HOMEWORK #10 Chapter 24

Transcription

1 HOMEWOK # hapte 4 7 Fin the capacitance o the paallel-plate capacito shown in Figue Pictue the Poblem We can moel this paallel-plate capacito as a combination o two capacitos an in seies with capacito in paallel. Expess the capacitance o two seies-connecte capacitos in paallel with a thi: s () whee s () Expess each o the capacitances,, an in tems o the ielectic constants, plate aeas, an plate sepaations: ( ), ( ), an ( ) Substitute in equation () to obtain: s Substitute in equation () to obtain:

2 76 Figue 4-46 shows ou capacitos connecte in the aangement known as a capacitance bige. The capacitos ae initially unchage. What must the elation between the ou capacitances be so that the potential ieence between points c an emains zeo when a voltage V is applie between points a an b? Pictue the Poblem Note that with V applie between a an b, an ae in seies, an so ae an 4. Because in a seies combination the potential ieences acoss the two capacitos ae invesely popotional to the capacitances, we can establish popotions involving the capacitances an potential ieences o the let- an ight-han sie o the netwok an then use the conition that V c V to eliminate the potential ieences an establish the elationship between the capacitances. Letting Q epesent the chage on capacitos an, elate the potential ieences acoss the capacitos to thei common chage an capacitances: Q V an V Q Divie the ist o these equations V () by the secon to obtain: V Pocee similaly to obtain: V 4 () V 4 Divie equation () by equation VV 4 () () to obtain: VV 4 I V c V then we must have: V V an V V4 Substitute in equation () an eaange to obtain: 4 9 n ai-ille paallel-plate capacito that has gap-with has plates which each have an aea. The capacito is chage to a potential ieence V an is then emove om the voltage souce. ielectic slab that has a ielectic constant o., a thickness, an an aea is then insete, as shown in Figue4-5. Let σ be the ee chage ensity at the conucto ielectic suace, an let σ be the ee chage ensity at the conucto ai suace. (a) Explain why the electic iel must have the same value insie the ielectic as in the ee space between the plates. (b) Show that σ σ. (c) Show that the inal

3 capacitance (ate the slab is insete) is.5 times the capacitance when the capacito is ille with ai. () Show that the inal potential ieence is V. (e) Show that enegy stoe ate the slab is insete is only two-this o the enegy stoe beoe insetion. Pictue the Poblem (b) We can expess the electic iels in the ielectic an in the ee space in tems o the chage ensities an then use the act that the electic iel has the same value insie the ielectic as in the ee space between the plates to establish that σ σ. In Pats (c) an () we can moel the system as two capacitos in paallel to show that the equivalent capacitance is /() an then use the einition o capacitance to show that the new potential ieence is V. (a) The potential ieence between the plates is the same o both halves (the plates ae equipotential suaces). Theeoe, E V/ must be the same eveywhee between the plates. (b) elate the electic iel in each σ E σ E egion to σ an : Expess σ an σ : σ E E an σ E E Divie the st o these equations by the n an simpliy to obtain: (c) Moel the patially ielecticille capacito as two capacitos in paallel to obtain: σ σ eq whee ( ) an ( ) Substitute o an an simpliy to obtain: eq.5ai-ille () Use the einition o capacitance to elate V, Q, an : Q V

4 Because the capacitos ae in paallel: Substitute o Q an simpliy to obtain: an (e) The enegy stoe ate the slab is insete is given by: Substituting o an V an simpliying yiels: V Qi Vi Q V V V U V V ( )( V) iv U i U i The pesence o the ielectic slab euces the potential ieence between the capacito plates an, hence, the enegy stoe in the capacito.

5 HPTE 5 86 In the cicuit shown in Figue 5-6, the batteies have negligible intenal esistance. Fin (a) the cuent in each banch o the cicuit, (b) the potential ieence between point a an point b, an (c) the powe supplie by each battey. Pictue the Poblem Let I Ω be the cuent elivee by the 7.-V battey, I Ω the cuent elivee by the 5-V battey, an I Ω, iecte up, the cuent though the.-ω esisto. We can apply Kichho s ules to obtain thee equations that we can solve simultaneously o I, I, an I. Knowing the cuents in each banch, we can use Ohm s law to in the potential ieence between points a an b an the powe elivee by both the souces. (a) pply Kichho s junction ule at junction a: I () Ω I Ω IΩ pply Kichho s loop ule to a loop aoun the outsie o the cicuit to obtain: 7. V (. Ω) IΩ (. Ω) I Ω () pply Kichho s loop ule to a loop aoun the let-han banch o the cicuit to obtain: (.Ω) I (.Ω) 7.V Ω I 5. V o.ω I.Ω I. () ( ) ( ) V Ω Ω Ω Solve equations (), (), an () simultaneously to obtain: I., I., Ω an I Ω. Ω (b) pply Ohm s law to in the potential ieence between points a an b: (c) Expess the powe elivee by the 7.-V battey: (c) Expess the powe elivee by the 5.-V battey: V ab P P 7 V 5 V 5.V. V (.Ω) I (.Ω)(. ) 5.V Ω I Ω. W I Ω. W ( 7. V)(. ) ( 5. V)(. )

6 6 In the cicuit shown in Figue 5-78,. MΩ, 5. MΩ, an. µf. The capacito is initially without chage on eithe plate. t t, switch S is close, an at t. s switch S is opene. (a) Sketch a gaph o the voltage acoss an the cuent in between t an t. s. (b) Fin the voltage acoss the capacito at t. s an at t 8. s. Pictue the Poblem We can apply both the loop an junction ules to obtain equations that we can use to obtain a linea ieential equation with constant coeicients escibing the cuent in as a unction o time. We can solve this ieential equation by assuming a solution o an appopiate om, ieentiating this assume solution an substituting it an its eivative in the ieential equation. Equating coeicients, equiing the solution to hol o all values o the assume constants, an invoking an initial conition will allow us to in the constants in the assume solution. Once we know how the cuent vaies with time in, we can expess the potential ieence acoss it (as well as acoss because they ae in paallel). To in the voltage acoss the capacito at t 8. s, we can expess the epenence o the voltage on time o a ischaging capacito ( is ischaging ate t. s) an evaluate this unction, with a time constant ieing om that oun in (a), at t 6. s. The iagam shows the cicuit shotly ate the switch is close. The iections o the cuents in the esistos an the capacito have been chosen as shown. S I V I Q I a (a) pply the junction ule at junction a to obtain: pply the loop ule to loop to obtain: I () I I Q I () pply the loop ule to loop to obtain: Q I ()

7 Dieentiate equation () with espect to time to obtain: Dieentiate equation () with espect to time to obtain: Using equation (), substitute o I in equation (5) to obtain: t o I t I Q I t I I t I (4) Q I I Q t t t o I I (5) t I t ( I I ) (6) Q t Solve equation () o I : I Q I Substitute o I in equation (6) an simpliy to obtain the ieential equation o I : To solve this linea ieential equation with constant coeicients we can assume a solution o the om: I t I I I t ( t) a be I (7) Dieentiate I (t) with espect to time to obtain: I t t t b t [ a be ] e Substitute o I an I /t to obtain: b e t t ( a be ) Equate coeicients o t e to obtain:

8 equiing the equation to hol o all a values o a yiels: I I is to be zeo when t : a b b a Substitute in equation (7) to obtain: I ( t) whee t ( e ) e t Substitute numeical values an evaluate : Substitute numeical values an evaluate I (t):- I (. MΩ)( 5. MΩ)(.µ F).4s. MΩ 5. MΩ.V t.49s ( t) ( e ). MΩ 5. MΩ t.4s (.49µ )( e ) Because an ae in paallel, they have a common potential ieence given by: ( ) ( ) ( ) ( )( )( t.49s.49s t V t I t MΩ e ) ( 7.4V)( e t ) V µ.49s Evaluate V at t. s: V ( s) ( 7.4V)( e.s ) 5.8V The voltage acoss the capacito as a unction o time is shown in the ollowing gaph. The cuent though the 5.-MΩ esisto ollows the same time couse, its value being V /(5. 6 ).

9 6 5 4 V (V) t (s) (b) The value o V at t. s has aleay been etemine to be: When S is opene at t. s, ischages though with a time constant given by: V (.s) 5.8 V 5.8V ( 5.MΩ)(. F) 5.s ' µ Expess the potential ieence acoss as a unction o time: V ( t. s) ' ( t) Ve ( t. s) 5.s ( 5.8V) e Evaluate V at t 8. s to obtain: V ( 8.s) ( 5.8V).6V e 6.s 5.s in goo ageement with the gaph.

10 HPTE 6 paticle has a chage q, a mass m, a linea momentum o magnitue p an a kinetic enegy K. The paticle moves in a cicula obit o aius pepenicula to a uniom magnetic iel B. Show that (a) p Bq an (b) K B q / m. Pictue the Poblem We can use the einition o momentum to expess p in tems o v an apply Newton s n law to the obiting paticle to expess v in tems o q, B,, an m. In Pat (b) we can expess the paticle s kinetic enegy in tems o its momentum an use ou esult om Pat (a) to show that K B q m. (a) Expess the momentum o the paticle: pply F ma to the obiting aial paticle to obtain: c p mv () v qvb m v qb m Substitute o v in equation () to obtain: (b) Expess the kinetic enegy o the obiting paticle as a unction o its momentum: qb p m m p K m qb Substitute ou esult o p om Pat (a) to obtain: K ( qb) m q B m 5 ectangula cuent-caying 5-tun coil, as shown in Figue 6-6, is pivote about the z axis. (a) I the wies in the z plane make an angle θ 7º with the y axis, what angle oes the magnetic moment o the coil make with the unit vecto ˆ i? (b) Wite an expession o n ˆ in tems o the unit vectos ˆ i an j ˆ, whee n ˆ is a unit vecto in the iection o the magnetic moment. (c) What is the magnetic moment o the coil? () Fin the toque on the coil when thee is a uniom magnetic iel B.5 T j ˆ in the egion occupie by the coil. (e) Fin the potential enegy o the coil in this iel. (The potential enegy is zeo when θ.) Pictue the Poblem The iagam shows the coil as it woul appea om along the positive z axis. The ight-han ule o etemining the iection o nˆ has been use to establish nˆ as shown. We can use the geomety o this igue to

11 etemine θ an to expess the unit nomal vecto nˆ. The magnetic moment o the coil is given by µ NInˆ an the toque exete on the coil by µ B. Finally, we can in the potential enegy o the coil in this iel om µ B U. y I 7 nˆ θ x (a) Noting that θ an the angle whose measue is 7 have thei ight an let sies mutually pepenicula, we can conclue that: (b) Use the components o nˆ to expess nˆ in tems o î an ĵ : (c) Expess the magnetic moment o the coil: θ 7 nˆ n iˆ n x.799ˆ i.6 ˆj y.8ˆ i.6 ˆj µ NInˆ ˆj cos7 iˆ sin 7 ˆj Substitute numeical values an evaluate µ : µ ( )( )( )( i.6 ˆ j) ( ) iˆ cm.799ˆ.5 m (.5 m ) ( ) iˆ.4 m (.5 m )j ˆ ˆj () Expess the toque exete on the coil: µ B Substitute o µ an B to obtain: {( ) iˆ.5 m (.5 m ) ˆj } (.5T) (.5 N m)( iˆ ˆj ) (.79 N m)( ˆj ˆj ) (.5 N m)kˆ ˆj (e) Expess the potential enegy o U µ B

12 the coil in tems o its magnetic moment an the magnetic iel: Substitute o µ an B an evaluate U: U {( ) ˆ.5 m i (.5 m ) ˆj }(.5T) (.5 N m)( iˆ ˆj ) (.79 N m)( ˆj ˆj ).8J ˆj

13 HPTE 8 4 In Figue 8-48, a conucting o that has a mass m an a negligible esistance is ee to slie without iction along two paallel ictionless ails that have negligible esistances sepaate by a istance l an connecte by a esistance. The ails ae attache to a long incline plane that makes an angle θ with the hoizontal. Thee is a magnetic iel iecte upwa as shown. (a) Show that thee is a etaing oce iecte up the incline given by F ( B l v cos θ)/. (b) Show that the teminal spee o the o is v t ( mg sinθ)/ ( B l cos θ). Pictue the Poblem The ee-boy iagam shows the oces acting on the o as it slies own the incline plane. The etaing oce is the component o F m acting up the incline, i.e., in the x iection. We can expess F m using the expession o the oce acting on a conucto moving in a magnetic iel. ecognizing that only the hoizontal component o the o s velocity v pouces an inuce em, we can apply the expession o a motional em in conjunction with Ohm s law to in the inuce cuent in the o. In Pat (b) we can apply Newton s n law to obtain an expession o v/t an set this expession equal to zeo to obtain v t. y θ F n F m θ mg x (a) Expess the etaing oce acting on the o: Expess the inuce em ue to the motion o the o in the magnetic iel: Using Ohm s law, elate the cuent I in the cicuit to the inuce em: F F m cosθ () whee Fm IlB an I is the cuent inuce in the o as a consequence o its motion in the magnetic iel. Blv cosθ Blv cosθ I

14 Substitute in equation () to obtain: (b) pply Fx max to the o: When the o eaches its teminal spee v t, v t : F Blv cosθ lb cosθ B l v cos θ B l v v mg sinθ cos θ m t an v B l v g sinθ cos θ t m B l vt g sinθ cos m θ Solve o v t to obtain: v t mgsinθ B l cos θ 7 Pio to 96, magnetic iel stengths wee usually measue by a otating coil gaussmete. This evice use a small multi-tun coil otating at a high spee on an axis pepenicula to the magnetic iel. This coil was connecte to an ac voltmete by means o slip ings, like those shown in Figue 8-6. In one speciic esign, the otating coil has 4 tuns an an aea o.4 cm. The coil otates at 8 ev/min. I the magnetic iel stength is.45 T, in the maximum inuce em in the coil an the oientation o the nomal to the plane o the coil elative to the iel o which this maximum inuce em occus. Pictue the Poblem We can apply Faaay s law an the einition o magnetic lux to eive an expession o the inuce em in the otating coil gaussmete. Use Faaay s law to expess the φ m inuce em: t Using the einition o magnetic lux, elate the magnetic lux though the loop to its angula velocity: Substitute o ( t) to obtain: φ an simpliy m φ ( t) NBcosωt m t NBω NBω sin [ NBcosωt] ( sinωt) ωt max sinωt

15 whee NBω max Substitute numeical values an evaluate max : ev min π a ev min 6s 4 ( 4)(.45T)(.4 m ) 8.475V max The maximum inuce em occus at the instant the nomal to the plane o the coil is pepenicula to the magnetic iel B. t this instant, φm is zeo, but is a maximum.