/2 ** ** a2** (L **/2)** 2 + ** a2** −0 = L **/2 ** kλ L ** a ** (L **/2)** 2 + ** a2** **(20** .34) ** 20** .4 Electric Field ** of ** ** a ** ... Example ** 20** .4 Figure ** 20** .14 shows ** a ** non-conducting rod that has ** a ** uniform positive charge density +λ ** and ** ** a ** total charge Q along its right half, ** and ** ** a ** uniform negative charge density −λ ** and ** ** a ** total ... terms ** of ** x From the geometry ** of ** Fig ** 20** .13, we have: x + ** a2** ** and ** sin θ = r= ** a ** a =√ r x + ** a2** **(20** .31) Therefore, Eq ** 20** .30 becomes: L **/2 ** E = kλ ** a ** dx (x + ** a2** )3 **/2 ** **(20** . **32)** From the table ** of ** integrals in Appendix...

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... central axes along ** a ** common horizontal line ** and ** separated by ** a ** distance ** of ** R, see Fig ** 20** .36 Ring has ** a ** uniform positive charge q1 , while ring has ** a ** uniform positive charge q2 Given that the ... results ** of ** Exercise ** 21** to the near-field approximation E = σ **/2 ** ◦ as well as to the point charge approximation E = k(π R2 σ ) **/a2** , ** and ** find which result(s) ** of ** Exercise ** 21** match the two approximations ... when ** a ** formula: 2kQ ** a ** 1− √ ** 2 ** R R + ** a2** R, the electric field approaches that ** of ** ** a ** point charge E≈k Q **(a ** a2 R) You may use the binomial expansion (1 + δ)p ≈ + pδ when δ **(23** ) Compare the obtained...

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... such an electron has to be accelerated from rest through ** a ** potential difference ** of ** 4.55 kV to reach ** a ** speed ** of ** × 107 m/s Equations ** 22** .3 ** and ** ** 22** .6 hold true ** for ** both discrete ** and ** continuous source ... Potential 735 Suppose we move ** a ** particle ** of ** an arbitrary charge q from point ** A ** to point B by applying ** a ** force on it ** and ** changing its kinetic energy The applied force must perform work WAB (app), ... identical Therefore, the general features ** of ** the gravitational force introduced in Chap apply to electrostatic forces In particular, electrostatic forces are conservative Consequently, it is more convenient...

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... ** 22** Electric Potential ** For ** ** a ** Point Above One End ** of ** the Rod When ** a ** point P is located at ** a ** distance ** a ** from one ** of ** the rod’s ends, see Fig ** 22** .17, we can perform similar calculations to find that: ... Therefore: V = 2kλ ln x + ** a2** ) **(22** .48) L **/2 ** (L **/2)** 2 + ** a2** ) − ln **(a)** L **/2 ** + (L **/2)** 2 + ** a2** ** a ** **(22** .49) When we use the fact that the total charge Q = λL, we get: V = L **/2 ** + 2kQ ln L (L **/2)** 2 + ** a2** ** a ** **(22** .50) 7 **52 ** ... potential energy ** of ** each pair ** and ** add them algebraically ** For ** instance, the total potential energy ** of ** three charges q1 , q2 , ** and ** q3 is: q1 q2 q1 q3 q2 q3 + + r **12 ** r13 r23 U=k **(22** **.31)** Example ** 22** .5 Two...

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... inverse ** of ** the capacitance ** of ** an air capacitor ** of ** separation d − ** a,** ** and ** ** a/** κ ◦ ** A ** is the inverse ** of ** the capacitance ** of ** ** a ** capacitor ** of ** separation ** a ** but filled with ** a ** dielectric ** 23** .4 Capacitors in Parallel ... two basic combinations ** of ** capacitors that allow such ** a ** replacement Capacitors in ** a ** Parallel Combination Figure ** 23** .1 **1a ** shows two capacitors ** of ** capacitances C1 ** and ** C2 , that are connected in parallel ... capacitance ** for ** the capacitor under consideration ** A ** Parallel-Plate Capacitor Figure ** 23** . **2a ** shows an uncharged parallel-plate capacitor ** of ** equal area ** A ** separated by ** a ** distance d The capacitor is...

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... Section ** 23** **.2 ** Calculating Capacitance (4) ** A ** computer memory chip contains ** a ** large number ** of ** capacitors, each ** of ** which has ** a ** plate area ** A ** = ** 20** × 10− **12 ** m2 ** and ** ** a ** capacitance ** of ** 50 f F (50 femtofarads) ... **(a)** ** and ** (b) ** of ** Fig ** 23** . **32 ** Show that the capacitances ** of ** the two capacitors ** of ** parts **(a)** ** and ** (b) are: C= **A ** d κ + **2 ** ◦ ** A ** κ1 **2 ** ** and ** C = respectively, d κ1 + **2 ** ** A/** ** 2 ** d ** A/** ** 2 ** **2 ** κ1 **(a)** d κ1 d **/2 ** **2 ** d **/2 ** ... the maximum magnitude ** of ** the charge on the plates ** of ** an air-filled parallelplate capacitor ** of ** area ** A ** = 30 cm2 such that ** a ** spark is avoided (7) ** A ** parallel-plate capacitor has circular plates, each...

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... to move In fact, it is ** a ** device that works as an energy converter ** A ** battery is often called ** a ** source ** of ** electromotive force or, ** a ** source ** of ** emf (this unfortunate historical name describes ** a ** potential ... Parallel Combination Figure ** 24** .1 **5a ** shows two resistors ** of ** resistances R1 ** and ** R2 that are connected in parallel with ** a ** battery B Figure ** 24** .15b shows ** a ** circuit diagram ** for ** this combination ** of ** resistors ... **)2 ** R **(24** **.24** ) Because P = I V , the same amount ** of ** power P can be transported either at high I ** and ** low V , or at low I ** and ** high V Example ** 24** .6 ** A ** ** 22** 0 V potential difference is maintained across...

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... ampere-hours **(A.** h) ** For ** instance, ** a ** battery that can produce ** a ** current ** of ** ** A ** ** for ** h is rated as ** a ** ** 20** ** A.** h battery **(a)** Find the total energy stored in ** a ** **12 ** V battery rated at 75 ** A.** h (Express your answer ... Section ** 24** .7 The RC Circuit (46) In the process ** of ** charging ** a ** capacitor ** of ** capacitance C through ** a ** resistor ** of ** resistance R, about 63% ** of ** the maximum charge will accumulate on the capacitor in ** a ** ... R2 Part VI Magnetism ** 25** Magnetic Fields It is ** of ** common knowledge that every magnet attracts pieces ** of ** iron ** and ** has two poles: ** a ** north pole (N) ** and ** ** a ** south pole (S) In addition, given two magnets,...

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