CHAPTER 30: Inductance, Electromagnetic Oscillations, and AC Circuits


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1 HAPTE 3: Inductance, Electromagnetic Oscillations, and A ircuits esponses to Questions. (a) For the maximum value of the mutual inductance, place the coils close together, face to face, on the same axis. (b) For the least possible mutual inductance, place the coils with their faces perpendicular to each other.. The magnetic field near the end of the first solenoid is less than it is in the center. Therefore the flux through the second coil would be less than that given by the formula, and the mutual inductance would be lower. 3. Yes. If two coils have mutual inductance, then they each have the capacity for selfinductance. Any coil that experiences a changing current will have a selfinductance. 4. The energy density is greater near the center of a solenoid, where the magnetic field is greater. 5. To create the greatest selfinductance, bend the wire into as many loops as possible. To create the least selfinductance, leave the wire as a straight piece of wire. 6. (a) No. The time needed for the circuit to reach a given fraction of its maximum possible current depends on the time constant, τ = /, which is independent of the emf. (b) Yes. The emf determines the maximum value of the current (I max = V /,) and therefore will affect the time it takes to reach a particular value of current. 7. A circuit with a large inductive time constant is resistant to changes in the current. When a switch is opened, the inductor continues to force the current to flow. A large charge can build up on the switch, and may be able to ionize a path for itself across a small air gap, creating a spark. 8. Although the current is zero at the instant the battery is connected, the rate at which the current is changing is a maximum and therefore the rate of change of flux through the inductor is a maximum. Since, by Faraday s law, the induced emf depends on the rate of change of flux and not the flux itself, the emf in the inductor is a maximum at this instant. 9. When the capacitor has discharged completely, energy is stored in the magnetic field of the inductor. The inductor will resist a change in the current, so current will continue to flow and will charge the capacitor again, with the opposite polarity.. Yes. The instantaneous voltages across the different elements in the circuit will be different, but the current through each element in the series circuit is the same.. The energy comes from the generator. (A generator is a device that converts mechanical energy to electrical energy, so ultimately, the energy came from some mechanical source, such as falling water.) Some of the energy is dissipated in the resistor and some is stored in the fields of the capacitor and the inductor. An increase in results in an increase in energy dissipated by the circuit.,,, and the frequency determine the current flow in the circuit, which determines the power supplied by generator. 9 Pearson Education, Inc., Upper Saddle iver, NJ. All rights reserved. This material is protected under all copyright laws as they 7
2 hapter 3 Inductance, Electromagnetic Oscillations, and A ircuits. X = X at the resonant frequency. If the circuit is predominantly inductive, such that X > X, then the frequency is greater than the resonant frequency and the voltage leads the current. If the circuit is predominantly capacitive, such that X > X, then the frequency is lower than the resonant frequency and the current leads the voltage. Values of and cannot be meaningfully compared, since they are in different units. Describing the circuit as inductive or capacitive relates to the values of X and X, which are both in ohms and which both depend on frequency. 3. Yes. When ω approaches zero, X approaches zero, and X becomes infinitely large. This is consistent with what happens in an ac circuit connected to a dc power supply. For the dc case, ω is zero and X will be zero because there is no changing current to cause an induced emf. X will be infinitely large, because steady direct current cannot flow across a capacitor once it is charged. 4. The impedance in an circuit will be a minimum at resonance, when X = X. At resonance, the impedance equals the resistance, so the smallest possible will give the smallest impedance. 5. Yes. The power output of the generator is P = IV. When either the instantaneous current or the instantaneous voltage in the circuit is negative, and the other variable is positive, the instantaneous power output can be negative. At this time either the inductor or the capacitor is discharging power back to the generator. 6. Yes, the power factor depends on frequency because X and X, and therefore the phase angle, depend on frequency. For example, at resonant frequency, X = X, the phase angle is º, and the power factor is one. The average power dissipated in an circuit also depends on frequency, since it depends on the power factor: P avg = I V cosφ. Maximum power is dissipated at the resonant frequency. The value of the power factor decreases as the frequency gets farther from the resonant frequency. 7. (a) The impedance of a pure resistance is unaffected by the frequency of the source emf. (b) The impedance of a pure capacitance decreases with increasing frequency. (c) The impedance of a pure inductance increases with increasing frequency. (d) In an circuit near resonance, small changes in the frequency will cause large changes in the impedance. (e) For frequencies far above the resonance frequency, the impedance of the circuit is dominated by the inductive reactance and will increase with increasing frequency. For frequencies far below the resonance frequency, the impedance of the circuit is dominated by the capacitive reactance and will decrease with increasing frequency. 8. In all three cases, the energy dissipated decreases as approaches zero. Energy oscillates between being stored in the field of the capacitor and being stored in the field of the inductor. (a) The energy stored in the fields (and oscillating between them) is a maximum at resonant frequency and approaches an infinite value as approaches zero. (b) When the frequency is near resonance, a large amount of energy is stored in the fields but the value is less than the maximum value. (c) Far from resonance, a much lower amount of energy is stored in the fields. 9. In an circuit, the current and the voltage in the circuit both oscillate. The energy stored in the circuit also oscillates and is alternately stored in the magnetic field of the inductor and the electric field of the capacitor. 9 Pearson Education, Inc., Upper Saddle iver, NJ. All rights reserved. This material is protected under all copyright laws as they 7
3 Physics for Scientists & Engineers with Modern Physics, 4 th Edition Instructor Solutions Manual. In an circuit, energy oscillates between being stored in the magnetic field of the inductor and being stored in the electric field of the capacitor. This is analogous to a mass on a spring, with energy alternating between kinetic energy of the mass and spring potential energy as the spring compresses and extends. The energy stored in the magnetic field is analogous to the kinetic energy of the moving mass, and corresponds to the mass, m, on the spring. The energy stored in the electric field of the capacitor is analogous to the spring potential energy, and corresponds to the reciprocal of the spring constant, /k. Solutions to Problems. (a) The mutual inductance is found in Example NNA 854 Tm A55. m M 3. H l.44 m (b) The emf induced in the second coil can be found from Eq. 33b. di I.A e M M 3. H 3.79 V dt t.98 ms. If we assume the outer solenoid is carrying current I, then the magnetic field inside the outer solenoid is B ni. The flux in each turn of the inner solenoid is B r nir. The mutual inductance is given by Eq. 3. N n l n Ir M M nn r I I l 3. We find the mutual inductance of the inner loop. If we assume the outer solenoid is carrying current I, then the magnetic field inside the outer solenoid is N B I. The magnetic flux through each l loop of the small coil is the magnetic field times the area perpendicular to the field. The mutual inductance is given by Eq. 3. NI N A sin NI N NN Asin BA sin A sin ; M l l I I l 4. We find the mutual inductance of the system using Eq. 3, with the flux equal to the integral of the magnetic field of the wire (Eq. 8) over the area of the loop. I w M wdr ln I I l l l r l 5. Find the induced emf from Eq di I.8 H. A e 5. A V dt t.36s 9 Pearson Education, Inc., Upper Saddle iver, NJ. All rights reserved. This material is protected under all copyright laws as they 7
4 hapter 3 Inductance, Electromagnetic Oscillations, and A ircuits 6. Use the relationship for the inductance of a solenoid, as given in Example 33. N A l.3h.3 m N 47 turns 7 l A 4 Tm A.m 7. Because the current in increasing, the emf is negative. We find the selfinductance from Eq di I t.s e e.5 V.566 H dt t I.5 A.8 A 8. (a) The number of turns can be found from the inductance of a solenoid, which is derived in Example N A 4 Tm A8.5m.9 H. H l.7 m (b) Apply the same equation again, solving for the number of turns. N A l.9 H.7 m N 8turns 7 l A 4 Tm A.5m 9. We draw the coil as two elements in series, and pure resistance and a pure inductance. There is a voltage drop due to the resistance of the coil, given by Ohm s law, and an induced emf due to the E inductance of the coil, given by Eq Since the current is I induced increasing increasing, the inductance will create a potential difference to b a oppose the increasing current, and so there is a drop in the potential due to the inductance. The potential difference across the coil is the sum of the two potential drops. di V I 3. A H3.6 A s.3v ab dt. We use the result for inductance per unit length from Example H m 9 55 H m 7 r 9 4 Tm A ln 55 H m r re.3 me.8 m l r r.3m. The selfinductance of an airfilled solenoid was determined in Example 33. We solve this equation for the length of the tube, using the diameter of the wire as the length per turn. on A oal n Al l d 3.H.8 m 7 4 Tm/A.6m d l 46.6 m 46 m r The length of the wire is equal to the number of turns (the length of the solenoid divided by the diameter of the wire) multiplied by the circumference of the turn. l 46.6 m D 3. m,49 m km d.8 m 9 Pearson Education, Inc., Upper Saddle iver, NJ. All rights reserved. This material is protected under all copyright laws as they 73
5 Physics for Scientists & Engineers with Modern Physics, 4 th Edition Instructor Solutions Manual The resistance is calculated from the resistivity, area, and length of the wire m,49m l.7k 3 A.45 m N A N d. The inductance of the solenoid is given by. The (constant) length of the l l 4 wire is given by l Nd, and so since d.5d, we also know that N.5N. The fact wire sol sol sol that the wire is tightly wound gives l Nd. Find the ratio of the two inductances. sol wire N N wire d d sol sol 4 l l l l sol sol sol sol wire N N l wire lsol wire d d sol sol 4 l l l sol sol sol N d N N d N l.5 3. We use Eq. 34 to calculate the selfinductance, where the flux is the integral of the magnetic field over a crosssection of the toroid. The magnetic field inside the toroid was calculated in Example 8. N N r NI N h r B hdr ln I I r r r r r r dr h 4. (a) When connected in series the voltage drops across each inductor will add, while the currents in each inductor are the same. eee di di di di eq eq dt dt dt dt (b) When connected in parallel the currents in each inductor add to the equivalent current, while the voltage drop across each inductor is the same as the equivalent voltage drop. di di di e e e dt dt dt eq eq Therefore, inductors in series and parallel add the same as resistors in series and parallel. 5. The magnetic energy in the field is derived from Eq Energy stored B u Volume B B.6 T Energy Volume r l.5 m.38 m 8.9 J 7 4 Tm A 6. (a) We use Eq. 46 to calculate the energy density in an electric field and Eq. 37 to calculate the energy density in the magnetic field. ue E 8.85 /N m. N/ 4.4 J/m Pearson Education, Inc., Upper Saddle iver, NJ. All rights reserved. This material is protected under all copyright laws as they 74
6 hapter 3 Inductance, Electromagnetic Oscillations, and A ircuits u B 7 4 Tm/A B. T.59 J/m.6 J/m (b) Use Eq. 46 to calculate the electric field from the energy density for the magnetic field given in part (a)..59 J/m u E u E 6. N/ E 6 3 u B 8 B 8.85 /Nm 7. We use Eq. 37 to calculate the energy density with the magnetic field calculated in Example B 4 T m A3.A I I 3 3 ub.6 J m 8 8.8m 8. We use Eq. 37 to calculate the magnetic energy density, with the magnetic field calculated using Eq. 8. u B 7 4 Tm/A5 A B I I m.6 J/m To calculate the electric energy density with Eq. 46, we must first calculate the electric field at the surface of the wire. The electric field will equal the voltage difference along the wire divided by the length of the wire. We can calculate the voltage drop using Ohm s law and the resistance from the resistivity and diameter of the wire. V I Il I E l l lr r u E I E r J/m 8 5A.68 m 8.85 /N m 3.5 m 9. We use Eq. 37 to calculate the energy density in the toroid, with the magnetic field calculated in Example 8. We integrate the energy density over the volume of the toroid to obtain the total energy stored in the toroid. Since the energy density is a function of radius only, we treat the toroid as cylindrical shells each with differential volume dv rhdr. B NI N I ub r 8 r r N I N I h r dr N I h r U ubdv rhdr ln r 8 r 4 r r 4 r. The magnetic field between the cables is given in Example 35. Since the magnetic field only depends on radius, we use Eq. 37 for the energy density in the differential volume dv rl dr and integrate over the radius between the two cables. U r I I r dr I r udv B rdr ln l l r r 4 r r 4 r 3 9 Pearson Education, Inc., Upper Saddle iver, NJ. All rights reserved. This material is protected under all copyright laws as they 75
7 Physics for Scientists & Engineers with Modern Physics, 4 th Edition Instructor Solutions Manual. We create an Amperean loop of radius r to calculate the magnetic field within the wire using Eq Since the resulting magnetic field only depends on radius, we use Eq. 37 for the energy density in the differential volume dv rl dr and integrate from zero to the radius of the wire. I Ir B d l Ienc Br r B U Ir I 3 I udv B rdr rdr 4 l l 4 6 t. For an circuit, we have I Imax e max. Solve for t. t t I I I I e e t ln I I max max I I.95 I t ln ln max I max I I.99I t ln ln max I max I I.999I t ln ln max I max (a) (b) (c) 3. We set the current in Eq. 3 equal to.3i and solve for the time. t / I.3I I e t ln (a) We set I equal to 75% of the maximum value in Eq. 39 and solve for the time constant. t / t.56 ms I.75I I e.847 ms.85 ms ln.5 ln.5 (b) The resistance can be calculated from the time constant using Eq mh ms 5. (a) We use Eq. 36 to determine the energy stored in the inductor, with the current given by Eq. Eq 39. V t / U I e (b) Set the energy from part (a) equal to 99.9% of its maximum value and solve for the time. V V t / U.999 e t ln (a) At the moment the switch is closed, no current will flow through the inductor. Therefore, the resistors and can be treated as in series. I I I e e, I3 (b) A long time after the switch is closed, there is no voltage drop across the inductor so resistors and 3 can be treated as parallel resistors in series with. 9 Pearson Education, Inc., Upper Saddle iver, NJ. All rights reserved. This material is protected under all copyright laws as they 76
8 hapter 3 Inductance, Electromagnetic Oscillations, and A ircuits (c) I I I3, e= I I, I I33 e I I e I I I e I I I I Just after the switch is opened the current through the inductor continues with the same magnitude and direction. With the open switch, no current can flow through the branch with the switch. Therefore the current through must be equal to the current through 3, but in the opposite direction. e e I, I, I (d) After a long time, with no voltage source, the energy in the inductor will dissipate and no current will flow through any of the branches. I I I3 7. (a) We use Eq. 35 to determine the emf in the inductor as a function of time. Since the exponential term decreases in time, the maximum emf occurs when t =. e di d t / I t / t / Ie e Ve max V dt dt e. (b) The current is the same just before and just after the switch moves from A to B. We use Ohm s law for a steady state current to determine I before the switch is thrown. After the switch is thrown, the same current flows through the inductor, and therefore that current will flow through the resistor. Using Kirchhoff s loop rule we calculate the emf in the inductor. This will be a maximum at t =. V V t / 55 I, ei e e e max V V 6.6kV 8. The steady state current is the voltage divided by the resistance while the time constant is the inductance divided by the resistance, Eq. 3. To cut the time constant in half, we must double the resistance. If the resistance is doubled, we must double the voltage to keep the steady state current constant. 44 V V 4 V 48 V 9. We use Kirchhoff s loop rule in the steady state (no voltage drop across the inductor) to determine the current in the circuit just before the battery is removed. This will be the maximum current after the battery is removed. Again using Kirchhoff s loop rule, with the current given by Eq. 3, we calculate the emf as a function of time. V V I I 5  t/ t/ t.k / 8mH. s t ei e I e V e V e V e e The emf across the inductor is greatest at t = with a value of e max V. 9 Pearson Education, Inc., Upper Saddle iver, NJ. All rights reserved. This material is protected under all copyright laws as they 77
9 Physics for Scientists & Engineers with Modern Physics, 4 th Edition Instructor Solutions Manual N A 3. We use the inductance of a solenoid, as derived in Example 33:. sol l (a) Both solenoids have the same area and the same length. Because the wire in solenoid is.5 times as thick as the wire in solenoid, solenoid will have.5 times the number of turns as solenoid. N A N N l N A N N l (b) To find the ratio of the time constants, both the inductance and resistance ratios need to be known. Since solenoid has.5 times the number of turns as solenoid, the length of wire used to make solenoid is.5 times that used to make solenoid, or l.5 l, and the wire wire diameter of the wire in solenoid is.5 times that in solenoid, or d.5d. Use this to wire wire find their relative resistances, and then the ratio of time constants. l lwire l wire wire A d wire wire d l d wire wire wire 3 l l l wire wire wire l d wire wire Awire d wire dwire ; = (a) The AM station received by the radio is the resonant frequency, given by Eq We divide the resonant frequencies to create an equation relating the frequencies and capacitances. We then solve this equation for the new capacitance. f f 55 khz 35pF.6nF f f 6kHz (b) The inductance is obtained from Eq f 6 H 3 4 f 4 55 Hz 35 F 3. (a) To have maximum current and no charge at the initial time, we set t = in Eqs. 33 and 35 to solve for the necessary phase factor. I Isin I( t) Isint Icost Q Qcos QQcost Qsin t Differentiating the charge with respect to time gives the negative of the current. We use this to write the charge in te of the known maximum current. dq I I I Q costicos t Q Q( t) sin t dt 9 Pearson Education, Inc., Upper Saddle iver, NJ. All rights reserved. This material is protected under all copyright laws as they 78
10 hapter 3 Inductance, Electromagnetic Oscillations, and A ircuits (b) As in the figure, attach the inductor to a battery and resistor for an extended period so that a steady state current flows through the inductor. Then at time t =, flip the switch connecting the inductor in series to the capacitor. 33. (a) We write the oscillation frequency in te of the capacitance using Eq. 34, with the parallel plate capacitance given by Eq. 4. We then solve the resulting equation for the plate separation distance. f x 4 A f A/ x (b) For small variations we can differentiate x and divide the result by x to determine the fractional change. dx 4 A fdf df x f dx 4 A fdf ; x 4 A f f x f (c) Inserting the given data, we can calculate the fractional variation on x. x Hz 6.% x MHz 34. (a) We calculate the resonant frequency using Eq f 8,45 Hz 8.5 khz.75 H45 F (b) As shown in Eq. 35, we set the peak current equal to the maximum charge (from Eq. 4) multiplied by the angular frequency. I Q V f 45 F 35 V 8,45 Hz (c) A 6.65 ma We use Eq. 36 to calculate the maximum energy stored in the inductor. 3 U I.75 H A 3.87 J 35. (a) When the energy is equally shared between the capacitor and inductor, the energy stored in the capacitor will be one half of the initial energy in the capacitor. We use Eq. 45 to write the energy in te of the charge on the capacitor and solve for the charge when the energy is equally shared. Q Q Q Q (b) We insert the charge into Eq. 33 and solve for the time. T T Q Qcost t cos Pearson Education, Inc., Upper Saddle iver, NJ. All rights reserved. This material is protected under all copyright laws as they 79
11 Physics for Scientists & Engineers with Modern Physics, 4 th Edition Instructor Solutions Manual 36. Since the circuit loses 3.5% of its energy per cycle, it is an underdamped oscillation. We use Eq. 45 for the energy with the charge as a function of time given by Eq Setting the change in energy equal to 3.5% and using Eq. 38 to determine the period, we solve for the resistance. Q T Qe cos cos E T T.35 ln(.35).3563 e E Q cos H F As in the derivation of 36, we set the total energy equal to the sum of the magnetic and electric energies, with the charge given by Eq We then solve for the time that the energy is 75% of the initial energy. Q I Q t Q t Q t U UE UB e cos t e sin t e Q Q t.75 e t ln.75 ln As shown by Eq. 38, adding resistance will decrease the oscillation frequency. We use Eq. 34 for the pure circuit frequency and Eq. 38 for the frequency with added resistance to solve for the resistance. (.5) H k 9.8 F 39. We find the frequency from Eq. 33b for the reactance of an inductor. X 66 X f f 383Hz 33 Hz.3 H 4. The reactance of a capacitor is given by Eq. 35b, (a) (b) X 6 X 9 f 6. Hz 9. F 6 6. f.7 X f. Hz 9. F 9 Pearson Education, Inc., Upper Saddle iver, NJ. All rights reserved. This material is protected under all copyright laws as they 8
12 hapter 3 Inductance, Electromagnetic Oscillations, and A ircuits 4. The impedance is X. The f extreme values are as follows. X max 6 Hz. F X 6, Hz. F min 6 6 eactance (k ) Frequency (Hz) The spreadsheet used for this problem can be found on the Media Manager, with filename PSE4_ISM_H3.XS, on tab Problem We find the reactance from Eq. 33b, and the current from Ohm s law. 3 X f 33.3 Hz.36 H V V IX I X V.339 A 3.3 A 43. (a) At, the impedance of the capacitor is infinite. Therefore the parallel combination of the resistor and capacitor behaves as the resistor only, and so is. Thus the impedance of the entire circuit is equal to the resistance of the two series resistors. Z (b) At,, the impedance of the capacitor is zero. Therefore the parallel combination of the resistor and capacitor is equal to zero. Thus the impedance of the entire circuit is equal to the resistance of the series resistor only. Z 44. We use Eq. 3a to solve for the impedance. V V V I I 3.A 6 Hz 94mH 45. (a) We find the reactance from Eq. 35b. X f 66 Hz8.6 F (b) We find the peak value of the current from Ohm s law. V, V I I A at 66Hz peak X (a) Since the resistor and capacitor are in parallel, they will have the same voltage drop across them. We use Ohm s law to determine the current through the resistor and Eq. 35 to determine the current across the capacitor. The total current is the sum of the currents across each element. 9 Pearson Education, Inc., Upper Saddle iver, NJ. All rights reserved. This material is protected under all copyright laws as they 8
13 Physics for Scientists & Engineers with Modern Physics, 4 th Edition Instructor Solutions Manual V V I ; I V f X Hz.35 F 6 I V f f I I V f V f 49 6 Hz.35 F.67 6.% (b) We repeat part (a) with a frequency of 6, Hz. 6 I 49 6, Hz.35 F % I I , Hz.35 F 47. The power is only dissipated in the resistor, so we use the power dissipation equation obtained in section 57. avg I P.8A 35 87W.9kW 48. The impedance of the circuit is given by Eq. 38a without a capacitive reactance. The reactance of the inductor is given by Eq. 33b. 3 (a) Z X 4 f Hz.6 H (b) Z X 4 f Hz.6 H The impedance of the circuit is given by Eq. 38a without an inductive reactance. The reactance of the capacitor is given by Eq. 35b. (a) Z X f 4 sig. fig. (b) Z X Hz 6.8 F f 4 6 Hz 6.8 F 5. We find the impedance from Eq V V Z 7 3 I 7 A 5. The impedance is given by Eq. 38a with no capacitive reactance. Z X f 4 4 6Hz f 6 Z Z f 4 f Hz Hz 9 Pearson Education, Inc., Upper Saddle iver, NJ. All rights reserved. This material is protected under all copyright laws as they 8
14 hapter 3 Inductance, Electromagnetic Oscillations, and A ircuits Hz f 46Hz 4 6Hz H 645 Hz.6 khz 5. (a) The current is the rmv voltage divided by the impedance. The impedance is given by Eq. 38a with no inductive reactance, Z X. f I V V Z 38 4 f Hz.8 F V.379 A.4 A V 543 (b) The phase angle is given by Eq. 39a with no inductive reactance. 6 X 6.Hz.8 F f tan tan tan 4 38 The current is leading the source voltage. (c) The power dissipated is given by P I.379A W (d) The voltage reading is the current times the resistance or reactance of the element. V I.379 A V 9 V sig. fig..379 A I V I X 78.88V 79 V 6 f 6. Hz.8 F Note that, because the maximum voltages occur at different times, the two readings do not add to the applied voltage of V. 53. We use the voltage across the resistor to determine the current through the circuit. Then, using the current and the voltage across the capacitor in Eq. 35 we determine the frequency. V, I I V, f I V, 3. V f 4 Hz 6 V V V,, Since the voltages in the resistor and capacitor are not in phase, the voltage across the power source will not be the sum of their voltages. 9 Pearson Education, Inc., Upper Saddle iver, NJ. All rights reserved. This material is protected under all copyright laws as they 83
15 Physics for Scientists & Engineers with Modern Physics, 4 th Edition Instructor Solutions Manual 54. The total impedance is given by Eq. 38a. Z X X f f Hz3. H 4 9. Hz6.5 F k The phase angle is given by Eq. 39a. f X X f tan tan 4. Hz3. H. Hz tan 6.5 F tan The voltage is lagging the current, or the current is leading the voltage. The current is given by Eq V 75V I 8.3 A Z (a) The current is the voltage divided by the impedance. The impedance is given by Eq. 38a with no capacitive reactance. Z X f. V V V I Z 4 f Hz.5H V.4 A (b) The phase angle is given by Eq. 39a with no capacitive reactance. X 6. Hz.5H f tan tan tan The current is lagging the source voltage. (c) The power dissipated is given by P I.4 A W (d) The voltage reading is the current times the resistance or reactance of the element. V I.4 A V V V I X I f.4 A 6. Hz.5H.5V Note that, because the maximum voltages occur at different times, the two readings do not add to the applied voltage of V. 9 Pearson Education, Inc., Upper Saddle iver, NJ. All rights reserved. This material is protected under all copyright laws as they 84
16 hapter 3 Inductance, Electromagnetic Oscillations, and A ircuits 56. (a) The current is found from the voltage and impedance. The impedance is given by Eq. 38a. Z X X f f. 6 Hz.35H Hz6 F V 45V I.565A.5A Z (b) Use Eq. 39a to find the phase angle. f X X f tan tan 6Hz.35H 6 6Hz 6 F tan tan 88.. (c) The power dissipated is given by P I.565A..5W 57. For the current and voltage to be in phase, the reactances of the capacitor and inductor must be equal. Setting the two reactances equal enables us to solve for the capacitance. X f X 7.8F f 4 f 4 36Hz.5H 58. The light bulb acts like a resistor in series with the inductor. Using the desired voltage across the resistor and the power dissipated by the light bulb we calculate the current in the circuit and the resistance. Then using this current and the voltage of the circuit we calculate the impedance of the circuit (Eq. 37) and the required inductance (Eq. 38b). P 75W V, V I.65A 9 V V I.65A V, Z f I V 4 V 9.88 H f I 6 Hz.65 A 59. We multiply the instantaneous current by the instantaneous voltage to calculate the instantaneous power. Then using the trigonometric identity for the summation of sine arguments (inside back cover of text) we can simplify the result. We integrate the power over a full period and divide the result by the period to calculate the average power. P IV I sint V sin t I V sint sintcos sincost IV sin tcos sintcostsin 9 Pearson Education, Inc., Upper Saddle iver, NJ. All rights reserved. This material is protected under all copyright laws as they 85
17 Physics for Scientists & Engineers with Modern Physics, 4 th Edition Instructor Solutions Manual T P PdT I V sin tcos sin tcos tsin dt T IV cos sin tdt IVsin sintcos tdt IV cos IV sin sin t IV cos 6. Given the resistance, inductance, capacitance, and frequency, we calculate the impedance of the circuit using Eq. 38b. X f66 Hz.5 H 3.67 X.57 f 6 66 Hz. F Z X X (a) From the impedance and the peak voltage we calculate the peak current, using Eq V 34 V I.5 A.3 A Z 5.95 (b) We calculate the phase angle of the current from the source voltage using Eq. 39a. X X tan tan (c) We multiply the peak current times the resistance to obtain the peak voltage across the resistor. The voltage across the resistor is in phase with the current, so the phase angle is the same as in part (b). V, I.5 A5 34 V ; 6.4 (d) We multiply the peak current times the inductive reactance to calculate the peak voltage across the inductor. The voltage in the inductor is 9º ahead of the current. Subtracting the phase difference between the current and source from the 9º between the current and inductor peak voltage gives the phase angle between the source voltage and the inductive peak voltage. V, IX.5 A V (e) We multiply the peak current times the capacitive reactance to calculate the peak voltage across the capacitor. Subtracting the phase difference between the current and source from the 9º between the current and capacitor peak voltage gives the phase angle between the source voltage and the capacitor peak voltage. V, IX.5 A.57 7 V Using Eq. 33b we calculate the impedance of the inductor. Then we set the phase shift in Eq. 39a equal to 5º and solve for the resistance. We calculate the output voltage by multiplying the current through the circuit, from Eq. 37, by the inductive reactance (Eq. 33b). X f75 Hz.55 H 6.48 X X 6.48 tan tan tan 5 9 Pearson Education, Inc., Upper Saddle iver, NJ. All rights reserved. This material is protected under all copyright laws as they 86
18 hapter 3 Inductance, Electromagnetic Oscillations, and A ircuits Voutput V I 9.7 V V IZ Z The resonant frequency is found from Eq The resistance does not influence the resonant frequency. f 6. H 38 F Hz 63. We calculate the resonant frequency using Eq. 33 with the inductance and capacitance given in the example. We use Eq. 33 to calculate the power dissipation, with the impedance equal to the resistance. f 65 Hz 6.3 H. F 9.V V V P IV cos V 34W (a) We find the capacitance from the resonant frequency, Eq f f H33. Hz (b) At resonance the impedance is the resistance, so the current is given by Ohm s law. Vpeak 36 V I 35.8 ma peak F 65. (a) The peak voltage across the capacitor is the peak current multiplied by the capacitive reactance. We calculate the current in the circuit by dividing the source voltage by the impedance, where at resonance the impedance is equal to the resistance. V V V V XI T f f (b) We set the amplification equal to 5 and solve for the resistance. T 3 9 f f 5 Hz 5. F 66. (a) We calculate the resonance frequency from the inductance and capacitance using Eq.33. f 46 Hz khz 9.55 H. F (b) We use the result of Problem 65 to calculate the voltage across the capacitor. V. V V 4 V 9 f 35. F46 Hz (c) We divide the voltage across the capacitor by the voltage source. V 4 V V. V 9 Pearson Education, Inc., Upper Saddle iver, NJ. All rights reserved. This material is protected under all copyright laws as they 87
19 Physics for Scientists & Engineers with Modern Physics, 4 th Edition Instructor Solutions Manual 67. (a) We write the average power using Eq. 33, with the current in te of the impedance (Eq. 37) and the power factor in te of the resistance and impedance (Eq. 39b). Finally we write the impedance using Eq. 38b. V V V cos P I V Z V Z Z (b) The power dissipation will be a maximum when the inductive reactance is equal to the capacitive reactance, which is the resonant frequency. f (c) We set the power dissipation equal to ½ of the maximum power dissipation and solve for the angular frequencies. V V P Pmax 4 We require the angular frequencies to be positive and for a sharp peak, 4. The angular width will then be the difference between the two positive frequencies. 68. (a) We write the charge on the capacitor using Eq. 4, where the voltage drop across the capacitor is the inductive capacitance multiplied by the circuit current (Eq. 35a) and the circuit current is found using the source voltage and circuit impedance (Eqs. 37 and 38b). V V V Q V IX X Z (b) We set the derivative of the charge with respect to the frequency equal to zero to calculate the frequency at which the charge is a maximum. 3 dq d V V 4 4 / 3 d d (c) The amplitude in a forced damped harmonic oscillation is given by Eq This is equivalent to the circuit with F V, k /, m, and b. 69. Since the circuit is in resonance, we use Eq. 33 for the resonant frequency to determine the necessary inductance. We set this inductance equal to the solenoid inductance calculated in Example 33, with the area equal to the area of a circle of radius r, the number of turns equal to the length of the wire divided by the circumference of a turn, and the length of the solenoid equal to the diameter of the wire multiplied by the number of turns. We solve the resulting equation for the number of turns. 9 Pearson Education, Inc., Upper Saddle iver, NJ. All rights reserved. This material is protected under all copyright laws as they 88
20 hapter 3 Inductance, Electromagnetic Oscillations, and A ircuits f lwire r N A r 4 fo l Nd Hz. F4 Tm A.m fol wire N 3 d. m 37 loops 7. The power on each side of the transformer must be equal. We replace the currents in the power equation with the number of turns in the two coils using Eq Then we solve for the turn ratio. Z p I N s p Pp IpZp Ps IsZs Z s I p Ns N 3 p Zp N Z 8. s s 7. (a) We calculate the inductance from the resonance frequency. f.398 H.4H 3 4 f 9 o 4 7 Hz. F (b) We set the initial energy in the electric field, using Eq. 45, equal to the maximum energy in the magnetic field, Eq. 36, and solve for the maximum current. V V I max Imax 9. FV.3984 H.8A (c) The maximum energy in the inductor is equal to the initial energy in the capacitor. 9,max U V. F V 6J 7. We use Eq. 36 to calculate the initial energy stored in the inductor. 5 U I.6H.5A 7.5 J We set the energy in the inductor equal to five times the initial energy and solve for the current. We set the current equal to the initial current plus the rate of increase multiplied by time and solve for the time. U J U I I.8mA.6H I I.8mA 5.mA I I t t.79s 78.mA/s 73. When the currents have acquired their steadystate values, the capacitor will be fully charged, and so no current will flow through the capacitor. At this time, the voltage drop across the inductor will be zero, as the current flowing through the inductor is constant. Therefore, the current through is zero, and the resistors and 3 can be treated as in series. 9 Pearson Education, Inc., Upper Saddle iver, NJ. All rights reserved. This material is protected under all copyright laws as they 89 5
21 Physics for Scientists & Engineers with Modern Physics, 4 th Edition Instructor Solutions Manual V V I I.4 ma ; I k 74. (a) The self inductance is written in te of the magnetic flux in the toroid using Eq We set the flux equal to the magnetic field of a toroid, from Example 8. The field is dependent upon the radius of the solenoid, but if the diameter of the solenoid loops is small compared with the radius of the solenoid, it can be treated as approximately constant. N N d 4 NI r B N d I I 8r This is consistent with the inductance of a solenoid for which the length is l r. (b) We calculate the value of the inductance from the given data, with r equal to half of the diameter. 7 4 T m/a55. m N d 58 H 8r 8.33 m 75. We use Eq. 34 to calculate the self inductance between the two wires. We calculate the flux by integrating the magnetic field from the two wires, using Eq. 8, over the region between the two wires. Dividing the inductance by the length of the wire gives the inductance per unit length. lr r B I I h l hdr dr I I r r r r l r lr lr lr r lr ln rln r ln ln ln h l r r r l r 76. The magnetic energy is the energy density (Eq. 37) multiplied by the volume of the spherical shell enveloping the earth. 4 B.5 T U ubv 4r h m 5. m.5 J 4 TmA 77. (a) For underdamped oscillation, the charge on the capacitor is given by Eq. 39, with. Differentiating the current with respect to time gives the current in the circuit. t dq t Qt ( ) Qe cos t ; It ( ) Qe cost sint dt The total energy is the sum of the energies stored in the capacitor (Eq. 45) and the energy stored in the inductor (Eq. 36). Since the oscillation is underdamped ( / ), the cosine term in the current is much smaller than the sine term and can be ignored. The frequency of oscillation is approximately equal to the undamped frequency of Eq t t Q I Qe cost Qe sint U U U t Qe Qe cos t sin t t 9 Pearson Education, Inc., Upper Saddle iver, NJ. All rights reserved. This material is protected under all copyright laws as they 9
22 hapter 3 Inductance, Electromagnetic Oscillations, and A ircuits (b) We differentiate the energy with respect to time to show the average power dissipation. We then set the power loss per cycle equal to the resistance multiplied by the square of the current. For a lightly damped oscillation, the exponential term does not change much in one cycle, while the sine squared term averages to ½. t t du d Qe Qe dt dt t t t Q e PI Q e sin tq e The change in power in the circuit is equal to the power dissipated by the resistor. 78. Putting an inductor in series with the device will protect it from sudden surges in current. The growth of current in an circuit is given is Eq V t max t I e I e The maximum current is 33 ma, and the current is to have a value of 7.5 ma after a time of 75 microseconds. Use this data to solve for the inductance. I I e e max I t t I max 6 75 sec5 t 7.5mA ln I ln I 33mA max 4.4 H Put an inductor of value 4.4 H in series with the device. 79. We use Kirchhoff s loop rule to equate the input voltage to the voltage drops across the inductor and t resistor. We then multiply both sides of the equation by the integrating factor e and integrate the t t t righthand side of the equation using a u substitution with u Ie and du die Ie dt di Vin I dt di dt t For / t, e. Setting the exponential term equal to unity on both sides of the equation gives the desired results. Vindt Vout t t t t Vine dt Ie dt du I e Vout e 8. (a) Since the capacitor and resistor are in series, the impedance of the circuit is given by Eq. 38a. Divide the source voltage by the impedance to determine the current in the circuit. Finally, multiply the current by the resistance to determine the voltage drop across the resistor. Vin Vin V I Z f 9 Pearson Education, Inc., Upper Saddle iver, NJ. All rights reserved. This material is protected under all copyright laws as they 9
23 Physics for Scientists & Engineers with Modern Physics, 4 th Edition Instructor Solutions Manual 3 mv Hz. 6 F 3 mv (b) epeat the calculation with a frequency of 6. khz. 3 mv55 V 3 mv 55 6 Hz. 6 F Thus the capacitor allows the higher frequency to pass, but attenuates the lower frequency. 8. (a) We integrate the power directly from the current and voltage over one cycle. T P IVdt Isin tvsin t 9 dt Isin tvcos tdt T sin t IV IV sin 4 (b) We apply Eq. 33, with 9. 9 Pearson Education, Inc., Upper Saddle iver, NJ. All rights reserved. This material is protected under all copyright laws as they 9 sin P I V cos9 As expected the average power is the same for both methods of calculation. 8. Since the current lags the voltage one of the circuit elements must be an inductor. Since the angle is less than 9º, the other element must be a resistor. We use 39a to write the resistance in te of the impedance. Then using Eq. 37 to determine the impedance from the voltage and current and Eq. 38b, we solve for the unknown inductance and resistance. f tan fcot V Z f fcot f f cot I V V = 5.5mH 5mH fi cot 6Hz 5.6A cot 65 f cot 6 Hz 5.5mH cot We use Eq. 38b to calculate the impedance at 6 Hz. Then we double that result and solve for the required frequency. Z f 35 6Hz.44 H 354 4Z Z f f.khz.44h 84. (a) We calculate capacitive reactance using Eq. 35b. Then using the resistance and capacitive reactance we calculate the impedance. Finally, we use Eq. 37 to calculate the current. X f 6.Hz.8 F Z X
24 hapter 3 Inductance, Electromagnetic Oscillations, and A ircuits V V I.38mA.4mA Z 5887 (b) We calculate the phase angle using Eq. 39a. X 474 tan tan (c) The average power is calculated using Eq P I V cos.4avcos W (d) The voltmeter will read the voltage across each element. We calculate the voltage by multiplying the current through the element by the resistance or capacitive reactance. V I.38mA 5.7k 6V V I X.38mA V Note that since the voltages are out of phase they do not sum to the applied voltage. However, since they are 9º out of phase their squares sum to the square of the input voltage. 85. We find the resistance using Ohm s law with the dc voltage and current. When then calculate the impedance from the ac voltage and current, and using Eq. 38b. V 45V V V 8 ; Z 3.58 I.5A I 3.8A Z f 69mH f 6Hz 86. (a) From the text of the problem, the Q factor is the ratio of the voltage across the capacitor or inductor to the voltage across the resistor, at resonance. The resonant frequency is given by Eq V I X f res Q V I res (b) Find the inductance from the resonant frequency, and the resistance from the Q factor. f H.5 H f 4. F. Hz Q H Q 35. F 87. We calculate the period of oscillation as divided by the angular frequency. Then set the total energy of the system at the beginning of each cycle equal to the charge on the capacitor as given by Eq. 45, with the charge given by Eq. 39, with cost cos tt. We take the difference in energies at the beginning and end of a cycle, divided by the initial energy. For small damping, the argument of the resulting exponential term is small and we replace it with the first two te of the Taylor series expansion. 9 Pearson Education, Inc., Upper Saddle iver, NJ. All rights reserved. This material is protected under all copyright laws as they 93
25 Physics for Scientists & Engineers with Modern Physics, 4 th Edition Instructor Solutions Manual Qe t Qe T Umax t t e t Qe t cos U Q e Q e U Q 88. We set the power factor equal to the resistance divided by the impedance (Eq. 38a) with the impedance written in te of the angular frequency (Eq. 38b). We rearrange the resulting equation to form a quadratic equation in te of the angular frequency. We divide the positive angular frequencies by to determine the desired frequencies. cos Z cos H55 F55 F F H 4.78 F ΩF ΩF rad s,. 7 rad s FH rad s.7 rad s f 4kHz and 33Hz 89. (a) We set V V sint and assume the inductive reactance is greater than the capacitive reactance. The current will lag the voltage by an angle. The voltage across the resistor is in phase with the current and the voltage across the inductor is 9º ahead of the current. The voltage across the capacitor is smaller than the voltage in the inductor, and antiparallel to it. (b) From the diagram, the current is the projection of the maximum current onto the y axis, with the current lagging the voltage by the angle. This is the same angle obtained in Eq. 39a. The magnitude of the maximum current is the voltage divided by the impedance, Eq. 38b. V It ( ) Isint sin t ; tan 9. (a) We use Eq. 38b to calculate the impedance and Eq. 39a to calculate the phase angle. X 754rad s.h 6.59 X 6 754rad s.4 F Z X X k X X tan tan t 9 Pearson Education, Inc., Upper Saddle iver, NJ. All rights reserved. This material is protected under all copyright laws as they 94
26 hapter 3 Inductance, Electromagnetic Oscillations, and A ircuits (b) We use Eq. 33 to obtain the average power. We obtain the voltage by dividing the maximum voltage by. The current is the voltage divided by the impedance. V V.95V 3 P I V cos cos cos cos W Z Z 3.4 (c) The current is the peak voltage, divided by, and then divided by the impedance. V.95V 5 I.87 A 9 A 3 Z 3.4 The voltage across each element is the current times the resistance or reactance of the element. 5 3 V I.87 A 3..67V 5 V I X.87 A 358.9V 5 4 V I X.87 A V 9. (a) The impedance of the circuit is given by Eq. 38b with X X and. We divide the magnitude of the ac voltage by the impedance to get the magnitude of the ac current in the circuit. Since X X, the voltage will lead the current by. No dc current will flow through the capacitor. V V Z I Z V I t sin t (b) The voltage across the capacitor at any instant is equal to the charge on the capacitor divided by the capacitance. This voltage is the sum of the ac voltage and dc voltage. There is no dc voltage drop across the inductor so the dc voltage drop across the capacitor is equal to the input dc voltage. Vout,ac V Q out V V We treat the emf as a superposition of the ac and dc components. At any instant of time the sum of the voltage across the inductor and capacitor will equal the input voltage. We use Eq. 35 to calculate the voltage drop across the inductor. Subtracting the voltage drop across the inductor from the input voltage gives the output voltage. Finally, we subtract off the dc voltage to obtain the ac output voltage. di d V V V sin t cos t dt dt V sint V Vout Vin V V V sint sint VV sintvv sint 9 Pearson Education, Inc., Upper Saddle iver, NJ. All rights reserved. This material is protected under all copyright laws as they 95
27 Physics for Scientists & Engineers with Modern Physics, 4 th Edition Instructor Solutions Manual V Vout,ac Vout V V sin t t (c) The attenuation of the ac voltage is greatest when the denominator is large. X X sin We divide the output ac voltage by the input ac voltage to obtain the attenuation. V V,out V,in V (d) The dc output is equal to the dc input, since there is no dc voltage drop across the inductor. V V,out 9. Since no dc current flows through the capacitor, there will be no dc current through the resistor. Therefore the dc voltage passes through the circuit with little attenuation. The ac current in the circuit is found by dividing the input ac voltage by the impedance (Eq. 38b) We obtain the output ac voltage by multiplying the ac current by the capacitive reactance. Dividing the result by the input ac voltage gives the attenuation. VX V,out V,out IX V X 93. (a) Since the three elements are connected in parallel, at any given instant in time they will all three have the same voltage drop across them. That is the voltages across each element will be in phase with the source. The current in the resistor is in phase with the voltage source with magnitude given by Ohm s law. V I () t sint (b) The current through the inductor will lag behind the voltage by /, with magnitude equal to the voltage source divided by the inductive reactance. V I () t sint X (c) The current through the capacitor leads the voltage by /, with magnitude equal to the voltage source divided by the capacitive reactance. V I () t sint X (d) The total current is the sum of the currents through each element. We use a phasor diagram to add the currents, as was used in Section 38 to add the voltages with different phases. The net current is found by subtracting the current through the inductor from the current through the capacitor. Then using the Pythagorean theorem to add the current through the resistor. We use the tangent function to find the phase angle between the current and voltage source. V V V V X X I I I I 9 Pearson Education, Inc., Upper Saddle iver, NJ. All rights reserved. This material is protected under all copyright laws as they 96
28 hapter 3 Inductance, Electromagnetic Oscillations, and A ircuits V () sin It t V V X X tan tan tan V X X (e) We divide the magnitude of the voltage source by the magnitude of the current to find the impedance. (f) Z V V I V 9 Pearson Education, Inc., Upper Saddle iver, NJ. All rights reserved. This material is protected under all copyright laws as they 97 The power factor is the ratio of the power dissipated in the circuit divided by the product of the voltage and current. V I, I V I VI V V 94. We find the equivalent values for each type of element in series. From the equivalent values we calculate the impedance using Eq. 38b. eq eq eq eq eq eq Z 95. If there is no current in the secondary, there will be no induced emf from the mutual inductance. Therefore, we set the ratio of the voltage to current equal to the inductive reactance and solve for the inductance. V V V X f.4 H I f I 6 Hz 4.3 A 96. (a) We use Eq. 4 to calculate the capacitance, assuming a parallel plate capacitor. K N m. 4 m oa.3 F. pf 3 d. m (b) We use Eq. 35b to calculate the capacitive reactance. 6 X M f Hz. F (c) Assuming that the resistance in the plasma and in the person is negligible compared with the capacitive reactance, calculate the current by dividing the voltage by the capacitive reactance. Vo 5 V 4 Io 4.7 A.4mA 6 X This is not a dangerous current level.
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