CHAPTER 23: ELECTROMAGNETIC INDUCTION, AC CIRCUITS, AND ELECTRICAL TECHNOLOGIES

Transcription

1 College Phyic Student Manual Chapter CHAPTE : EECTOMAGNETC NDUCTON, AC CCUTS, AND EECTCA TECHNOOGES. NDUCED EMF AND MAGNETC FUX. What i the value of the magnetic flux at coil in Figure.5 due to coil? Uing the equation Φ BAcoθ, we can calculate the flux through coil, ince the coil are perpendicular: Φ BAco θ BAco9. FAADAY S AW OF NDUCTON: ENZ S AW 7. erify that the unit of Δ Φ / Δ t are volt. That i, how that T m /. ΔΦ The unit of will be: [ Φ] [ ] T m m Δ N m A N m C ( N A m) o that T m /. A lightning bolt produce a rapidly varying magnetic field. f the bolt trike the earth vertically and act like a current in a long traight wire, it will induce a voltage in a loop aligned like that in Figure.57(b). What voltage i induced in a. m diameter loop 5. m from a. A lightning trike, if the current fall to zero in 5.µ? (b) Dicu circumtance under which uch a voltage would produce noticeable conequence. NΔΦ (a) We know E, where the minu ign mean that the emf create the current and magnetic field that oppoe the change in flux, and Φ BA ( πr )B. 7

2 College Phyic Student Manual Chapter Since the only thing that varie in the magnetic flux i the magnetic field, we can µ then ay that ΔΦ π r ΔB. Now, ince B the change in magnetic field πd µ occur becaue of a change in the current, or Δ B Δ. Finally, ubtituting πd NΔΦ thee into the equation E give: E Nπr µ Δ πd µ Nr Δ d 7 ( π T m/a)( )(.5 m) (. A) 5 ( 5. m)(.5 ) 5 (b) An example i the alternator in your car. f you were driving during a lightning torm and thi large bolt of lightning hit at 5. m away, it i poible to fry the alternator of your battery becaue of thi large voltage urge. n addition, the hair at the back of your neck would tand on end becaue it become tatically charged.. MOTONA EMF. f a current flow in the Satellite Tether hown in Figure., ue Faraday law, enz law, and H- to how that there i a magnetic force on the tether in the direction oppoite to it velocity. The flux through the loop (into the page) i increaing becaue the loop i getting larger and encloing more magnetic field. return path v tether hip Thu, a magnetic field (out of the page) i induced to oppoe the change in flux from the original field. Uing H-, point your finger out of the page within the loop, then your thumb point in the counterclockwie direction around the loop, o the induced magnetic field i produced by the induction of a counterclockwie current in 7

3 College Phyic Student Manual Chapter the circuit. Finally, uing H-, putting your right thumb in the direction of the current and your finger into the page (in the direction of the magnetic field), your palm point to the left, o the magnetic force on the wire i to the left (in the direction oppoite to it velocity).. EDDY CUENTS AND MAGNETC DAMPNG 7. A coil i moved through a magnetic field a hown in Figure.59. The field i uniform inide the rectangle and zero outide. What i the direction of the induced current and what i the direction of the magnetic force on the coil at each poition hown? (a) The magnetic field i zero and not changing, o there i no current and therefore no force on the coil. (b) The magnetic field i increaing out of the page, o the induced magnetic field i into the page, created by an induced clockwie current. Thi current create a force to the left. (c) The magnetic field i not changing, o there i no current and therefore no force on the coil. (d) The magnetic field i decreaing out of the page, o the induced magnetic field i out of the page, created by an induced counterclockwie current. Thi current create a force to the right. (e) The magnetic field i zero and not changing, o there i no current and therefore no force on the coil..5 EECTC GENEATOS. What i the peak emf generated by a.5 m radiu, 5- turn coil i rotated one- fourth of a revolution in.7 m, originally having it plane perpendicular to a uniform magnetic field. (Thi i rev/.) Uing the information given in Exercie.: Δθ rev N 5; A π r ( π rad) π and.7, (.5 m) ; and B.5 T, 75

4 College Phyic Student Manual Chapter we get: ω Δθ ( / )( π ) rad rad/. Therefore, E. BACK EMF ( 5)( )(.5 m) (.5 T)( 7.7 rad/) k NABω π 9. Suppoe a motor connected to a ource draw. A when it firt tart. (a) What i it reitance? (b) What current doe it draw at it normal operating peed when it develop a back emf? (a) Uing the equation. A, we can determine given the voltage and the current:. Ω (b) Again, uing, we can now determine the current given that the net voltage i the difference between the ource voltage and the back emf:. A. Ω. The motor in a toy car i powered by four batterie in erie, which produce a total emf of.. The motor draw. A and develop a.5 back emf at normal peed. Each battery ha a.ω internal reitance. What i the reitance of the motor? Since the reitor are in erie, we know the total internal reitance of the batterie. Ω. Therefore, i ( ) E E..5, o that + ʹ ʹ + ʹ. A.7 TANSFOMES (. Ω). Ω 7

5 College Phyic Student Manual Chapter. A caette recorder ue a plug- in tranformer to convert to., with a maximum current output of ma. (a) What i the current input? (b) What i the power input? (c) thi amount of power reaonable for a mall appliance? N (a) Uing the equation and p, we can determine the primary p N p p p Np p current:, o that N. (. A). A. ma p p (b) P (. A). W in p p (c) Ye, thi amount of power i quite reaonable for a mall appliance..9 NDUCTANCE 55. Two coil are placed cloe together in a phyic lab to demontrate Faraday law of induction. A current of 5. A in one i witched off in. m, inducing a 9. emf in the other. What i their mutual inductance? Δ Uing the equation E M, where the minu ign i an expreion of enz law, we can calculate the mutual inductance between the two coil: M E. 5. A Δ ( ) ( ) 9..8 mh. A large reearch olenoid ha a elf- inductance of 5. H. (a) What induced emf oppoe hutting it off when A of current through it i witched off in 8. m? (b) How much energy i tored in the inductor at full current? (c) At what rate in watt mut energy be diipated to witch the current off in 8. m? (d) n view of the anwer to the lat part, i it urpriing that hutting it down thi quickly i difficult? Δ (a) Uing the equation E, we have 77

6 College Phyic Student Manual Chapter Δ E ( 5. H) ( A) ( 8. ).5. k 5 ind (b) Uing E ( 5. H)( A).5 J (c) Uing the equation ΔE P, we have ΔE P Δ t 5.5 J.5 W.5 MW - 8. (d) No, it i not urpriing ince thi power i very high. 8. Unreaonable eult A 5. H inductor ha A of current turned off in. m. (a) What voltage i induced to oppoe thi? (b) What i unreaonable about thi reult? (c) Which aumption or premie i reponible? Δ (a) E ( 5. H) ( A).5. (b) The voltage i o extremely high that arcing would occur and the current would not be reduced o rapidly. (c) t i not reaonable to hut off uch a large current in uch a large inductor in uch an extremely hort time.. CCUTS 9. f you want a characteritic time contant of., and you have a 5 Ω reitor, what value of elf- inductance i needed? Uing the equation, τ we know τ (. )( 5Ω) 5 H 75. What percentage of the final current flow through an inductor in erie with a reitor, three time contant after the circuit i completed? 78

7 College Phyic Student Manual Chapter t We ue the equation e τ, becaue the problem ay, after the circuit i t completed. Thu, the final current i given by: e τ, where t τ o that: t τ e e.95 The current i 95.% of the final current after time contant.. EACTANCE, NDUCTE AND CAPACTE 8. What capacitance hould be ued to produce a. MΩ reactance at. Hz? Uing the equation X C, we can determine the neceary capacitance: πfc 9 C. F. nf πfx C π (. Hz)(. Ω) 87. (a) An inductor deigned to filter high- frequency noie from power upplied to a peronal computer i placed in erie with the computer. What minimum inductance hould it have to produce a. kω reactance for 5. khz noie? (b) What i it reactance at. Hz? (a) Uing the equation πf, X (b) Again uing X π f, or πf πf, π (. Ω) (.5 Hz). H. mh π f π. C SEES AC CCUTS - (. Hz)(. H) 8. Ω 95. What i the reonant frequency of a.5 mh inductor connected to a. µ F capacitor? 79

8 College Phyic Student Manual Chapter Uing the equation f, we can determine the reonant frequency for the π C circuit: f.5 Hz. khz - -5 π C π 5. H. F ( )( ). An C erie circuit ha a.5 Ω reitor, a µ H inductor, and an 8. µ F capacitor. (a) Find the circuit impedance at Hz. (b) Find the circuit impedance at 5. khz. (c) f the voltage ource ha 5., what i at each frequency? (d) What i the reonant frequency of the circuit? (e) What i at reonance? (a) The equation πf give the inductive reactance: π f π Hz - - ( )(. H) 7.5 Ω The equation X C give the capacitive reactance: πfc X C π f C π -5 ( Hz)( 8. F).58 Ω Finally, the equation ( ) circuit: Z + Z + X give the impedance of the C C ( ) (.5 Ω) + ( 7.5 Ω.58 Ω).7 Ω X C (b) Again, πfgive the inductive reactance: X - ( Hz)(. H). Ω π 5. X C give the capacitive reactance: πfc X C π - - ( 5. Hz)( 8. F).979 Ω And ( ) Z + X give the impedance: C Z (.5 Ω) + (. Ω.979 Ω).7Ω 8

9 College Phyic Student Manual Chapter (c) The current i found uing the equation. For f Hz, Z 5..9 Ω. A and for 5. f 5. khz,.5 A.7 Ω (d) The reonant frequency i found uing the equation f : π C f 5. Hz 5. khz π C π -7-5 (. H)( 8. F) (e) At reonance, X X, o that Z and 5..5 Ω. A reduce to: 8