Chapter 22 Solutions


 Amberly Joseph
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1 PHY49 Chapte Solutions Poblem : A +5 micoc chage is located 4 cm fom a +3 micoc chage The magnitude of the electostatic foce on the lage chage and on the smalle chage (in N) is espectively Answe: 5 5 Solution: The magnitiude of the electostatic fo is given by 9 KQ Q (899 Nm / C )(5µ C)(3µ C) F 53N (4cm) emembe the foce on due to is equal and opposite to the foce on due to Poblem : Two point paticles have chages q and q and ae sepaated by a distance d Paticle q expeiences an electostatic foce of millin due to paticle q If the chages of both paticles ae doubled and if the distance between them is doubled what is the magnitude of the electostatic foce between them (in millin)? Answe: Solution: The magnitude of the initial electistatic foce is Kq q F i mn d The magnitude of the final electostatic foce is K(q )(q ) F f Fi mn (d) Poblem 3: Two identical point chages +Q ae located on the yaxis at y+d/ and yd/ as shown in the Figue A thid chage q is placed on the xaxis At what distance fom the oigin is the net foce on q a maximum? d +Q x +Q q F θ F Solutions Chapte Page of 6
2 PHY49 Answe: d /( ) Solution: The net foce on q is the supeposition of the foces fom each of the two chages as follows: KQq F KQq F F F + F ( x KQqx + ( d / ( cos θxˆ sin θyˆ ) ( cos θxˆ + sin θyˆ ) ) KQq ) 3 / xˆ cos θxˆ whee I used x + (d/) and cosθ x/ To find the maximum we take the deivative of F and set it equal to zeo as follows: df dx KQq ( x + ( d / ) ( d / ) x ) KQq ( x + ( d / ) ) 5/ Solving fo x gives d /( ) x ) 3/ ( x 3x + ( d / ) ) 5/ Poblem 4: Two gam balls hang fom lightweight insulating theads 5 cm long fom a common suppot point as shown in the Figue When equal chages Q ae place on each ball they ae epelled each making an angle of degees with the vetical What is the magnitude of Q in micoc? Answe: Solution: equiing the x and y components of the foce to vanish yields QE Q T mg θ Q Solutions Chapte Page of 6
3 PHY49 KQ Tsinθ QE Tcosθ mg Eliminating the tension T and solving fo Q gives mg tan 4mg sin tan Q K K and Q θ θ θ 4( kg)( 9 8m / s )( 5m) sin tan Nm / C 3 µ C Poblem 5: Fou point chages q Q Q and Q fom a squae with sides of length as shown in the Figue What is the magnitude of the esulting electostatic foce on the chage q due to the othe thee chages? Answe: 4KqQ/ Solution: The electostatic foce on q is the vecto supeposition of the following thee foces F and the net foce is thus 3 F tot The magnitude is given by KqQ F xˆ KqQ F yˆ KqQ xˆ + ( ) Q yˆ KqQ ( xˆ yˆ) ( ) KqQ KqQ F tot 4 Q 3 q Q tot Solutions Chapte Page 3 of 6
4 PHY49 Poblem 6: Two gam chaged balls hang fom lightweight insulating theads m long fom a common suppot point as shown in the Figue If one of the balls has a chage of micoc and if the balls ae sepaated by a distance of 5 cm what is the chage on the othe ball (in micoc)? Answe: 37 Solution: The conditions fo static equilibium on ball Q is KQQ T sinθ x T cosθ mg and dividing the st equation by the second yields KQQ tan x mg θ Solving fo the poduct of the two chages gives x mg (5m) ( kg )(98m / s Q Q K (m )(899 Nm / C ) whee I used tanθ x/() Thus C Q 368µ C 6 C Q m ) θ x 5 cm m T Q θ mg 5 KQ Q /x C Poblem 7: A chage +Q is fixed two diagonally opposite cones of a squae with sides of length On anothe cone of the squae a chage of +5Q if fixed as shown in the Figue On the empty cone a chage is placed such that thee is no net electostatic foce acting on the +5Q chage What chage is placed on the empty cone? Answe: Q +Q +5Q +Q q Solutions Chapte Page 4 of 6
5 PHY49 Solution: Setting the magnitude of the net fo on the +5Q chage due to the two +Q chages equal to the magnitude of the foce on the +5Q chage due to the chage q gives KQ( 5Q) Kq( 5Q) ( ) and solving fo the magnitude of q q (and noting that q must be negative) yields q Q Poblem 8: Thee point chages Q Q and 4Q fom a tiangle with sides of length and as shown in the Figue What angle does the esulting electostatic foce on the chage Q make with the positive xaxis (in degees)? Answe: 7 Solution: We see fom the figue that and hence θ 7 o KQ /() tanθ 4KQ / Poblem 9: Two identical point chages +Q ae located on the yaxis at y+d/ and yd/ and a thid chage Q is located at the oigin as shown in the Figue The thee chaged +Q +Q and Q fom an electically neutal system called an electic quadupole A chage q is placed on the xaxis a distance x d fom the quadupole At what is the net foce on q due to the quadupole? Answe: KQq 57 x ˆ d 4Q 8 d Q Q +Q +Q 4KQ / x Q F 3 θ q KQ / F F θ F Solutions Chapte Page 5 of 6
6 PHY49 Solution: The net foce on q is the supeposition of the following thee vecto foces: KQq F ( xˆ yˆ cos θ sin θ ) KQq F ( cos xˆ sin yˆ θ + θ ) KQq F3 xˆ x whee x + (d/) and cosθ x/ Adding the foces gives F F + F + F KQq ( x KQq x ( + ( d / x) 3 KQq d (5/ 4) ) 3/ which at x d yields KQq F d ( + (/ ) ) 3/ x + ( d / ) xˆ 3/ xˆ ) 3/ KQq xˆ 57 xˆ d x Poblem : In the pevious poblem what is the magnitude of foce on the chage q when x becomes vey lage compaed to the quadupole sepaation d (Hint: take the limit of the quadupole foce on q when x << d) Answe: 3KqQd /(4x 4 ) Solution: Fom the pevious poblem we see that the magnitude of the quadupole foce is given by F KQq x ( + ( d / x) KQq x 3/ 3KQqε 3 ( ( + ε) 3/ ) 4 whee I used (+ε) 3/ ~  3ε/ with ε (d/x) x ) KqQd 4x xˆ Solutions Chapte Page 6 of 6
7 PHY49 Poblem : Chapte 3 Solutions What is the magnitude (in millin) and diection of the electostatic foce on a  micoc chage in a unifom electic field given by E ( N / C) xˆ? Answe: in the x diection Solution: The electostatic foce is given by F qe ( µ C)( N / C ) xˆ mnxˆ Poblem : Two point chages +8 nanoc and  nanoc lie on the xaxis and ae sepaated by 6 metes as shown in the Figue What is the magnitude of the electic field (in N/C) at a point P on the xaxis a distance of 6 metes to the ight of the negative chage? Answe: zeo Solution: The electic field at the point P is the supeposition of the electic fields fom each of the two chaged as follows: whee Q nc and 6 m KQ E x $ K4Q E x KQ x $ $ ( ) E E + E tot +8 nc 6 m  nc 6 m P Poblem 3: Two chages Q and Q ae sepaated by distance and lie on the xaxis with Q at the oigin as shown in the figue At a point P on the xaxis a distance /3 fom Q the net electic field is zeo What is the atio Q /Q? Answe: 5 Solution: In ode fo the net electic field to be zeo it must be tue that Q /3 P Q Solutions Chapte 3 Page of 4
8 PHY49 KQ KQ which implies that Q /Q /4 ( / 3) ( / 3) Poblem 4: A paticle with a chage to mass atio of C/kg stats fom est in a unifom electic field with magnitude E N/C How fa will the paticle move (in m) in seconds? Answe: Solution: We know that q F ma qe so that the acceleation is a E m Thus the distance taveled is d at q m Et (C / kg)(n / C)(s) m Poblem 5: A lage insulating sheet caies a unifom suface chage of density  micoc/m A poton (with a mass of 67x 7 kg) is eleased fom est at pependicula distance of m fom the sheet How much time (in micosec) elapses befoe the poton stikes the sheet? Answe: 55 Solution: We know that F ee Ma so that a ee/m We also know that if the paticle stats fom est then the distance taveled d at / Solving fo t gives d dm 4dMε t a ee eσ 4(m )(67 (6 9 whee I used E σ/(ε ) kg)(885 C 6 C)( C / m /( Nm ) )) 55µ s 7 Solutions Chapte 3 Page of 4
9 PHY49 Poblem 6: At a distance of mete fom an isolated point chage the electic field stength is N/C At what distance (in m) fom the chage is the electic field stength equal to 5 N/C? Answe: 4 Solution: We know that E KQ/ and E KQ/ so that E E o E N / C ( m) m E 5N C Hence m 4 m / Poblem 7: The magnitude of the electic field 3 m fom a point chage Q is equal to N/C What is the chage Q (in C)? Answe: Solution: We know that KQ E and solving fo gives E (N / C)(3m) Q C 9 K 899 Nm / C Poblem 8: A neon sign includes a long neonfilled glass tube with electodes at each end acoss which an electic field of N/C is placed This lage field acceleates fee electons which collide with and ionize a potion of the neon atoms which emit ed light as they ecombine Assuming that some of the neon ions (mass 335x 6 kg) ae singly ionized (ie have chage e6x 9 C) and ae acceleated by the field what is thei acceleation (m/s )? Answe: 96x Solution: We know that F qe ma so that Solutions Chapte 3 Page 3 of 4
10 PHY49 a qe m 9 ( 6 C)( N / C) kg 96 m/ s a b c d e Poblem 9: Of the five chage configuations (a)(e) shown in the Figue which one esults in the maximum magnitude of the electic field at the cente of the squae? Each configuation consists of a squae with eithe +Q Q o no chage at each cone? Answe: d Solution: The magnitude of the electic field at the cente of each squae is given by KQ ( a) E KQ ( b) E 4KQ ( c) E KQ ( d) E 4 ( e ) E whee is the length of the side of the squae The squae with the lagest Efield at the cente is (d) Poblem : Two nonconducting infinite paallel ods ae a distance d apat as shown in the Figue If the ods have equal and opposite unifom chage density λ what is the magnitude of the electic field along a line that is midway between the two ods? d +λ λ Solutions Chapte 3 Page 4 of 4
11 PHY49 Answe: 8Kλ/d Solution: The magnitude of the electic field fom a single infinite od is given by E Kλ/ Hence the magnitude of the electic field along the line midway between the two ods is given by E Kλ Kλ 8Kλ + d / d / d Poblem : Two point chages +8 nanoc and + nanoc ae sepaated by 6 metes What is the magnitude of the electic field (in N/C) midway between them? Answe: 6 Solution: The electic field at the point P is the supeposition of the electic fields fom each of the two chaged as follows: K (8nC ) E xˆ (3m) K (nc ) E xˆ (3m) K (6nC ) Etot E + E xˆ (3m) The magnitude of the net electic field is 9 K(6nC) (899 Nm / C)(6 E tot (3m) 9m Poblem : Two point chages Q and q lie on the xaxis and ae sepaated by a distance of mete as shown in the Figue The chage Q lies at the oigin and the net electic field (on the xaxis) fom the two chages is zeo at the point x 75 m If the C) 9 6N / C xcomponent of the electic field fom the two chages E x 3 N/C at the point x m (on the xaxis) what is Q (in nanoc)? Answe: 4 Q +8 nc m E at x 75 m 6 m m q P 3 m + nc E 3 N/C Solutions Chapte 3 Page 5 of 4
12 PHY49 Solution: The fist condition (net electic field zeo at x 75 m) gives KQ Kq which implies q Q / 9 ) ( 75m) (5m The second condition (E 3 N/C at x m) gives KQ (m) and solving fo Q yields Kq + (m ) KQ (m ) N / C 36(m ) (3N / C) Q 4nC 9 3(899 Nm / C ) Poblem 3: Two point chages +4 micoc and  micoc lie on the xaxis and ae sepaated by a distance of mete as shown in the Figue If the +4 micoc lies at the oigin at what point (o points) on the xaxis is the net electic field zeo? +4 µc Answe: x m Solution: The fist condition that must be satisfied fo the net electic field to be zeo is that the magnitude of the Efield fom the +4 mc chage must be equal to the magnitude of the Efield fom the mc chage This implies that 4 x Solving fo the distance x yields x ± ( x m) ( x m) m o x m and x /3 m The second condition that must be satisfied is that the vectos must point in opposite diections and is tue only at the point x m (at x /3 m they point in the same diection) x  µc Solutions Chapte 3 Page 6 of 4
13 PHY49 Poblem 4: A  C point chage lies on the xaxis a distance of mete fom an infinite nonconducting sheet of chage with a unifom suface chage density σ C/m which  C lies in the yzplane as shown in the Figue At what point (o points) along σ C/m the xaxis is the net electic field zeo? Answe: x56 m Solution: The fist condition that must be satisfied fo the net electic field to be zeo is that the magnitude of the Efield fom the infinite sheet must be equal to the magnitude of the Efield fom the C point chage This implies that σ ε K Q ( x m) o Kε Q x m) σ x ( ± 564) m ( Q π σ Solving fo the distance x yields The second condition that must be satisfied is that the vectos must point in opposite diections and is tue only at the point x 564 m (at x 436 m they point in the same diection) m xaxis λ de θ P θ y/ x dq λ Poblem 5: Two nonconducting semiinfinite ods lies along the xaxis as shown in the Figue One od lies fom x / to x infinity and the othe od lies fom x / to and x infinity The ods have chage unifomly distibuted along thei length with a linea density λ What is the magnitude of the electic field at a point P located on the yaxis at y /? Answe: Kλ/ Solutions Chapte 3 Page 7 of 4
14 PHY49 Solution: Fom the symmety in the poblem we see that the xcomponent of the electic field at the point P is zeo and that the ycomponent of the electic field fom the two ods is twice that fom one of the ods as follows E y Kλ / de y ( x y + y KdQ cosθ Kλ dx 3/ ) y y / + ( / ) whee I used dq λdx and cosθ y/ with x + y Plugging in y / yields Kλ E y / 4Kλ ( / ) / Kλ 7 Poblem 6: A positive chage 8Q is distibuted unifomly along a thin cicula ing of adius If the ing has a chage Q located at its cente (see Figue) at what point z along the axis of the ing (othe than infinity) is the net electic field zeo? Answe: / 3 + ( / ) Solution: The magnitude of the net electic field on the zaxis will be zeo povided K( 8Q) z ( z + ) KQ z 3 / Taking the /3 powe of both sided yields o z ( z + ) / 4z z + and solving fo z gives z / 3 Q Total Chage +8Q on ing zaxis Solutions Chapte 3 Page 8 of 4
15 PHY49 Poblem 7: Two identical point chages +Q ae located on the yaxis at y+d/ and yd/ as shown in the Figue What is the magnitude of the electic field on the xaxis a distance x d fom the oigin? Answe: 43KQ/d Solution: The net electic field at the point P is the supeposition of the foces fom each of the two chages as follows: KQ E cos KQ E cos E E + E ( x KQx + ( d / ) ( θ xˆ sin θ yˆ ) ( θ xˆ + sin θ yˆ ) KQ ) 3 / xˆ cos θ xˆ whee I used x + (d/) and cosθ x/ Setting x d yields KQ E (5/ 4) d 43 3/ d KQ d +Q +Q x P E θ E E tot Poblem 8: In the pevious poblem what is the magnitude of the electic field on the x axis when x becomes vey lage compaed to the sepaation d between the chages (Hint: take the limit of the electic field when x >> d)? Answe: KQ/x Solution: Fom the pevious poblem we see that E ( x KQx + ( d / ) ) 3/ x KQ ( + ( d / x) ) 3/ KQ x Solutions Chapte 3 Page 9 of 4
16 PHY49 P E Semiinfinite od Chage on od Q E Poblem 9: A od caying a total chage Q unifomly distibuted along its length (constant λ) lies on the xaxis a distance fom a semiinfinite od with the same unifom chage density λ as the finite od as shown in the Figue What is the magnitude of the net electic field at the point P on the xaxis a distance fom the end of finite od? Answe: 5KQ/(6 ) Solution: The net electic field at the point P is the supeposition of the foces fom each of the two ods (finite and semiinfinite) as follows: KQ KQ E E xˆ xˆ od x ( x + ) K λ KQ E E semi xˆ xˆ 3 3 KQ 5 KQ E E E xˆ xˆ whee I used λ Q/ and I used the fact that the point P was a distance 3 fom the end of the semiinfinite od +λ + Poblem : A thin plastic od is bent so that it makes a thee quates of a cicle with adius as shown in the Figue If two quates of the cicle cay a unifom chage density +λ and one quate of the cicle caies a unifom chage density λ what is the magnitude of the electic field at the cente? Answe: K λ / Solution: Fist I calculate the electic field at the cente of a semicicle as with chage density λ as follows: + + +λ λ E E  λ +  SemiCicle de θ Solutions Chapte 3 Page of 4
17 PHY49 E x / K cosθdθ λ π π / whee I have used de KdQ/ with dq λds λdθ and de x decosθ Now I calculate the electic field at the cente of a quatecicle with chage density λ as follows: E x / 4 K cosθdθ λ π π / 4 Kλ Kλ The net electic field at the cente of the theequate cicle is the supeposition of the fields fom the semicicle and the quatecicle as follows: K λ E ˆ E semi x K λ E ˆ ˆ E quate x y K λ E E (3 ˆ ˆ + E x y ) o o whee I used sin 45 cos 45 / The magnitude of the net electic field is thus Kλ Kλ E 9 + λ QuateCicle de θ Poblem : A nonconducting semiinfinite od lies along the xaxis as shown in the Figue (one end at x and the othe at xinfinity) The od has chage unifomly distibuted along its length with a linea density λ What is the xcomponent of the electic field E x at a point P located a distance y above the end of the od? Answe: Kλ/y λ dq x P θ de θ de x y Solutions Chapte 3 Page of 4
18 PHY49 Solution: Fom the figue we see that E x Kλ de x ( x x + y KdQ sinθ ) 3/ dx Kλ y whee I used dq λdx and sinθ x/ with x + y Poblem : An electic dipole is placed on the yaxis with +Q at y and Q at yd as shown in the figue What is the magnitude of the electic field at a point P located at xd on the xaxis? Answe: 74KQ/d Solution: We supeimpose the two E fields as follows: E E KQ E xˆ d KQ E xˆ yˆ d KQ + E ( ) xˆ d o o whee I have used sin 45 cos 45 / Solving fo the magnitude yields yˆ KQ KQ E d d +Q Q d d θ E P E Solutions Chapte 3 Page of 4
19 PHY49 Poblem 3: As shown in the figue a ball of mass M kg and chage Q3 C is suspended on a sting of negligible mass and length m in a nonunifom electic field E( x) axxˆ whee a37 N/(Cm) is a constant If the ball hangs at a nonzeo θ fom the vetical what is θ? (Hint: gavity pulls the ball down with acceleation g98 m/s ) Answe: 6 o Solution: equiing the x and y components of the foce to vanish yields T sinθ QE T cosθ Mg Eliminating the tension T gives sin θ QE Qax Qa sin θ cosθ Mg Mg Mg and thus Mg (kg )(98m / s ) cosθ 5 Qa (3C )(37 N /( Cm))(m ) so that θ 6 o yaxis θ T x θ Mg xaxis E Poblem 4: Two semicicula unifom nonconducting loops each of adius ae chaged oppositely to +Q and Q espectively The loops ae now joined end to end to fom a cicula loop as shown in the figue What is the magnitude of the electic field at the cente of the cicula loop? Answe: 4KQ/(π ) +Q E Q Solutions Chapte 3 Page 3 of 4
20 PHY49 Solution: Fist I calculate the electic field at the cente of a semicicle as follows: E x / λ π K cosθdθ π / Kλ KQ π whee I have used de KdQ/ with dq λds λdθ and de x decosθ and Q πλ Now fo two semicicles of opposite chage the two electic fields add giving E KQ π E + E + KQ π dq 4KQ π θ de Q Chage on od Q E point E od Poblem 5: x A od caying a total chage Q unifomly distibuted along its length (constant λ) and a point chage Q both lie on the xaxis and ae sepaated by a distance as shown in the Figue What is the magnitude of the net electic field at P on the xaxis a distance x /3 fom the end of the od? Answe: zeo Solution: We supeimpose the two Efields (od and point) as follows: At x /3 we have E KQ E E E po int od x KQ xˆ x( x + ) KQ ( x) + E od x( x + po int ) xˆ ( x) KQ 9 9 E ˆ x 4 4 xˆ Solutions Chapte 3 Page 4 of 4
21 PHY49 Chapte 4 Solutions Poblem : Conside a spheical conducting shell S of adius on which chage +Q is placed Without touching o distubing it this shell is now suounded concentically by a simila shell S of adius on which chage Q is placed (see Figue) What is the magnitude of the electic field in the egion between the two shells ( < < )? Answe: KQ/ Solution: Gauss' aw tells us that Shell S has chage Q Shell S has chage +Q d t Φ E E da Q enclosed ε If we choose ou Gaussian suface to be a Suface sphee with adius such that < < then we get E()4π Q/ε and solving fo E gives E KQ/ Poblem : In the pevious poblem what is the electic field inside shell S ( < )? Answe: zeo Solution: Gauss' aw tells us that Φ E E da Q enclosed ε If we choose ou Gaussian suface to be a Suface sphee with adius such that < then we get E()4π since Q enclosed and solving fo E gives E Poblem 3: A solid insulating sphee of adius has chage distibuted unifomly thoughout its volume What faction of the sphee s total chage is located within the egion < /? Answe: /8 Solution: The amount of chage Q() within the adius < is given by Solutions Chapte 4 Page of 8
22 PHY Q( ) dv ρ4π d π ρ 3 ρ whee ρ is the volume chage density and dv 4π d Thus Q( ) Q whee Q 4π 3 ρ/3 is the sphee s total chage Fo / (/) 3 /8 Poblem 4: A solid insulating sphee of adius has a nonunifom volume chage distibution given by ρ() a whee a is a constant What is the total chage Q of the insulating sphee? Answe: πa 4 Solution: The amount of chage Q within the sphee is given by Poblem 5: 3 4 ρ 3 ( ) dv ( a)4π d 4πa d Q( ) πa In a cetain egion of space within a distibution of chage the electic field is given by E( ) aˆ It points adially away fom the oigin and has a magnitude E() a whee a 5N/(Cm) How much electic chage (in nanoc) is located inside a shell with an inne adius of 5 metes and an oute adius of metes? Answe: 46 Solution: Gauss' aw tells us that Q enclosed ε E da ε (4π E( ) 4π a K ( 3 3 ) E( )) 4πε ( 3 (5N /( Cm))((m ) (5m) Nm / C E( ) 3 E( )) ) 46nC Solutions Chapte 4 Page of 8
23 PHY49 Poblem 6: In a cetain egion of space the electic field is given by E( ) ( a / ) ˆ It points adially away fom the oigin and has a magnitude E() a/ whee a 9Nm/C How much electic chage (in nanoc) is located inside a sphee with adius 5 metes? Answe: 5 Solution: Gauss' aw tells us that Φ E E da Q enclosed ε At the suface of a sphee of adius this Suface implies that E()4π Q/ε and solving fo Q and using K/(4πε ) gives E( ) a (9Nm / C)(5m) Q 5nC 9 K K 899 Nm / C Poblem 7: In a cetain egion of space within a distibution of chage the electic field is given by E( x) axxˆ It points in the x diection and has a magnitude E x (x) ax whee a 5N/(Cm) How much electic chage (in nanoc) is located inside the cube with sides of length m shown in the Figue? Answe: 6 Solution: Gauss' aw tells us that Q enclosed ε ( E x ε ( ) (885 E da E C x 3 () ) ε a / Nm )(5N / Cm)(m) zaxis 3 O yaxis 6nC xaxis Solutions Chapte 4 Page 3 of 8
24 PHY49 Poblem 8: Conside a cube of sides m as shown in the figue Suppose that a nonunifom electic field is pesent and is given by E( x) ( a + bx) xˆ whee a N/C and b5 N/(Cm) What is the total net chage within the cube (in picoc)? Answe: +354 Solution: Gauss' aw tells us that Φ E E da Q enclosed ε At the suface of the cube Suface φ E (E()E()) and hence Q ( E( ) E()) ε (5N /( Cm))(m) 3 ( a + b a) ε (885 C zaxis 3 b ε /( Nm)) 354 pc O yaxis xaxis Poblem 9: A solid insulating sphee of adius has a chage +Q distibuted unifomly thoughout its volume (the volume chage density ρ is constant) The insulating sphee is suounded by a solid spheical conducting shell with inne adius and oute adius 3 as shown in the Figue The conducto is in static equilibium and has a zeo net chage What is the Insulato chage +Q magnitude of the electic field at the point 5/ inside the conducto? Answe: zeo Solution: The electic field inside a conducto in static equilibium is zeo! Poblem : In the pevious poblem what is the magnitude of the electic field at the point 4 outside the conducto? Answe: KQ/(6 ) Solution: Gauss' aw tells us that 3 Conducto Net chage zeo on conducto Solutions Chapte 4 Page 4 of 8
25 PHY49 Φ E E da Q enclosed ε Fo a Gaussian sphee of adius < 3 this Suface implies that E()4π Q/ε and solving fo E() and using K/(4πε ) gives E() KQ/ and plugging in 4 yields E KQ/(6 ) Poblem : A thundestom foms a mysteious spheical cloud with a adius of m Measuements indicate that thee is a unifom electic field at the suface of the cloud that has a magnitude of N/C and is diected adially inwad towad the cente of the cloud What is the total net chage within the cloud (in micoc)? Answe: 445 Solution: Gauss' aw tells us that Φ E E da Q enclosed ε At the suface of the cloud Suface E()4π Q/ε and solving fo Q gives Q E ( N C m / )( ) 9 K 899 Nm / C 44 5µ C Poblem : A 6 micoc chage is at the cente of a cube cm on a side What is the electic flux (in Nm /C) though one face of the cube? (Hint: think symmety and don t do an integal) Answe: 3x 5 Solution: The total flux though the cube is φ E Q/ε and symmety tell us that the same amount of flux goes though each of the six faces of the cube Hence Φ face Φ 6 E Q enclosed 6ε 6 6 C 6(885 C / Nm ) 3 5 Nm / C Solutions Chapte 4 Page 5 of 8
26 PHY49 Poblem 3: A solid insulating sphee of adius has chage distibuted unifomly thoughout its volume (the volume chage density ρ is constant) The insulating sphee is suounded by a solid spheical conducto with inne adius and oute adius as shown in the figue The net chage on the conducto is zeo What is the magnitude of the electic field at the point 3 outside the conducto? Answe: 4πKρ/7 Solution: Gauss' aw tells us that Φ E E da Q enclosed ε Fo a spheical suface outside the Suface conducto ( > ) we have E()4π Q/ε ρ4π 3 /(3ε ) and solving fo E() gives E( ) Fo 3 we get E 4πKρ/7 4πKρ 3 3 Poblem 4: In the pevious poblem what is the magnitude of the electic field at the point 3/ inside the conducto? Answe: zeo Solution: The electic field is zeo inside a conducto in static equilibium This is ou definition of a conducto! Poblem 5: In the pevious poblem what is the magnitude of the electic field at the point / inside the insulato? Answe: πkρ/3 Solution: Gauss' aw tells us that Φ E E da Q enclosed Fo a spheical suface inside the insulato ( < ε Suface ) we have E()4π Q()/ε ρ4π 3 /(3ε ) and solving fo E() gives Insulato ρ chage/unit volume Conducto Zeo net chage Solutions Chapte 4 Page 6 of 8
27 PHY49 Fo / we get E πkρ/3 4 E( ) πkρ 3 Poblem 6: A solid insulating sphee of adius has a chage +Q distibuted unifomly thoughout its volume (the volume chage density ρ is constant) The insulating sphee is suounded by a solid spheical conducto with inne adius and oute adius as shown in the Figue The conducto is in static equilibium and has a net chage +Q What is the magnitude of the electic field at the point / inside the insulating sphee? Answe: KQ/( ) Solution: Gauss' aw tells us that Insulato ρ chage/unit volume Conducto Zeo net chage Φ E E da Q enclosed ε Fo a spheical suface inside the insulato Suface with adius ( < ) we have E()4π Q()/ε ρ4π 3 /(3ε ) and solving fo E() gives 4 E ( ) ρ 3 KQ π K 3 whee I used Q 4π 3 ρ/3 Fo / we get E KQ/( ) Poblem 7: In the pevious poblem what is the magnitude of the electic field at the point 3/ inside the conducto? Answe: zeo Solution: The electic field inside a conducto in static equilibium is always zeo Solutions Chapte 4 Page 7 of 8
28 PHY49 Poblem 8: A point chage +Q is located at the cente of a solid spheical conducting shell with inne adius of and oute adius of as shown in the Figue In addition the conducting shell has a total net chage of +Q How much chage is located on the oute suface ( ) of the conducting shell? Answe: Q Solution: We know that the net chage on the conducto is given by the sum of chage on the inne and oute sufaces Q net Q in + Q out Gauss' aw tells us that Φ E E da Q enclosed ε If we choose ou Gaussian suface to be a Suface sphee with adius such that < < then E (inside conducting mateial) which implies that Q enclosed Q in +Q Thus Q in Q and Q out Q Poblem 9: In the pevious poblem what is the magnitude of the electic field in the egion < inside the hole in the conducting shell? Answe: KQ/ Solution: Gauss' aw tells us that Φ E E da Q enclosed ε If we choose ou Gaussian suface to be a Suface sphee with adius such that < then we get E()4π Q/ε and solving fo E gives E KQ/ +Q Net chage +Q on conducto Solutions Chapte 4 Page 8 of 8
29 PHY49 Chapte 5 Solutions Poblem : If the potential diffeence between points A and B is equal to V A V B 3x 4 Volts how much wok (in millijoules) must be done (against the electic foce) to move a change paticle (Q 3 micoc) fom point A to point B without changing the kinetic enegy of the paticle? Answe: 9 Solution: W AB Q(V B V A ) (3x 6 C)(3x 4 V) 9x  J 9 micoc We know that the wok done against the electic field must be negative since the paticle will fall fom high potential to low potential Poblem : Eight identical point chages Q ae located at the cones of a cube with sides of length What is the electic potential at the cente of the cube? (Take V at infinity as the efeence point) Answe: 9KQ/ Solution: The potential at the cente of the cube is given by whee I used d 8KQ 6 KQ KQ Vcente 94 d 3 3 / Poblem 3: What is the electic potential enegy of the configuation of eight identical point chages Q at the cones of the cube with sides of length in the pevious poblem? Answe: 8KQ / Solution: The total electic potential enegy is given by KQ KQ Ucube QV i i QVcone 4Q i 3 whee V cone is the electic potential at the cone of the cube due to the othe 7 chages Solutions Chapte 5 Page of
30 PHY49 Poblem 4: Two point chages Q Q and Q Q ae on the xaxis at x + and x  and a thid point chage Q 3 Q is on the yaxis at the point y as shown in the Figue What is the electic potential diffeence V V V between the point P at the midpoint of the line between Q and Q 3 and the point P at the oigin? Answe: 5KQ/ Solution: The electic potential at P is given by KQ V 3 At P the electic potential is V KQ + / KQ d 4 QQ + x  whee d ( / ) + ( ) / Thus Q3Q KQ 4 KQ KQ V V yaxis y d QQ x xaxis Poblem 5: Two point chages Q Q and Q Q ae on the xaxis at x + and x  and a thid point chage Q 3 Q is on the y axis at the point y as shown in the pevious Figue What is the electic potential enegy of this chage configuation? QQ x  yaxis Q3Q QQ x Answe: 9KQ / Solution: The electic potential enegy of the chage configuation U is given by xaxis Solutions Chapte 5 Page of
31 PHY49 KQ KQ KQ KQ U Poblem 6: In the poblem 4 how much wok must be done (against the electic foce) to move the chage Q 3 fom the point y on the yaxis to the oigin esulting in the configuation shown in the Figue? Answe: 6KQ / Solution: The potential potential enegy of this new chage configuation U is given by U and thus the wok equied is KQ KQ + 5 KQ ( ) 5 KQ KQ KQ W U U 59 Poblem 7: Thee identical point chages Q ae located at the cones of a ight tiangle with sides of length as shown in the Figue What is the electic potential at the point P that bisects the hypotenuse? (Take V at infinity as the efeence point) Answe: 44KQ/ Solution: The electic potential at the point P that bisects the hypotenuse is given by KQ KQ KQ V / / whee I used Q Q P Q Solutions Chapte 5 Page 3 of
32 PHY49 Poblem 8: What is the electic potential enegy of the configuation of thee identical point chages Q located at the cones of a ight tiangle with sides of length in the pevious poblem? Answe: 7KQ / Solution: The potential potential enegy of the chage configuation U is given by U KQ + KQ + KQ KQ 7 Poblem 9: In poblem 7 how much wok is equied to move the chage located at the 9 o angle of the equilateal tiangle to the point P that bisects the hypotenuse? Answe: 88KQ / Solution: The potential potential enegy of this new chage configuation U is given by KQ KQ U + / 4 KQ 5 + and thus the wok equied is W U U 4 KQ 5 KQ 88 KQ Q KQ Q Q Solutions Chapte 5 Page 4 of
33 PHY49 Poblem : Thee identical chaged paticles (Q 3 micoc) ae placed at the vetices of an equilateal tiangle with side of length m and eleased simultaneously fom est as shown in the Figue What is the kinetic enegy (in millij) of one of the paticles afte the tiangle has expanded to fou times its initial aea? Answe: 45 Solution: Hee we use enegy consevation as follows: KQ Ei + 3 KQ Ef 3KE + 3 whee KE is the kinetic enegy of one of the paticles and whee i and f (since the aea inceased by facto of 4) Setting E i E f yields KE KQ KQ ( 899 Nm / C )( 3 C) ( m) 9 6 Q Q Q 4 5mJ Poblem : A unifom electic field E V/m points in the x diection as shown in the Figue Point A has coodinates (m) and point B has coodinated (mm) whee points ae labeled accoding to P(xy) What is the potential diffeence V B V A (in Volts)? Answe:  Solution: The potential diffeence is equal to the following line integal B C B V V E dl E dl E dl B A A A C A C m B E Solutions Chapte 5 Page 5 of
34 PHY49 whee I have chosen the path shown in the Figue Evaluating the line integal along AC gives zeo since E is pependicula to the path and along the path CB E is paallel to the path so that and hence V B V A V B C C A E dl E dl Ed ( V / m)( m) V Poblem : A unifom electic field E4 V/m cosses the xaxis at a 4 o angle as shown in the Figue Point B lies on the xaxis 5 cm to the ight of the oigin A What is the potential diffeence V B V A (in Volts)? Answe: 53 Solution: We know that B VB VA E d and hence VB V A E cosθ (4 V/m) cos(4 o ) (5 m) 53 V A 4 o A 5 cm B E Poblem 3: A nonunifom electic field is given by E ( x) ax xˆ whee a V/m 3 as shown in the Figue Point B lies at xy3 m and point A is at the oigin (xy) What is the potential diffeence V B V A (in Volts)? Answe: 9 Solution: The potential diffeence is equal to the following line integal A 3m B C E(x) Solutions Chapte 5 Page 6 of
35 PHY49 B C B V V E dl E dl E dl B A A whee I have chosen the path shown in the Figue Evaluating the line integal along CB gives zeo since E is pependicula to the path and along the path AC E is paallel to the path so that C A E dl and hence V B V A 9V 3 B C A E dl ax dx 3 a(3m) 3 C 9V Poblem 4: The potential along the xaxis is given by V(x) axbx whee a V/m and b V/m At what value(s) of x (in m) is the electic field equal to zeo? Answe: Solution: The electic field is minus the deivative of the potential Solving fo x gives Poblem 5: dv E ( ) x a bx dx a V / m x m b ( V / m ) The potential is given by V(xy) a/(x +y ) whee a Vm What is the magnitude of the electic field (V/m) at the point P (xy) (mm)? Answe: 4 Solution: The thee components of the electic field ae given by Solutions Chapte 5 Page 7 of
36 PHY49 E E x y V xa /( x + y x V ya /( x + y y V Ez z and the magnitude of the electic field is E E E E x + y + z 3/ ( x a + y At the oigin x m and y m the magnitude of the electic field is (Vm ) E V / m 4V / m 3 ( m) Poblem 6: The potential is given by V(xyz) ax+by+cz whee a 3 V/m b 4 V/m and c 5 V/m What is the magnitude of the electic field (in V/m) at the oigin P(xyz)()? Answe: 5 Solution: The thee components of the electic field ae given by V Ex a x V Ey b y V Ez cz z and the magnitude of the electic field is E Ex + Ey + Ez a + b + 4c z At the oigin z and the magnitude of the electic field is E a + b (3V / m) + ( 4V / m) 5V / m ) ) ) Solutions Chapte 5 Page 8 of
37 PHY49 Poblem 7: Two thin concentic cicula plastic ods of adius and have equal and opposite unifom chage densities λ and +λ espectively distibuted along thei cicumfeences as shown in the Figue What is the electic potential at the cente of the two cicles? (Take V at infinity as the efeence point) Answe: zeo Solution: The electic potential at the cente of the two cicles is given by KQ KQ Kπ λ Kπ λ V + πk ( λ λ) whee the chage on a cicula od is Q πλ +λ λ Poblem 8: A positive chage Q is distibuted unifomly along a thin cicula ing of adius A negatively chaged paticle (chage Q) is at the cente of the ing as shown in the Figue What is the electic potential at a point on the zaxis of the ing a distance fom the cente (Take V at infinity as the efeence point) Answe: 93KQ/ Total Chage +Q on ing Solution: The net potential is the sum of the potential of the ing plus the potential of the point chage as follows: KQ KQ Vz () z + z which at z becomes KQ KQ V( ) 93 Q zaxis Solutions Chapte 5 Page 9 of
38 PHY49 Poblem 9: An infinitely long thin od lies along the yaxis and has a unifom linea chage density λ 3 micoc/m as shown in the Figue If point P (a) and point P (ab) whee P (xy) what is the potential diffeence V V V (in kilovolts)? Answe: 374 Solution: We know that the electic field λ 3µC/m yaxis fom and infinite od points away fom the od and has a magnitude given by E() Kλ/ The potential diffeence is equal to the following line integal a a 3 E Kλ/ b xaxis V 3 V V E dl E dl E dl whee I have chosen the path shown in the Figue Evaluating the line integal along 3 gives zeo since E is pependicula to the path and along the path 3 E is paallel to the path so that and hence V V 3 V (899 E dl Nm 3 a a Kλ ln() 9 / C E dl Kλ dx Kλ ln() x )(3 6 3 C / m)ln() 374kV Solutions Chapte 5 Page of
39 PHY49 dq Chage Q on od Poblem : A positive chage Q is distibuted unifomly along a thin od of length as shown in the Figue What is the electic potential at a point P on the axis of the od a distance fom the end of the od? (Take V at infinity as the efeence point) Answe: 693KQ/ Solution: The electic potential at the point P is given by Kλdx KQ V Kλ ln() ln() 693 x whee I used dq λdx x P KQ Poblem : A cicula plastic od of adius has a positive chage +Q unifomly distibuted along onequate of its cicumfeence and a negative chage of 3Q unifomly distibuted along the est of the cicumfeence as shown in the Figue What is the electic potential at the cente of the cicle? (Take V at infinity as the efeence point)? Answe: KQ/ Solution: The electic potential at the cente of the cicle is given by 3Q KQ 3KQ KQ V Poblem : The electic field at the suface of a solid spheical conducto (adius cm) points adially outwad and has a magnitude of V/m Calculate the electic potential (in Volts) at the cente of the conducting sphee (Take V at infinity as the efeence point) Answe: +Q Solutions Chapte 5 Page of
40 PHY49 Solution: At the suface of the conducto we know that Hence V V KQ KQ suface Esuface V E ( V / m)(m) V cente suface suface Poblem 3: A solid insulating sphee of adius has chage a total chage Q distibuted unifomly thoughout its volume (the volume chage density ρ is constant) If the electic potential is zeo at infinity and Volts at the cente of the sphee what is the potential at the suface (in Volts)? Answe: 8 Solution: The electic field inside the insulato ( < ) is given by E() KQ/ 3 and the potental diffeence between the cente and the suface is V suface Vcente E d We know that V suface KQ/ which means that KQ d 3 3 KQ V cente and Vsuface Vcente 8V 3 KQ Solutions Chapte 5 Page of
41 PHY49 Chapte 6 Solutions Poblem : What is the capacitance C of a solid spheical conducting shell with inne adius and oute adius? Answe: /K Solution: The electic potential of the solid spheical conducto is given by V KQ KQ whee is the suface of the conducto Hence C Q V K Poblem : An enegy of 9x 3 Joules is stoed by a capacito that has a potential diffeence of 85 Volts acoss it Detemine the capacitance of the capacito (in micof) Answe: 5 Solution: We know that U C( V ) and hence 3 U (9 J ) C 5µ F ( V ) (85V ) Poblem 3: Capacito # with capacitance C and initial chage Q is connected in paallel acoss an initially unchaged capacito with capacitance C (#) Afte the system comes to equilibium what is the chage on capacito #? Answe: Q/3 Solution: Afte the two capacitos ae connected thei chages must satisify Solutions Chapte 6 Page of 4
42 PHY49 Q + Q Q Q Q C C whe the fist condition comes fom chage consevation and the second comes fom the fact that the potential dop acoss each capacito is the same since they ae connected in paallel Solving fo Q yields Q Q 3 Poblem 4: A chaged conducting sphee with a adius of 5 cm has a potential of 8 kilovolts elative to infinity What is the electic field enegy density (in J/m 3 ) at a point nea the suface outside the conducto? Answe: Solution: At the suface of the conducto we know that KQ KQ Vsuface Esuface Hence the electic field enegy density is V u ε E ε (885 C /( Nm ))(8 (5 m) 3 V ) J / m Poblem 5: The electic field enegy density nea the suface of an isolated chaged conducting sphee with a adius of 5 cm is J/m 3 What is the electic potential (in kilovolts) at the cente of the conducto? Answe: 75 Solution: At the suface of the conducto we know that KQ KQ Vsuface Esuface and the electic field enegy density is 3 Solutions Chapte 6 Page of 4
43 PHY49 Thus V cente V (5m) suface u ε E u ε 3 (J / m ) 885 C /( Nm 75kV ) whee I have used the fact that the potential is constant thoughout the conducto Poblem 6: In a cetain egion of space the electic field is given by E Ayx$ whee A V/m The electic field points O in the x diection and has a constant magnitude given by E(y)Ay What is the total amount of electic potential enegy (in millijoules) contained in the electic field within the cube with sides of length xaxis metes shown in Figue? Answe: 47 Solution: The enegy stoed in the electic field lines is given by 5 U udv εe dy ε A y dy ε A 6 and plugging in the numbes gives U (885x  C /(Nm ))( V/m ) ( m) 5 /6 47 mj zaxis yaxis Solutions Chapte 6 Page 3 of 4
44 PHY49 Poblem 7: Given an openai paallelplate capacito of plate aea A and plate sepaation d as shown in the Figue A flat slice of conducting mateial (thickness b; b<d) of the same aea A is now caefully inseted between the plates paallel to them without touching If the chage Q on the capacito was unchanged duing this opeation must the metal slice be foced between the plates o will it be pulled in by electostatic foces and what is the new capacitance? Answe: ε A/(db) pulled Solution: Initially the capacito has E Q/(ε A) V Ed C i ε A/d and stoed enegy U i Q Qd C ε A Afte the conducting mateial is inseted we have two capacitos in seies with εa εa C ε C d b d b ( ) and A C eff ( d b ) The stoed enegy is now U f Q Q ( d b) C ε A i < U eff Since the final stoed enegy is less than the initial stoed enegy the conducting mateial will be pulled in by the electostatic foce with the diffeence in stoed enegy doing the wok d i +Q Q b Solutions Chapte 6 Page 4 of 4
45 PHY49 Chapte 7 Solutions Poblem : Wie B is thee times longe than wie A and the two wies have the same volume If both wies ae made of the same substance and if the esistance of wie A is 9 Ohms what is the esistance of wie B (in Ohms) Answe: 8 Solution: We know that the esistance of a wie of length coss sectional aea A and esistivity ρ is given by ρ A whee V is the volumn V A Thus B ρ V ρb ρ(3a) ρa 9 9A 9(9Ω) 8Ω V V V B A Poblem : How much cuent passes though the Ω esisto in the cicuit shown in the Figue? I Answe: 6I/ Solution: Since the potential dop acoss each esisto is the same (ie paallel) we know that I I I 3 I I I and in addition I I + I I Hence 33 and solving fo I gives I A + 3 I I I + + I I I + 3 I I Ω Ω 3 Ω Solutions Chapte 7 Page of 3
46 PHY49 Poblem 3: Conside a long staight cylindical wie with adius caying the nonunifom cuent density J()a whee is the distance fom the cente axis of the wie How much total cuent I passes though the wie? Answe: πa 4 / Solution: The cuent I is the flux associated with the cuent density as follows: I J da 3 4 ( a ) π d πa d πa Poblem 4: Conside a long staight cylindical wie with adius m is caying the nonunifom cuent density which points along the axis of the wie and has a magnitude J() a/ whee is the distance fom the cente axis of the wie If a A/m how much total cuent I passes though the wie (in A)? Answe: 6 Solution: The cuent I is the flux associated with the cuent density as follows: a I J da πd πa π (A/ m)(m) 6A Solutions Chapte 7 Page of 3
47 PHY49 A B cut Poblem 5: Conside the infinite chain of esistos shown in the Figue Calculate the effective esistance (in Ohms) of the netwok between the teminals A and B given that each of the esistos has esistance Ohm Answe: 7 Solution: We fist cut the infinite chain A A at the point shown in the figue and eplace the infinite chain to the ight of the cut by its effective esistance We then combine B B the thee esistos in seies and then combine and + in paallel giving which implies that and ( + 3) 73 + Solutions Chapte 7 Page 3 of 3
48 PHY49 Chapte 8 Solutions Poblem : Conside the cicuit consisting of an EMF 3 Ω and thee esistos shown in the Figue How much cuent flows though the 4 4 V 4 Ω Ω esisto (in Amps)? Answe: 3 I I Solution: The oiginal cicuit is equivalent to a cicuit with and EMF and one esisto with esistance tot given by whee eff 3Ω fom tot Ω + 3Ω + 3Ω 6Ω 3 eff eff + 4Ω Ω 3Ω Thus the cuent I 4V/6Ω 4A Now we know the I I + I and that V I V I and hence I I (4A) 3A + 6 Ω I Poblem : A constant powe P is supplied to a tansmission line of esistance by a powe station If the powe dissipated in the tansmission line is Watts when the powe station delives an EMF of Volts how much powe is dissipated in the tansmission line (in W) when the powe station delives an EMF of Volts (assuming the same powe is supplied in both cases)? Answe: 5 Solution: The powes supplied by the station is given by P station I V I is constant so that I I V V V Solutions Chapte 8 Page of 5
49 PHY49 whee V V and V V Now the powe disapated in the line is P I so that I V P P P P 5W I V 4 The powe station loses less enegy in the line if it tansmits powe at a highe voltage! Poblem 3: A capacito C with an initial chage Q dischages though a esisto How many time constants τ C must elapse in ode fo the capacito to lose /3 of its chage? Answe: Solution: Fo an C cicuit with the chage Q(t) given by Qt () Qe t / τ and τ log( Q( t) / Q ) τ log(/ 3) τ t whee τ C and whee I used Q(t)/Q /3 Poblem 4: The capacito in the Figue is initially unchaged Afte the switch is closed (EMF V) what is the magnitude of the potential diffeence acoss the capacito (in V) when the chage on the capacito eaches onethid of its maximum value? Answe: 4 Solution: The potential diffeence acoss the is V C Q/C εc/(3c) ε/3 4 V whee I used Q max εc V C Poblem 5: A chaged capacito C has a esistance connected acoss its teminals to fom the C cicuit shown in the Figue If it takes seconds fo the capacito to loose onehalf of its stoed enegy how long does it take (in s) fo it to loose 9% of its initial chage? Answe: 33 C Solutions Chapte 8 Page of 5
50 PHY49 Solution: The stoed enegy in the C cicuit is given by and solving fo τ gives Qt () Ut () Ue t / C τ C t t ln( Ut ( ) / U) ln( 5 ) whee t s The chage as a function of time is Thus Qt () Qe t t / τ and t τ ln( Q( t) / Q ) t ln( ) ( s)ln( ) 33s ln( 5 ) ln( 5 ) Poblem 6: A capacito C is (fully) chaged by connecting it to a Volt EMF and then is allowed to dischage though a Ω esisto as shown in the Figue? What is the cuent I in the cicuit (in A) when the stoed enegy in the capacito has dopped to 5% of its initial value? Answe: 35 τ Solution: The chage Q(t) and stoed enegy U(t) as a function of time is given by Q( t) Qe Q U ( t) U C t / C e Q e t / C U t / τ e t / τ whee U Q /(C) and Q CV Hee V is the potential acoss the capacito at t (equal to the EMF that fully chaged the capacito) If t a is the time when the stoed enegy eaches 5% of its initial value then t a / τ e and taking the squaeoot gives The cuent at time t a is given by C I t a / τ e Ω Solutions Chapte 8 Page 3 of 5
51 PHY49 I V V t I e t a / τ 5 a ) A 3 5A Ω ( whee I used I V / Poblem 7: A "leaky" capacito's faulty insulation allows chage to pass slowly fom one plate to the othe Suppose that a chaged leaky micof capacito's potential diffeence dops to onefouth its initial value in sec What is the equivalent esistance between the capacito plates in megaohms? (Hint: the leaky capacito foms an C cicuit) Answe: 7 Solution: The leaky capacito foms an C cicuit with the chage Q(t) given by Qt () Qe t / τ Solving fo the esistance gives t Cln( Q( t) / Q ) and τ C ( s) 6 ( F)ln( 5 ) t ln( Qt ( ) / Q) 7MΩ C C left Poblem 8: Conside the dischaging of the two capacitos shown in the Figue Both capacitos have equal capacitance C and both have an initial chage of 8 micoc When the switches ae closed the capacito an the left will dischage though a single esisto with esistance while the capacito on the ight dischages though two equal esistos with esistance connected in paallel If the switches ae closed simultaneously how much chage is ight Solutions Chapte 8 Page 4 of 5
52 PHY49 on the ight capacito (in micoc) when the left capacito s chage is 4 micoc? Answe: Solution: The chage as a function of time on the two capacitos is given by Q Q left ight ( t) Q e ( t) Q e t / C t / C whee I used the fact the the effective esistance on the ight side is eff / Hence Q( t) Q ight Q( t) Q Thus when the left side Q(t)/Q 4/8 / the ight side has Q(t)/Q /4 so that Q ight µc left Solutions Chapte 8 Page 5 of 5
53 PHY49 Poblem : Chapte 9 Solutions A chaged paticle Q 5 C entes a egion with a unifom magnetic field B xˆ + 3yˆ + 4zˆ (in Tesla) If its velocity is given by v xˆ + 3yˆ + zˆ (in m/s) what is the magnitude of the magnetic foce on the paticle (in N)? Answe: 36 Solution: We know that the magnetic foce on a paticle is xˆ F Qv B Q yˆ 3 3 zˆ Q(6xˆ 4 yˆ) 4 The magnitude of the foce is F Q (5) 5 N 3 6 N Poblem : A chaged paticle Q5 C and mass M gams entes a egion with a unifom magnetic field B xˆ yˆ (in Tesla) What is the magnitude of the magnetic foce on the paticle (in N) if its velocity is v 4xˆ yˆ (in m/s)? Answe: zeo Solution: We know that the magnetic foce on a paticle is xˆ F Qv B Q 4 yˆ zˆ Note that v is in the same diection as B and hence the magnetic foce is zeo Poblem 3: One end of a staight wie segment is located at (xyz) () and the othe end is at (mm3m) A cuent of Amps flows though the segment The segment sits in a unifom magnetic field B xˆ yˆ (in Solutions Chapte 9 Page of 5
54 PHY49 Tesla) What is the magnitude of the magnetic foce (in N) on the wie segment? Answe: 67 Solution: We know that the magnetic foce on a wie segment is F I B xˆ I yˆ zˆ 3 I( 3xˆ + 6yˆ 5zˆ )Tm with the magnitude of the foce given by F I Tm ( A)8367 Tm 6 73 N Poblem 4: An electic powe tansmission line located an aveage distance of m above the eath's suface caies a cuent of 8 Amps fom east to west in a egion whee the eath's magnetic field is 8 gauss due noth at 6 o below the hoizontal What is the magnitude of the foce pe mete on the line (in millin/m)? Answe: 64 Solution: We know that the foce on a staight wie is given by F I B In this case and B ae pependicula and hence F 4 3 IB (8 A)(8 T ) 64 N / m Poblem 5: A chaged paticle Q C taveling in the xdiection with velocity v v xˆ entes a egion of space that has an electic field in the ydiection given by E E yˆ with E Volts and a magnetic field in the z diection given by B B zˆ with B Tesla If the paticle expeiences no net foce and continues with the same speed and diection what is its speed v (in m/s)? Answe: 5 Solution: The electic foce is given by Solutions Chapte 9 Page of 5
55 PHY49 F E QE yˆ and the magnetic foce is F E Qv B Qv B yˆ and setting F E + F B implies that E V v 5 m / s B T Poblem 6: A chaged paticle taveling in the ydiection with a momentum of kg m/s entes a egion of space that has a unifom Tesla magnetic field in the zdiection as shown the Figue If the paticle entes the magnetic field at the point A and then exits the magnetic field at the point B located a distance of 5 m to the left of the point A what is the chage of the paticle (in millic)? Answe: 4 Solution: The adius of the cicula path followed by the chaged paticle calculated by setting the foce equal to the mass times the centipetal acceleation as follows mv Q vb and solving fo the magnitude of the chage gives mv p (kgm / s) Q 4 mc B db (5m)(T ) whee I used p mv and d We know that the sign of the chage must be negative since the paticle moves to the left 5 m Bout Solutions Chapte 9 Page 3 of 5
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