Anr-4:lPhysics 48 (Fall20l1) --_:_==:_

_,\v. Chapter 29: Magnetic Fields Due to Carrents

"Nothing can bring you peace but yourself. " - Ralph lValdo Emerson

"The foolish man seeks happiness in the distance, the wise man grows it under his feet. "James Oppenheim

"Happiness is not a stcrtion ),ou arrive at, but a manner of traveling." - Margaret B. Runbeck

Reading: sections 764 - 781

Outline:

= Biot-Savart Lawmagnetic fie1d from a long straight wiremagnetic field from a infinite straight wiremagnetic field at the center ofa circular arc

+ force between two parallel cunents= Ampere's Law

using Aqtp"t"'. tu*= solenoids and toroids+ current carrying coil as a magnetic dipole (read on your own)

Problem Solving Techniques

A few of the eariy problems deal with the fieid of a long shaight wire. You should be able to find themagnitude and direction of the field at any point in space, given the current in the wire. UseB : /hi/21v to find the magnitude and the right-hand rule to find the direction. You may be asked forthe total fie1d of two or more long straight wires. You will then need to carry out vector addition.

Once you have found the magnetic field you may be asked for the force it exerts on a movingcharge. Use qi x E . Don't forget to include the sine of the angle between i and .D when youcalculate the magnitude. Know how to use the right-hand rule to find the direction ofthe force. Besure to take into account the sign ofthe charge.

Some problems ask you to use the Biot-savart law to compute the magnetic field of a current. Dividethe current into infinitesimal eiements, write the expression for the field ofan element, then integrateeach component over the current. You will need to write the integrand in terms of a single variable.If the wire is straight place it along the x-axis, say, and use the coordinate x ofpoints along the wireas the variable ofintegration. If the wire is a circular loop use the angle made by a radial line with acoordinate axis as fhe variable ofintepration.

ln many cases you may think of an electrical circuit as composed of finite straight-1ine and circular-arc segments, each of which produces a magnetic field. You can then calculate the field produced byeach s"egment and vectorially sum the individual fields to find the total field. Use the result given inProbleri 11 if you need to frnd the field on the perpendicular bisector of a finite straight wire. Tofrnd the field at some other point you sill need to integlate the BiofSavart 1aw. Use the proceduregiven in Section 1 for an infinite wire but replace the limits of integration with finite values. lJseB : t4i d l4nR for the field of a circular arc at its center of curvature'

Some problems ask you to use what you leamed in the last chapter to calculate the force that onewire exefis on another.

Arnpere's law problems take three forms. The most straightforward give the currents and ask you to

find the value of {f . ai around a given path. You must pay attention to the directions of thecurrents as they pierce the plane of the path and carefu11y observe which currents are encircled bythe path and w-hich are not. Other problems ask you to use Ampere's law to calculate the magneticfieli. Carefully choose the Amperian path you will use and pay attention to the evaluation of theAmpere's law integral in terms of the unknown field. Still other problems give you the magnetrcheli as a function ofposition and ask for the current through a given region. Carry out the lineintegral ofthe tangential component ofthe field around the boundary of the region and equate theresult to /41, then solve for l.

You should know how to compute the magnetic fields of some special current configurations (inaddition to a long straight wire): a solenoid, a toroid, and a magnetic dipole. The field of a.n idealsolenoid is given by B : ltnni inside the solenoid andby B = 0 outside. The fieid inside is aiong thecylinder axis. The field of a toroid is given by B : pnNl2tn at a point inside, a distance r fiom thecenter. The field inside the hole and outside the toroid is zero. The field lines inside are circles that at

concentric with the toroid. The field of a magnetic dipole is given by B = po F lZtz3 at a point on theaxis defined by the direction ofthe dipole moment, a distance z from the dipole. You should reca11from the last ciapter how to find the dipole moment of a curent loop (both magnitude and direction)and how to compute the torque of a uniform magnetic field on a dipole'

Questions and Example Problems from Chapter 29

Question ITie figure below shows four arrangements in which long parallel wires carry equal currents directlyinto oiout ollhe page at the comers ofidentical squares. Rank the arrangements accolding to themagnitude of the net magnetic field at the center of the square, greatest

l a ra r? l \

first.

o-----o q-----ot q l q - l!K - 2 r l \ ' ,D I, . ' - / ' t : xa li "6\ ' l ; z/t x ;O*--*"O 0-*- - -a

9- _- - -6i.,Y i +-----9i \ i| 3,,X :. t1. , / , / , \ |t , ' / ' \ |o"-*--o4 (,!j

d**---oq t , t J 3 4 t h \ 3 L l ( , ) 3

(e)

q. --n A,\&,

A--1 ),,t6

(0)

C-+ c r* "d Ps,

A- tik Vry

I *8 -+ ,c'"R &"4^; "" d' itua "6E 4wJTI ;2!AI a)''v\'.r'J )JL)t\4iJ p'aih n

A,n""ta "7 B- rr *;tl 4 ar\c

-.I-

ntagaAa

,u^.t, p-go

.1, c d-x

F-"i-J

B.*r -r, ^%yny',o -et*uea;e" l[ro Ml,il4j B, pr"di, v1\u)'Atdn)

K=l - (l" I t- a.

= ad3';J8,r i . (a-.)

f= q* -- z/.o er,

lJr*= d----+ I=(-

J J o L r

?.. rr(-r

A, L ,

4..-Tr a

_

L \o La

2..ir C^

?

{-c - ( .

f,/,t =

Question 2The Figurebeiow shows three cirouits, each consisting of two radial lengths and two concentric arcs,one ofradius r and the other ofradius R > r. The circuits have the same current through them and thesame angle between the two radial lengths. Rank the circuits according to the magnitude ofthe netmagnetic field at the center, greatest first.

Problem 2In the figure be1ow, a current i: 10 A is set up in a long hairpin conductor formed by bending a wireinto a semicircle of radius R = 5.0 mm. Point b is midway between the straight sections and sodistant from the semicircle that each section can be approximated as being an infinite wire. What are -the (a) magnitude and (b) direction (into or out ofthe page) ofB at point a ard the (c) magnitude and

Problem 3In the figure below, a wire forms a semicircle of radius R: 9.26 cm and two (radial) straight .-_/segments each of length L: 13.1 cm. The wire carries current i = 34.8 mA. What are the(a) magriitude and (b) direction (into or out of the page) ofthe net magnetic field at the semicircle'scenter of curvature C?

-0eNt "J^a"{r,f ^rfl'c,n ; JB = ,^t:- {jiiad=o

4r F\B=o

?r I

t_l t- tt - ' l t-

i Jr a,,"* tta'6?.

,*$ll,.,& afi"a,'4t,i; ;ggt*

zu"kd gtnni,lwi 6,n1 p,= n,;r1 "37;F,

E-.. = Xt *) = dlf - (. ln"r;. ' I ' , /n)(134 : i t .o' . w) atery"4n iIr N\Q- "' \ etifi I r1 A -rr ( 5oo^ 16 >.;

d t

dB= &qf

--o ----) B =o

o ) |

-) "1 lfi, u,^bt 4

q r;^tvc^-t'a-t o'vL "A b= & d = n a",*,"'vt

4::j_-_A,; = @ --l"tt*tdnl-tanA 4R 4(o.oqa.6m)

I B'= l' lrr16-zTr"hJtuo'.+ Il r

(d) direction ofB at point b.1- t'

i ,o^rtT dil

A,4tl* rt-'Artf ,Ory'r',,atr} Ir i aos

:::--- --lE-

(-F

oc

f l=tcl

B=

Problem 4In the figure below, point P2 is at perpendicular distance R:25.1 cm from one end of straight wireof length L = 13.6 cm canytng current l= 0.693 A. (Note that the wire is not long.) What is themagnitude of the magretic freld atPz?

?','4t rd=* t#iIl!

x

ds--. Jx4i^e, = -B-.v"

CA\x")fx= l**x*

a-- A"t ( anLP "t;- ) (Fril'u

l#-,-=;fiju" g:

Problem.5

form a square of edge length a : 8.50 cm. Each wire carries 15.0 A, and all the currents are out ofthe page. In unit-vector notation, what is the net magnetic field force per meter of wire length onwire 1?

O.45"

. ] .J P -L

q 7-q

3

.1.Lr lF- a--*Sl \ ^ I

ld ' - a .

I! !

r { . l@-. . n - . *@

xF,= +#"*ffi,

X L = F^ r,E ,*"e

= 4' Lt i r"'r*;J-T q

f l f - I)Ar,lL : ( l 5 .oA l =

l \a

w

r l dt l -Y t 1

dg t"-" 'ie/ln",@fu4* V'Aa^r*

"tr pa.ag

--.) p.= 4';A (. r 4 t )

rl".n +a"; Ai?6n'^/- L

( Jsi"g\)n t

dx6;t*

JB= j*- +:: ---) B- +?

- l '

l-]a* lO- T ru* a6

t rr $a'*' .b&a"een )' 6a-""'-0{at ouuvrz,da'-to pu.r-*,.1t , e t t ! . ' , "

X J ' U l - = J 4 n , L t L ^ .' - t

^ lat rd

p*fM cu^",f4 "fr-,}

t\?41&

!l-

r

rr \

t

tFr= -trr-tr"-^e

: z(Ltn Io I

(1,. t \ ,1c ' l i / ," ,) t - ( ' t ,z. l ' ' to " Ni,,) 1

, . a

1t ] , , .L La j rq

( o.ots. , )

Problem 6The figure beiow shows two very long straight wires (in cross section) that each carry a current of4.00 A tlirectly out of the page. Distance d1 : 6.00 m and distance d2:4.00 m. What is themagnitude ofthe net magnetic field at point P, which lies on a perpendicular bisector ofthe wires?

PJ3,

d -- qO-g = 9o-3r".1' -- S3.1.U./ Z=:-

i? = 4u-P -AD\'x".t I'tT I-

- r1 . (5 .ooq) r

r ' t 'I r - - l - - J - , n 'I r, I I r* i r| : J . - - f - ' !t J r ' 9 t

f ^ | --'+ \

'sa 'dse *

O( nr"o)

q b ' . i 5 '

I B"ls'

(r .+1. r ,i

(

- = () Bu(,At,. VL

I

tr-- i= S'ootli ", /l -;::".{(

34 : 1e* ---r o= 3t".1"d a. l,oo r'\

1\t/u'-- a B.o"nfl

Problem 7Show that a ruriform magnetic field cannot drop abruptiy to zero (as is suggested by the lack of fie1dlines to the right ofpoint a in the figure below) as one moves perpendicular to E , say along thehorizontal arrow in the figure. (Hint: Apply Ampere's law to the rectangular path shown by thedashed lines.) In actual magnets "fringing" of the magnetic field lines always occurs, which meansthat .E approaches zero in a gradual manner. Modify the held iines in the figure to indicate a morerealistic situation.

- ) A ^ t 1 ' t 'A . - - 4 I l -om6 = :9-z:-nNn$PF4I a\L af f J / -Tf f I

do = 8,4^d* B.d^,d B^"* = (4'i. ro'to/n) (q.oa*) _o,v\ 53-1.

= 5. ll,* ld tT ,pr""""\

Dq 'd> "

r \ l -f-! .-{ 3 ( t>\ A- --

t_, ,L , - , / ' ( L : \ , .

\ B, 'd s,, t \

o( B, r lu , ;

,/ -A r -) ..-.r\B"ds"+\Bu"dr,

o

( B* - ! ds " )

es a current of 18.0 A. The magnetic fie1dwire forming the solenoid.

et-\

j

i>

r Problem 8A solenoid 1.30 m long and 2.

, inside the solenoid is 23.0 mT

l --1"3a^\ ,

C= d /^ - - I "3ocr )| . d \- - I , jo ^ lo - tm

? r < ^ nL -- | o'

Problem 10The figure below shows an arrangement known as a Helmholtz coi1. It consists of two circularcoaxial coils, each of 200 tums and radius R = 25.0 cm, separated by a distance s : R. The two coiiscarry equal currents i: 12.2 mA in the same direction. Find the magnitude of the net magnetic fieldat P, midway between the coils.

) J]* l"'ae.,&'/ g*94 t At-r*,an* z aLxg)#G u^#n'r -f- +*n o rcc"L 4, tt l*r* i",VsJr,. )4 '. 3Gi _ Nl ,r,1. ( .r,

2-l 4"*z)"/ ' '\

=) ai poa^,* P o d,ut*t'*n z= Alr. 4"E"' a^4O& t fu nog..-fui yA; $*n Qle\ /od^l' P*ift 'nr #v + x-

^ f N l^" t8", - tlJ^.d= l [ ; ; ; l ] !12. 1- 4 L K + L ' v r / J I

, n a= N,UoL l l

r - ^ 2 t \ 7 / ^l . n / 4 )

B = (r.oo) ( ry n - | o " -rr,r /* ) ( r e.?..1o > n ) ( o. r s .) \ ( B.rr" td'Problem 11 I s

(o.er.)'/q7Y"In the figwe below, cunent t : 56.2 mA is set up in a loop having two radial le,ngths and twosemicircles of radii a : 5.72 cm andb :9.36 cm with a common center P. What are the (a)magritude and (b) direction (into or out of the page) of the magrretic field at P and the (c) magnitudeand (d) direction of the loop's maenetic dipole moment?

. tr) i IJ J-\)

(a) d= o .{.-'L fi,ar4ntat4.*A a* VB- U-L d,t'c/^'trt o5r't^cuu+ o^+

4nR 1 d'h-* b,*a"'t@"l^n q".3-

4iLql fb

+U='tlta +d (t.*)D^* =

B^ur=@ I{L*-,r..nLlI ' 1

5Ja-*v;'.J

E^* = 4 ,1r7* lo-? t: