( )( )( ) ( ) + ( ) ( ) ( )

Size: px

Start display at page:

Download "( )( )( ) ( ) + ( ) ( ) ( )"

Dulcie Cathleen Reed
5 years ago
Views:

1 3.7. Moel: The magnetic fiel is that of a moving chage paticle. Please efe to Figue Ex3.7. Solve: Using the iot-savat law, ( ) + ( ) qvsinθ 1 T m/a C. 1 m/s sin m 1. 1 m 15 = = = T The ight-han ule applie to the poton points out of the page. Thus, Moel: Assume the wies ae infinitely long. Please efe to Figue Ex Solve: The magnetic fiel stength at point 1 is At points an 3, at 1 = top + bottom =, out of page +, into page at 1 5 ( T, out of page) top 15 = kˆ T at 1 = = ( 1 T m/a)( 1 A) cm ( 4 ) cm 1 m 6 1 m + = = + = bottom 4 at, into page, into page. 1 T, into page 5 at 3 =, into page +, out of page = T, out of page

2 3.15. Moel: Assume the wies ae infinitely long. The fiel vectos ae tangent to cicles aoun the cuents. The net magnetic fiel is the vectoial sum of the fiels top an. Points 1 an 3 ae at a istance = cm fom both wies an point 3 is at a istance = 1 cm. bottom Solve: The magnetic fiel at points 1,, an 3 ae ˆ ˆ ( cos45 sin 45 ˆ ˆ ) ( cos45 sin 45 ) 1 = top + bottom = i j + i + j 7 1 T m/a 1 A 1 cos45. 1 T 1 m ˆ ˆ 4 = i = i = iˆ ( 1 7 T m/a)( 1 A) ˆ ˆ ˆ 4 ˆ = i + i = i = 4. 1 i T 1 1 m 4 ˆ ˆ cos45 sin 45 cos45 ˆ sin 45 ˆ. 1 ˆ T 3 = i + j + i j = i Moel: The aius of the eath is much lage than the size of the cuent loop. Solve: (a) Fom Equation 3.9, the magnetic fiel stength at the suface of the eath at the eath s noth pole is 7 ( 1 T m/a)( 8. 1 A m ) 6 ( ) z m 5 = = = T 3 3 This value is close to the value of T given in Table 3.1. (b) The cuent equie to pouce a ipole moment like that on the eath is ( eath ) = A = π 8 1 A m = π( m) = A Assess: This is an extemely lage cuent to un though a wie aoun the equato. 3.. Moel: The magnetic fiel is that of a cuent flowing into the plane of the pape. The cuent caying wie is vey long. Please efe to Figue Ex3.. Solve: Divie the line integal into thee pats: f i s = s + s + s left line semicicle The magnetic fiel of the cuent-caying wie is tangent to clockwise cicles aoun the wie. is eveywhee pepenicula to the left line an to the ight line, thus the fist an thi pats of the line integal ae zeo. Along the semicicle, is tangent to the path an has the same magnitue = / at evey point. Thus f i ight line π s = + L + = = = = 7 (4 1 T m/a)(. A) 6 ( π ) T m whee L = π is the length of the semicicle, which is half the cicumfeence of a cicle of aius.

3 3.3. Moel: The magnetic fiel is that of thee cuents. Please efe to Figue Ex3.3. Solve: Ampee s law gives the line integal of the magnetic fiel aoun the close path: 6 s = though = T m = + = 1 7 T m/a 6 A 4 A + ( 1 3 ) ( 3 ) T m 3 + = 7 ( A) 1 T m/a 3 = 1. A Assess: The ight-han ule was use above to assign positive signs to 1 an 3 an a negative sign to Moel: A magnetic fiel exets a foce on a moving chage. Please efe to Figue Ex3.7. Solve: (a) The foce on a chage moving in a magnetic fiel is F = qv = (qvsinα, iection of ight-han ule) on q The iection of the foce on a negative chage is opposite the iection etemine by the ight-han ule. A positive chage moving to the ight with own gives a foce into the page. (b) A negative chage moving paallel to the fiel has no foce an no eflection Moel: A magnetic fiel exets a magnetic foce on a moving chage. Please efe to Figue Ex3.3. Solve: (a) The foce on the chage is 19 7 F = qv = C 1. 1 m/s cos45 iˆ + sin45 kˆ.5 iˆ T on q ( m/s ) = ˆ ˆ ˆ ˆ ˆ ( + ) = + (b) ecause the coss pouct i ˆ i ˆ in the equation fo the foce is zeo, F on q = N C.5 T i i k i j N Moel: A chage paticle moving pepenicula to a unifom magnetic fiel moves in a cicle. Solve: The fequency of evolution of a chage moving at ight angles to the magnetic fiel is f cyc 31 6 q mf kg 45 1 Hz cyc 3 = = = = T 19 m q C Solve: (a) Fom Equation 3.19, the cycloton aius is (b) Fo the poton, 31 6 ( kg)( 1. 1 m/s) ( C)( 5. 1 T) mv = = =.114 m = 11.4 cm q elec ( kg)( 5. 1 m/s) ( C)( 5. 1 T) = = 1.4 m poton 19 5

4 3.4. Moel: The toque on a cuent loop is ue to the magnetic fiel. Solve: (a) We can use the ight-han ule to fin the foce iection on the cuents at the top, bottom, left, an ight segments of loop 1 an loop in Figue Ex3.4. We see that F top an F bottom ae equal an opposite to each othe. Similaly F left an F ight ae equal an opposite. Thus, F top + F bottom = an F left + F ight =. Since the top-bottom o left-ight foces act along the same line, they cause no toque. Thus, both the loops ae in static equilibium. (b) Now let us otate each loop slightly an e-examine the foces. The two foces on loop 1 still give F net =, but now thee is a toque that tens to otate the left loop back to its upight position. This is a estoing toque, so this loop position is stable. ut fo loop, the toque causes the loop to otate even futhe. Any small angula isplacement gets magnifie into a lage isplacement until the loop gets flippe ove. So, the position of loop is unstable Moel: An electic cuent pouces a magnetic fiel. Solve: (a) The fiel of a househol wie is ( 1 7 T m/a)( 1 A) m 6 = = = 1. 1 T = 1. T (b) The eath s fiel is eath = T = 5 T, so wie eath = 1. T 5 T =. =.%. (c) The fiel of a tansmission line is ( 1 7 T m/a)( A) m 6 = = =. 1 T =. T This is twice the fiel of pat (a), which woul pobably not be significantly wose. () Let s estimate that a fetus is 1 cm ( 4 inches) fom a 1 A cuent. Hee the fiel is ( 1 7 T m/a)( 1 A).1 m This is twice the fiel of pat (a). 6 = = =. 1 T =. T

5 3.5. Moel: Use the iot-savat law fo a cuent caying segment. Please efe to Figue P3.5. Solve: (a) The iot-savat law (Equation 3.6) fo the magnetic fiel of a cuent segment s is ˆ s = whee the unit vecto ˆ points fom cuent segment s to the point, a istance away, at which we want to evaluate the fiel. Fo the two linea segments of the wie, s is in the same iection as ˆ, so s ˆ =. Fo the cuve segment, s an ˆ ae always pepenicula, so s ˆ = s. Thus s = Now we ae eay to sum the magnetic fiel of all the segments at point P. Fo all segments on the ac, the istance to point P is =. The supeposition of the fiels is whee L = θ is the length of the ac. (b) Substituting θ = in the above expession, L θ = s = = ac loop cente = = This is Equation 3.7, which is the magnetic fiel at the cente of a 1-tun coil Moel: The magnetic fiel is that of a cuent in the wie. Please efe to Figue P3.59. Solve: As given in Equation 3.6 fo a cuent caying small segment s, the iot-savat law is ˆ s = Fo the staight sections, s ˆ = because both s an ˆ point along the same line. That is not the case with the cuve section ove which s an ae pepenicula. Thus, s θ θ = = = whee we use s = θ θ fo the small ac length s. ntegating to obtain the total magnetic fiel at the cente of the semicicle, π θ = 4 = 4 π = π π 4 π

6 3.61. Moel: The magnetic fiel is that of the cuent which is istibute unifomly in the hollow wie. Ampee s integation paths ae shown in the figue fo the egions m < < 1, 1 < <, an <. Solve: Fo the egion m < < 1, s = though. ecause the cuent insie the integation path is zeo, = T. To fin though in the egion 1 < <, we multiply the cuent ensity by the aea insie the integation path that caies the cuent. Thus, ( 1 ) though = 1 π π whee the cuent ensity is the fist tem. ecause the magnetic fiel has the same magnitue at evey point on the cicula path of integation, Ampee s law simplifies to ( 1 ) ( 1 ) 1 s = s = ( ) = = 1 Fo the egion <, though is simply because the loop encompasses the entie cuent. Thus, s = s = = = Assess: The esults obtaine fo the egions > an 1 < < yiel the same esult at =. Also note that a hollow wie an a egula wie have the same magnetic fiel outsie the wie.

7 3.73. Moel: A magnetic fiel exets a magnetic foce on a length of cuent caying wie. Please efe to Figue P3.73. Solve: The above figue shows a sie view of the wie, with the cuent moving into the page. Fom the ighthan ule, the magnetic fiel points own to give a leftwa foce on the cuent. The wie is hanging in static equilibium, so Fnet = Fmag + W + T = N. Consie a segment of wie of length L. The wie s linea mass ensity is =.5 kg/m, so the mass of this segment is m = L an its weight is W = mg = Lg. The magnetic foce on this length of wie is F mag = L. n component fom, Newton s fist law is F = T sinθ F = T sinθ L = N T sinθ = L net Diviing the fist equation by the secon, x mag Fnet = T cosθ W = T cosθ Lg = N T cosθ = Lg y T sin tan L θ gtanθ = θ = = T cosθ.5 kg/m 9.8 m/s tan1 = = =.864 T Lg g 1 A =.864 T, own. The magnetic fiel is Moel: Assume that the cm is lage compae with the effective aius of the magnetic ipole (ba magnet). Solve: Fom Equation 3.9, the on-axis fiel of a magnetic ipole at a istance z fom the ipole is = z ipole 3 The magnetic ipole moment can be obtaine fom the magnitue of the toque. We have Thus, the magnetic fiel pouce by the magnet is τ.75 Nm τ = = = =.15 Am sinθ.5 T sin9.15 Am = 1 T m/a = T 7 6 ipole 3 (. m)

8 3.75. Moel: A cuent loop pouces a magnetic fiel. Solve: The fiel at the cente of a cuent loop is loop = /. The electon obiting an atomic nucleus is, on aveage, a small cuent loop. Cuent is efine as = q/ t. Duing one obital peio T, the chage q = e goes aoun the loop one time. Thus the aveage cuent is avg = e/t. Fo cicula motion, the peio is m C 3 T = = = s 6 avg = = A 16 v. 1 m/s s Thus, the magnetic fiel at the cente of the atom is 7 3 ( 1 T m/a)( A) ( ) = = = 1.5 T m avg cente 11

9 3.8. Moel: The magnetic fiel is that of a conucting wie that has a nonunifom cuent ensity. Solve: (a) Consie a small cicula isk of with at a istance fom the cente. The cuent though this isk is ntegating this expession, we get i = JA = J ( ) = J 3 J J J = i = = = J = (b) Applying s = though to the cicula path of integation, we note that the wie has pefect cylinical symmety with all the chages moving paallel to the wie. So, the magnetic fiel must be tangent to cicles that ae concentic with the wie. The enclose cuent is the cuent within aius. Thus, (c) At =, J s = i = ( ) J 3 = = 3 = = = 3 This is the same esult as obtaine in Example 3.3 fo the magnetic fiel of a long staight wie.

Chapter 28: Magnetic Field and Magnetic Force. Chapter 28: Magnetic Field and Magnetic Force. Chapter 28: Magnetic fields. Chapter 28: Magnetic fields

Chapter 28: Magnetic Field and Magnetic Force. Chapter 28: Magnetic Field and Magnetic Force. Chapter 28: Magnetic fields. Chapter 28: Magnetic fields Chapte 8: Magnetic fiels Histoically, people iscoe a stone (e 3 O 4 ) that attact pieces of ion these stone was calle magnets. two ba magnets can attact o epel epening on thei oientation this is ue to