Solutions nl for
ECOMGTISM: PRINCIPS PRINCIPS D APPLICATIONS Paul aiDale R. Corson
by Paul rrain University of Montreal
Free d Co S Frcisco
Cpy rit
© 1979
by Paul ain
m
N pt f this bk be reprduced by mechanical, phtaphic, r electrnic rcess, r in the f f phnaphic recrding, recrdi ng, nr y it be stred in a retrieval sy stem, transitted , r theise cpies fr plic r private use withut written peissin frm the plisher
p
Printed in the United States f merica
71671152 71671152 9 8 7 6 5 4 3 2 1 ISBN
Cpy rit
© 1979
by Paul ain
m
N pt f this bk be reprduced by mechanical, phtaphic, r electrnic rcess, r in the f f phnaphic recrding, recrdi ng, nr y it be stred in a retrieval sy stem, transitted , r theise cpies fr plic r private use withut written peissin frm the plisher
p
Printed in the United States f merica
71671152 71671152 9 8 7 6 5 4 3 2 1 ISBN
Contents
Notes
1 Chater 2 Chater 3 Chater 4 hater 5 Chater 6 Chater 7 Chater Chater 9 ater 1 Chater 11 Chater 12 Chater 13 Chater 14 Chater 15 Chater 16 Chater 17 Chater 18 Chater 19 Chater 2 Chater
1 7 11 15 2 28 34 37 4 44 49
52 56 62 66 67 71 8 85 87
NTES 1 . So as to save space, simple matematical expressions are tped on a single line Wenever te order te operat ions is not indi cated explicit l b means o parenteses , te are perormed in te olloing order: multiplications , divisions , additions and subtrac tions Example s : l 1 1 l/ab ab l/a + b a + b , l / ( a + b ) a+b 1 l / ( a + b / c + d e/ g) 1 + ab b a + c + de g 2 Pr ograms or drawing t e curves in tis Manual it a HP9 82 cal culator and a 9862A plotter are available ree o carge rom te undersigned 3 Reerence is made , occa sionall , to a table o integrals b Digt Te ull reerence is "Tables o ntegrals and ter Mate matical Data" b Herbert rist ol Digt (Macmillan)
Paul Lorrain Dpartement de psique Universit de Montral Montral, Canada
CHTER 1 (1.2) -
A . B AB cos
9
x
4 - 6 30
0
1 -2 ( 1 2 )
�
A . B = AB cos
2 1 8 + 1 1 5 ;
AB
l ( 4+9 1) ( 1 3 6 + 1 )
231
0 . 6 5 0 , 1 3 0 . 5
cos
� (1.2)
- -
A. (B + C)
-+
A On,
A Om + AO '
-+
A. B + A . C
A On
1-4 ( 1 . 2 ) -+
+-
C2
=
D2 C 2 +D 2
=
-+
-
C.C A A + B . B + 2A B 2 A + B 2 2B cos A2 + B 2 2AB cos 2 ( A2 B 2 ) , C 2 - D 2 4AB cos =
B
1 -6 ( 1 . 3 ) -
A
x
-
B is normal to the p lane of A
ad B .
I ts magnitude is the area
show hatched.
Then
-
I A
B) . l
x
-
is the a se of the par alle lepi ped , multipl ied by its height , or its
A
1
vlue. iilarl, A . ( B x C is als the vle f the parallelepiped. 1-7 1.3 The x-cpet is A (B z+ C z - A z (B + C AB z -AzB + A C z -Az C r the x-cpet f x x The sae applies t the - ad z- c pets.
l - 1. 3 Fr the x- cpet, a bx c-b cx - a z b zc x-b xc z b xa xc x+ a c + a zc z - c x axb x+a b + a zb z The crrespdig equatis fr the - ad z - cpets ca be fud b rtatig the subscripts. 1 4 d; / dt is perpedcular t . The r is a cs tat . Als, d/dt . 2 ; . d; / dt 0 d/ dt r 2 2rdr/dt The dr/dt 0 ad r cstat 1-10 1. 4 x 500 cs 30 t 500 si 3 0 t -4 . 9 0t 2 2 250t - 4. 90t 433t 4 3 3 t + 250t - 4 . 9 0 t 2 v 433 i + 250 - 9 . t j - 9 . 0
=
r =
1.5 . ; Ax+A +A z /x + / +
=
12 1 . 5 A x /x +A / +Az / x + +
=
=A
1. 5 a l/r /X l/r + / 1/ r + / z ' l /r he re r x
=
2
N /X l /r - 1 / r 2 rx - 1/r 2 x - /r -x /r 3 y syetry, /Y /r y-y /r 3 , /z 1 / r z-z r 3 ice x-x is the x- cpet f , ad x- r is the x-cpet f 1 , etc, l/r 1 /r 2 this case , l/r /X l/r + Y l /r + / /X l /r -1/r 2 rX -1r2 x- r-- /r 3 ad siilarly fr the ther derivatives The lr -1r 2
=
1-14 15 i a j A r x f x Y f/ Y f/z f/x A r r x f r is zer, s ice r x is perpediclar t r c A f is zer fr the sae reas
= +
1- 15 1 8 a /XX + /yy + /z3 The flx f r is r . r 1 4r2 4r3 r , sig the divergece there, fr a sphere radis r, rd4r 3 rr 1 da s -
- -
1 - 1 1 8 v f /x f Ax + /y f AY + /z fA x x Y Y fA + A f
=
=
1 - 1 1 8 a f r , /X x2 +y 2 +z 2 2x, etc v f /X 3xr2 + /Y yr2 + /z 2zr 2 3r 2 + 3x 2x + r 2 + Y 2y + 2r 2 + 2 z 2 z r 2 + x 2 + 2y 2 + 4z 2 12x2 + 8y 2 + 1z 2 12
= =
3
f = r r i +y r j +z rk r = 3 + + = i + y j + zk 3 i + yj + zk f A + A f r + 3i r + y + z = 0
+
] lL [LL 2 = [L 2]
[
c v . f
=
S f is epressed in meters squared
H ll- Rdyd H
+R H -R R - d
-R
+R +R = H 2 - dx - H d H R = HR R - d -R -R - 7 Calculate the vlume in the cant here y z are all psitive
R (R2 _ Z2 ) � (R2 _y 2 _ z 2 ) �
V = J
0
0
R - R R - z dz
dX dY d
R
2 2 (R _Z )
�
2
2
1
R - Y - dy dz
0 0
R3 - R3 3 = R3
- 0 d s = dfd and is zer n the cylindrical surface N the crss - sectin f the cylinder is B dfXdXJ B d = f - fa B a Thus dfXdXJ dX = f - fa f a a
j
f
- 2 . Set F K/r K/r 2 . Then
K /r - K/r
- 2 2 . T he ork do done curve is zero even taking into accont the curvature of the arth. Then the gravitationa fied is conservative. - 2 3 . 2 Since the fied is conservative
r .d +
a over
over Q
a a over over Q
- 2 4 . 2 Since the vaue of the integra is independent of the path the fied fied defi defined ned the - 2 5 . 2 Since the force is azimut 2F " ° and from Stokes's theorem x F s o the forc fo rcee is nonconservative. The cur is cacuated as foos 2 2 . 5 Fx =- F s in e=- F y F Fcos Fx/r Fx/x 2 here F K x 2 2 . So
=
xF
=
K
i J
i
/
/x
/ z
K 2 2 2 x + y .3 x + 2 .3 ° . 3 Kx 2 x + K 2 2 . 2 2 0. 3 O 3 3 x + ( x + )
2 2 0. 3 ( x + y )
[
2 _
x 2 + y2
. 4K k k _-
r
5
.3 K
2 2 1 . 3 ( x + y )
]
2Y
-6 (lll)
i
7
A f f r
z zfr
y yfr
[zfy
- yfz i + Nfy frry f fr r yr f fz z frzr S [zyrfr - yzr yzr f fr r ] +
-7 (ll)
i
j
fA
y fAy
f A
[y fAz
{
z fA z
- fAy ] +
A Y - A + A zfy - Ayf + f
- (o)
· AYD z - AzDY + y Az D - A D z +/zAx Dy - AYD / z - Az DAzy - Ay /z + Dy /z - A y + D zAy -A D /y z - Dz o z /y - DYz - AYD z AzDy - D y
9 (ol) A
A
y y AY
z z Az
1-0 1 . 1 2 E
. dt =
c
x E ) a
- x 10 -2 2 - x 1 0 V = 2 0V
1 1 11) 2 f) = 2/ x 2
+ 2 / 2 + 2 / z2 ) f/ x) + j .. 2 2 2 2 2 2 / x) / x + / + / z ) f + j ... 2 f)
CHPTER 2 2-1 2 1 COULO ' S LAW a Ee = mg , E mg/e 9 . 1 x 10-1 x 9 . 8/1 . 6 x 10-19 . 6 x 10-11 Vm 1 m e/ o r 2 , r 2 e / E , r b) E
=
2-2 2 . 1) SEP ATON OF PHOSPHATE FRO QUART Let x be the horzontal coornate an the vertcal coornate, ownwar x = ) QE/m) t 2 , = )gt 2 x/ QE/mg = Q /m) E /g) 10 - x 10 /9 . 8) 0. 2x 100 Reference : A.D oore, Electrotatc an t Applcaton, Wle, 197.
R
m
2- 2 . E LECTRC FELD NTENS T E E 2 B metr, te vertcal component cancel an E -2 co Q a 2 a 2 ) 2 /2 = - Q 2 o a
= +
+
..
7
2- 2 . ) ELECTRIC FIELD INTENSI The charge n the rng 2rr Each pon n the rng at a tance a 2 r 2 rom P . B m metr , E along the ax .
2
arr /2 o a 2 r 2 /2
E R
E
a 2 o
o
rr / a 2 r 2 ) / 2
+
a/2 o ) 1/ a 2 R2 ) l /a
/2 o when a « R
2-5 2 . 5) CATHODE- TUBE O cour e not . In approa one pl ate an elec tron gan knetc energ b long potental energ, lke a bo allng n the gravtatonal el o the earth. 2-6 2 . 5 ) CATHODE-A TUBE Let V be the acceleratng voltage an e the abolute value o the elec tronc charge . Then 1/2)mv 2 eV, v 2eV/m) . I the tance travele D, the tme o lght Dm/2eV) . Durng that tme the electron all b a tance 1 / 2 ) g t 2 . 9 D m/2eV . 9 0 . 2 ) 2 9 . 1 x 1 0-1 / 2 x 1 . 610 -19 x 5 x 1 0 -16m . 1 . 1 x 10 A atom ha a ameter o the orer o 10-10 m .
2 - 7 2 . 5) MACROS COPIC PRTICLE G 1 . 6 5 x x 2 x 8 . 85 x 10 -12 x 10 -12 / ) 1 x 10 / 10 -2 ) 1 . 8 x 10 -16 C -16 kg m /) 10 -18 /8) x 1000 5 . 2 x 10 1/2)mv 2 1 . 5 x 10 x 1 . 8 x 10 16 , v Reerence A. D . Moore , p 59 .
8
89 m/
( 2 5) E /mg
ELECTROSTATIC SPYING
( Q /m) ( E/ g )
4 3 E / g: 10 / 9. 8 ' 10
Reference : AD Mre, pp 71 , 250, 259 , 262 2-9 ( 2 5 ) THE RUTHERFORD EERINT a) At the distance f clsest apprach, Q Q 2 / 4 r is equal t the kinetic energ : Q l Q 2 /4 r 7 6 8 x l 6 x 1 6 x l -19 r 2 x 7 9 x ( 1 6 x l -19 ) 2 / x 8 8 5 x 10 -12 x 7 6 8 x 1 6 x l -1 2 9 6 fm b ) Q Q 2 /4 r 2 7 6 8 x 1 6 x l -1 /2 9 6 x l -15 4l 5 N c) a ( 41 5 / 4 x 1 7 x l - 2 7 ) /9 8 6 2 x l 2 6 g ' s
=
2-10 (2 5) ELECTROSTATIC SEED-SORTING a) 2 x 10 -2 ( 1 / 2 ) g t + 0 01) 2 - t 2 5 ( 0 0 2 t + 1 0-4 ) , t 02 s The pper pe a must hve fal len thrugh a di s tanc e f gt 2 /2 , r ab ut 200 b) The average mass f ne pea is 2000 /l00 x 600 x 24 2 2 x 1 0 -4 k Thus 4 x 10 - 2 ( 1 /2) (QE/m) t .2 , where t ' is the time interval during
J
m
which a pea is deflected t' ( 2 x 4 x l - 2 x 2 2 x l -4 /1 5 x l -9 x 5 x l 5 ) ! 0 1 5 7 s The plates have a length L v t + gt ' 2 /2 , with v2 2g x 0 2 , v L 2 x 15 7 + 4 9 x 15 7 2 450
=
=
= 2m/s
m
Reference: FlurescenceActivated Cell-Srtin, Scientific erican, March 1976, p 108 2-11 ( 2 5 ) CYLINDRICAL ELEC TROS TATIC NALYSE R mv2 /R QE QV/a , v QVR/ma) Reference : Jur Phs E , Sc i Instr 40 (1977)
�
=
2 12 ( 2 5) PALEL-PLATE ALYSE R Reference: Rev Sci Instr 142 (19 71) , Rev Sci Instr 48 , 454 (1977)
�,
2-1 ( 2 5 ) CYLINDRICAL ND PARALLEL-PLATE NALYSERS COARED In the cyindrica anayser, v (QR/ma) S (1/2)mv 2 /Q ( R / 2 a ) V Fr a given instrent, R and a are fixed and V is a measure f the given rati In the parae-p ate anayser , f rm Prb 2 -12 , (1/2)mv 2 /Q QV /Q (a/2b) S V is a measure f the se rati
2-14 (2 5) ION RUSTER Cnsider a sateite f mass M and vecity V in a regin where gravtatina frces are negigib e Te sateite ej ects m 'kg/ se c backwards at a vec ity v with resp ec t t the sate it e The mmentum f the system (M tta ejected mass) is cnstant Then, with res pec t t a fix d reference frame , ca ing p the t ta mment f the ejected fu, ( d /dt) (M) + (dp/ dt) 0 , r MdV / d t + VdM/ d t + m' Vv) 0 , MdV / d t - m ' V + m ' V - m ' v 0 , M(dV/dt) m ' v It i s this quantity that is caed the tus t Nte that the trust is nt the fr ce (d /dt ) (M) n the as t equatin a) F m ' v , ( 1 / 2 ) m ' v 2 IV, m' (I/nem
=
v2
=
=
(2mV/ne) b ) ( 2 x 1 7 x - 2 7 x 5 x 4 / 1 6 x -19 ) ! 0 1 c) F m ' v , ( 1 / 2 ) m ' v2 P F d)
2IV/ ( I/ne )m F
�
( 2m ' p )
=
2 p /v
Q/4 R, Q
=
�
�
2m ' /I P 4 x 8 8 5 x 12 x x 5 x 0 4
2P / ( 2 IV/ m , )
4 RV 5 6 x 1 0-5 s
2 6 x -2N
5 5 6 x -6V
t Q/I 1 0Q Reference: RG Jahn, Physics f Eectric Prpusin 2-1 5 ( 5 ) COLLOID THRUS TER Reference: Te Eectr ica Pr pusin Sp ace Veices , A Brit and B Makin, Cntemprar Phsics 14 197 p 25 ee as S tati c Eect rific ain 19 75 , The Insti tute f Physi cs , Lndn , 175 , p 44
1
HAPTER -1 ( 1 ) NGLE SUB TENDED BY A LINE AT A POINT e 2 arc tan (a/2b)
=
-2 ( 1) SOLID NGLE SUB TENDED BY A DISK AT A POINT The rng f radu and wdth dr ub tend at P a l d angle 2rdr/(b 2 + r 2 ) c e 2drb/(b 2 r 2 ) / 2
+
2 1- ( 1 a 2 /b 2 If b a, If b
) 2 J
0,
n
2
If
n
, a 2
- ( 2) GAUSS ' S W N Gau ' law can nl tell u that the net flux f E emt ted b a dple er , nce the net charge er Fr exampl e , the average radal E er
-4 ( 2
SURFAE DENSITY OF ELETRONS ON A HARGED BODY a) E 8 8 5 x l 12 x x l 0 6 2 7 x l -5 /m 2 b ) Each atm c cupe an area f ab ut x 10-10 ) 2 meter 2 Thu te nber f atm per quare meter abut 10 19 c) The nber f electrn per quare meter 2 7 x 10-5 /1 6 x 10-19 ,
=
11
r 1 7 x 1014 r 1 7 x l -5 •
The umb er f fre e e letr s p er atm s 1 7 x 10 14 / 10 1
THE ELECTRIC FIELD IN A NUCLEU -15 R 1 2 5 x 1 0 ( 1 2 7 ) 1 / 6 2 8 x l -15m At the eer, ( / 2 ) R2 /R / ) (R2 / 2 e/8 R x 5 x 1 6 x l -1 / 8 x 8 8 5 x l -12 x 6 2 8 x 10 - 15 At the surfae, 7 V ( / ) (R2 /2 - R2 / 6) R2 / Veter / 1 5 1 2 x 10 V , 2 5 x 1 6 x 10 -1 / 4 x 8 85 x l -12 ( 6 2 8 x l -15 ) 2 E Q / 4T R
=
(2)
=
=
v =
=
o
=
0
0
=
=
1 x 10 21V /m
- 6 ( 2) THE PACE DERIVATI OF E x , EY , E Frm Gauss ' s law, Ex / x E y / y E / / 0 , E y ie x Ey / , Ex / E / x , E y / x
=
=
=
0
- 7 ( 2) PHYCALLY IO IBLE FIELD We set E Ek If 0 , · 0 ad E/ Als, x 0 ad E/x E / y if 0, E is uifrm If 0 , · / ad E/ / A l s , x E 0 ad E/x E/y 0, as befre The is a fut f , but idepedet f x ad
=
= == =
=
0 =
=
==
12
0
4)
ION BE
2 2 Fr om Laplace ' s equation, 3 V / 3x
/ A ,
- / V / x
V - ( / ) xB Se V a a x 0 B Als, V - ( / ) a Aa ad A V /aa/ Fally, V ( V / a a/ ) x - x /2
=
=
E
=
=
-dV/dx
V /aa/2 x/ A IFO ND A NON UNIFO FIE D
-10 (4)
£
a) V 1000 x E -1000 See Fs a ad ) V/x -10 4 , V/x -10 4xA, V -10 x / xB Se V a a x 0 B Se V 100 a x 01 e A 4 2 V - 10 x /2 1 00x See Fs ad d
=
=
=
=
1
= 00
3 - 11 3.4) VACUUM DODE 43 ) X-23 /9S -2V/'x2 o 4 3 V ( 1 Vo / s ) ( 3/4x 4 /3 + Since V Vo at x s then A v ( 1/mv eVo v J - ( 4E V 9s ) (2eV /m)
'V/'x
=
00
0
-.335 x 10 -6Vo 3 /
=
= 0
( 3 . 7)
0
2 -
2
2 3 /2
3/
-
-
G
I
0
T
-
t
e
P� " I :I
3-14 ( 3 . GES a) A t some ont P on the con uctng lane E Q /4 o (D + r ) cos Q D/4 D + r ) 3/ + r 3 E -QD/(D o
O
GES
EA - ( Q4 o a EB - (Q/ o a E C Q /4 o (a + 4a ) (a + a ) ED �Q /4 o (a + 4a ) J (-ai + aj J E tot (Q/4 0 a ( - + /5) - ( 1.ll Q /4 o a ) j
0
V x/s) 4/3 + Vo x/s 4/3 eVo /m _ ( 5 / ) em V 3 / s ) o
T
3- l3
(4V / 4/ 3 3x l 3 + A
Q
=
"
e .
b ) - D/ (D + ) /
o
-D
/(D + ) /
-
o
CHA PTE R - 1 ( 1 ) THE PE ITT IVI T O REE SPACE om Cou ombs a [ o ] [ /L ] = [ / (L)LJ = [ / / C ) L ] [ C/L] h th bacts cat that a coc oy th th msos a h L C sta fo oc Lgth a Capactac Not that L s a gy / C - ( 1 ) THE EARTHS ELECTRI C IELD a) C R 7 1 x 10 - 700 b) R R E ( 6 x 0 6 x / 9 x 0 9 x 0 C c) V R E/ R E R 6 x 0 8 V Rfc: Rcha yma, Lc tus o Phs cs Ch Aso Ws y
0
0
0
0
( ) PAALLEL- PTE CA PACITO R I f th a p ats C 8 8 x 10 - 1 ( A/ t ) Wth fou p ats C 88 x 10 -1 (A / t ) , tc o N p ats C 8 8 x 10- ( N- ) At 4-3
8 . 85 (N- l ) A /t) pF
- ( ) PAALLEL- PLATE CACITOR Th p at spaato mght b 1 m Th 10 -1 A 10 - , o o squa ctmt
1
8 . 8 x 10 -12A/0 -
4- ( 4 Z) PALLEL- PLATE CACI TO R C'
=
C C / ( C + C ) = E S / ( +b ) a b o a b
=
E S / ( - s ' ) O
Th capactanc s agr but t s npnt of th poston of cou g p1
8 ( ) ELECTROSTATI C ENE R
a) /WZ b ) W WZ
=
( Q V l / Z ) / ( Q V / Z ) z
( Q I V /Z ) / ( Q V/ Z) Z
=
4- 9
( 4 . 3)
LECTROSTAT I C ENE RG
V l / V
Z
Q /QZ
C l C V Z
=
Cl/C
Z
a ) The energy that is dissipated Ql
Z
Q
Z
Z
]
Ql + Q
Z + Z C Z CZ Z ( C + CZ
.
b) Th nrgy sspat b ou hatng n t rsstanc R of th rs t Q an Q Z th chargs at t = 0 Q an Q Z t chargs at t . C schargs to C Z Thn Q /C - Q Z C Z
=
IR , Q + Q Z
=
Q O + Q ZO 16
=
Q,
1
=
dQ Z /dt ,
Sc Q
= Q0 at t
d Q /dt 2
[
=
0 A Q 0 - 1/ C + 1/C 1 [ - ( 1 /C l+ l / C 2 ) /RJ exp [ - ( 1 / C l+ l /C 2 ) t /RJ
Q 20 -
[-Q 0 ( 1/C l+ l/C ) / R + Q / C lRJ xp [- ( 1/C l+ 1/C ) t/ RJ xp [ - ( 1/ C l+l/ C ) t / RJ / C R- Q / C R
[ -Q 20 / C R + Q / C RJ 2 10 l
W
[ Q lO
2
f ( dQ /dt) R
o
=
l
20
2 ]
- ( 1/C l+ 1 /C ) /R R( O- l )
( Q l /CQ0 /C ) ( Q l C-Q 0 C l ) 2C C2 ( C + C2) ( 1/C l + l/C ) l l Ths s th sult fou u a xcpt that th tal chags a o call Q l a Q 0 sta of Q l a Q 410 ( 4 . ) ROTON B O R R a ) W ( 1/ ) (1/ ) / o ) R / - / 6 ) 4
o
)
WG c) ) If
o
Us th aov sult placg 1/4 o y G G /5 R x 6 . 6 7 x l - ll x ( . x l0 / 5 x 1 7 4 x l 0 6 1 4 x l ( 1 0 0 0 / 1 7 x l - 7 1 6 x - 19 9 6 x l l C/m R s th aus of th sp of potos 4 R 5 /15 1 4 x 10 R 5 o x 1 . 4 x l 9 / 4 ( 96 x 10 10 1/5 0.17
=
0
411 4 . 5 ) ELECTOSTATI C MOTOR fc: A .D. Moo, Elctostatcs a ts Appl catos .
1
4-12 ( 4 5 ) ELECTROSTATC PESSE a ) V = Q /4"o R , E = Q / 4" o R2 , V = E R = x 10 6 / 0 05 = 1 5 x 10 5 V b) Th prssr s 2 /2"o ( Q /4 R2 = ( Q /4" R) 2 ( " 2 /R2 ) /2 ( 1 5 x 10 5 ) 2 " /2 R2 40Pa 4 x 10 - 4 amosphr
:
h
e
4-1 ( 4 5 ) PALLEL - PLATE CA PACTOR Lt ach p a hav a ara S Th h capac tac chags by C ( " o S / s ) = - "o S /s 2 ) s Lt s b po s tv Th capacac crass a a charg V ("oS /s 2 ) Vs rturs to h attry Tus -" E2 Ss WB - ( " o Th rgy stor th f crass by W ( "oE 2 sS/2) = ( " oV 2 S /2s) = - ( " o V 2 S /2 ) s /s 2 = -"o E 2 Ss/2 Th mchaca or o o th sysm s s S (" oE 2 / 2 ) s " o E 2 S /2
4 - 14 ( 45 ) PAALLEL- PLATE CACTO R S h prcg prob m Hr, WB W (" E2 sS/2 ) (" E2 S/2) s s ( " oE 2 S /2) s, "oE 2 S / 2
o
a E s costa
4- 15 45) OSCLLATN G PARALLEL- PLATE CACTOR a ) Th rgy sor h capacor s Q V/2 = CV 2 /2 = ( o A /x) V2 / 2 Lt x x o Th th oa potta rgy s W mg ( x-x o ) (x-x o ) 2 /2 oA ( 1 /x-/x o ) V2 /2 VQ , hr Q s th charg f to h batry bcaus of h cras capactac: VQ = -V2 C - V2 "oA ( 1/x - 1/x o ) Th battry gas rgy f C s gav Thus W mg (x-x o ) ( x -x o ) 2 /2 - oA ( 1/x - 1/x o ) V 2 /2 ( x-x o ) mg ( x -x )/2 AV2 /2xx
>
+
+
+
+
+
+
18
x-xo /x] x 2 /2
+ -x /2 x V2 /2x
There are three doward ore ad at euiiriu (x-x o o V2 /2x 2 . o at eui riu dW/dx (x-xo o V2 / 2x 2 0
+
v
The reatio F -(d/dx (b) oe ro the onseratn o eer or a a dipaeet ear euii riu. e tt dW/dx Kx-x we aue tat the W (x ure approxiates a paraoa W (K/2 (x-xe 2 i the reio ear the poit o stae uiiriu Thu K ( d 2W/dx 2 eq - 2 /x 3 e - V2 /x 3 6 . 1 6 z.
w
-16 . H-VOTGE GENETOR a The hare de it o the pat e ad hee E reai otat whe the pate are eparated The the reae i eer i 2 S (- / 2 . The ehaia wor doe i te ore E 2 S/2 E o o u tipied -
€
Reeree D Moore Eet rot ati ad it ppi ato Chapter 8 -17 . a C ' V
NK-ET RNTER 2 V/ R /R 2 o 3 /3 2R / R 2 32 R / 3 32 R A / 3 Q I
d
1000 Q/ 2 V/ 5 0 0 R
Q/ e
O
l
Q/
6 o VR /3000 R2 /R /3 2 R
tn ( R / R ) 2 l
8 8 x 10-12 x 100/ 00 2 x 10- 2 x 10- 3 /2 x 10 - 10 x 10 - 10 /
1
3
8x-C
dropet rea th deet dur Dur that te t ueted to a traere ore E ad t aeera 2 Te traee deto - to QE/ or 8 x 10 x 10 or 80 / at 2 /2 = 80 (x -3 2 /2 = 0 . 6 The traere eot at the ar ed o te deet pate at = 8 0 x x -3 = 0 . 3 2 / Spea ue o the BM oura o Reear ad Deeop et auar 19 7 7 .
CHTER (.1 c
=
CONDCTON N NFORM DM
c -ax/s , E o
J / c = J / (c ax / s ) o
-2 ( 2 SSTE FLM Let the hae a area a 2 ad a the t . R = a/at /t.
Te
-3 ( 2 SSTOET The thrut ' hre ' the a eted per eod d the exhaut eot. Se e the o uto o ro . 2- 1. Te ' 2 /2 3000 =6000/' ' 6000 ' 1 Reeree: Roert ah h o etr ropuo p 13.
B
- ( . 2 OLE LOSSE S V / = V 2 10 x 2 V = 18 V
2
- ( 4 VOLTGE DVDER The urret lowi throuh Rl a R2 i Vi ( R R2 Vo R2 V o /V i R2 / (Rl R2
=
-6 ( 4 OTENTOTER See rob - - 7 SLE CRCUT V ' V V R / (R R -9 LFER a Rl a R2 arr the ae urret I (Vi Vi / Rl (Vi - Vo / R2 (Vi Vo / /R l Vo ( -1/ - l /R2
-R2 / Rl (R l R2 /
Vo /V i V /V o i
�
-R / R if A 2 l
»
1 and
- R2 /Rl / l 1 / (R2 /Rl / The ai R /R l ut if R / R « A . 2 l
theeore be uh le tha
rsts
expx e
1xx
4 /2 ! x 3 / 3 ! x /4 1 1 l/ 1/6 1/24
-11 ( TETEDRON a B etr the urret throu CB a B are eual Te poeti al at C a D i hal-wa betee the po tet ial at a B b R/ R i paral lel with the R be twee a B betwee oe a B i R
Te reitae
-12 ( CUBE a B etr poit BED are at the ae potetial oit FC are
at anthe ptenta
b) The esstance f A t BE s R/3 That fm BE t FC s R/ 6 That f FC t G s R/ 3 The ess tance is 5R/6
5-13 (55) CE a) P ints FB Banches F and can be eithe emved sh t ccuted b) Reve ths e b anches Aund the inne squae , we have a es s tance 2R/2 R Aund the ute squae , the es is tance is 3R Thus we have R and 3R n paae and the ess tance s 3R/ 4 .
5-14 (55) CUBE stt the cube as in Fg a Then, by syety, B and E ae at the same ptentia and can be sh ted Smay , C and can be sh ted Nw edaw the igue as n b and c The ess tance t the ight f the dtted ne s 0 . 4 R and RA R( 1 4R) / (R+1 4R) 1 . 4 R / 2 4 083R
G
D 2
5 15 5 ) LINE FAUT LOCATION et the ength the ine be and et a be the resistance ne meter wire Then 2 a x + Rs = 5 5 0 / 3 8 145 . 5 , Rs 45 5 2 ax , 2 a x + Rs 2ax) / Rs + 2ax)
=
550/2
639,
axRs + 2ax)+ Rs ax) 3 8 9 Rs + 2ax) , a s + 2 a2 x 2 a2 x 2 + Rs ax) = 3 8 . 9 Rs + 6 . 39 a 6 39ax . Canceing the axRs t erms and subs ti tting the vaue Rs in the irst equatin, 2 a2 x 2 a2 x 2 + 145 . 5 2ax) a 38 . 9145 5 2ax) + 6 3 9a 6 . 3 9ax , 2 a 2 x 2 52 . 8 ax + 6 9 a 55 5 . Nw a 1 / 5 8 x 1 . 5 x -3 2 2 . 4 39 x 3 Sving, x 818 kimeters
516 5 ) NIFO RES ISTIVE NET a) Frm Kirchf ' s vtage aw, the sum the currents wing int is zer Thus VAV ) /R + VB V ) /R + VC V ) /R + VV ) /R 0 , Vo
=
(V +V +V +V ) / 4. A B C D
b) Fr a threedimensina circuit we have 6 resistrs cnnected t 0 and
51 5 9) POTENTIA IVIER Fr the eft hand mesh, V -I R-(I -I ) R i l l 2
=
O.
Fr the midde mesh (I -I ) R + I R + I R 2 2 l 2
Then 2
=
O.
Vi /5R, V = V/5R) R
: ViS
2
(59) S ILE CIRCU IT WITH WO (Rl + r ) I , - rI V , - rI l + (R+ r ) I =
I
=
1-
=
SOUR CES V
(R-R l ) V / [Rl R+ r (Rl +R ) ]
DELTA- S TAR NSOTI ONS Equatg th voltags VA-VB R ( ID-I C ) RA ( IB -I C ) + � ( IA- I C ) VB -V C Ra ( ID-IA) � ( I C - IA) + R ( IB -IA) V C -VA ( ID-IB ) RC ( IA- IB ) + RA ( I- IB ) Rrtg - IARB - IB RA + I CRA+ R) + IDR 0 - IA ( �+R C-R - IB R - I C + IDRa 0 - IARC + IB (RC +RA\ ) - I CRA + ID\ lmatg ID from Eqs 4 a 5 - IA-IB RA + I C (RA+ -R C ) + (R / Ra ) [ - IA ( �+R C-R + IB RC + I C IA [ - � - (R /R ( +R C-R + IB[ -RA + ( RRC 510)
Thus � + Re / R �+RC - Re � + (R/R)� + RA - Re =
0,
=
RA /RC
=
RRa
(1)
() () ( 4) (5) (6)
= 0 ( 7)
(10)
( ll ) Oly to of ths quatos ar pt. Combg th frst to
+ (RA /R C ) + RA
Or , setting
R
=
Re' Re
l/ G , G
=
=
°
( �RC + R C RA + RA ) /RC
GAGB / ( GA+GB +GC )
4
(12)
( l)
-20 10 )
ETA-ST TRNS FTIONS
5K
Redaw the ccut as n Fg a and tanso the eft hand deta nto a s ta, as n Fg b , wth 4000 x 1000/ 000 4/)1000, R2 8/)1000, R3 2/)1000, 2 89 k R -21 12) OUTPUT SISTNCE OF A BRIGE CIRCUIT The output esstance s the esstance one woud masue at the temnas o the votmete the souce wee epaced y a shot ccut Ths s R/2 + R/2 R -2 2 12 ) INTEA RES IS TNCE OF AUTOMOBIE BATTER The headghts , ta ght s , e t c daw ab out 1 A Hence the nte na sst ance o the ba ttey s about 1/ 1) . Ths s much too age , be caus e a c ankng moto daws , s ay 20 0 A A noma automobe at tey has an ntena esstance o the ode of 10 -2 . S tandad Handb ook fo Eectca Engnees , Se ctons 21 and 24 -23 14 ) IS CHARGING A CPACITOR THROUGH A RE SI STOR Fom Kcho ' s votage aw, Q/ C RQ/dt Thus RdQ/t - Q/C 0 , Q Q o exp-tC) , V Vo exp-t/RC) 100 exp -t) -24 14) Vo
Q/C
GEATOR
It/C
2
5 - 5 ( 5 1 4) CHARGIN G A CACITOR THROU A S I STO Th gy supp by th souc s CV Ws Vt V ( Q /t) t V Q CV . 0 0 0 Th gy sto th capacto fo t + s CV / . Th gy sspat th ssto s R [ (V/R) Xp (-t/RC ) ] t CV /
o
5 - 6 ( 514) RC TNS IENT a ) I V/ R + CV/t Vs IR + V R1 C dV/ dt + ( l +R l / RZ ) V
=
(V/ R + CV/t ) R + V
Vs
Sc V 0 at t 0 , V [Vs / (1+R / R ) ]{ 1 -xp [ - ( 1 +R / R ) t/ R1 C ] } [R / (R+R ) ]{ 1-xp [-(R +R ) t/R1R C ] }Vs Th tm costat s R1 R C / (R+R ) o C/ ( /R /R ) . For t
+ 0,
V
=
RZV s / ( Rl +RZ ) .
b ) No at t 0 , V RVs / ( R+R ) Th capacto schags thouh R a
5 - 7 (14) RC DIENTIATIN G CIRCUIT V Q/ C + RIQ /C V o RI RQ /tRCV t
/
A
6
IFFEENTIATING A SQURE AVE
5-28
5-30 ( 5 . 14) RC INTEGRTING CIRCUIT The cuent fwng nt te capact s I . V RI + Q / RdQ/dt + Q/ RdQ/dt , t V Q/ (/RC) V d t
o
5-31 ( 5 . 14)
INTEGRATING CIRCUIT
5-32 ( 5 . 14) INTEGTING CIRCUIT (V-V A) / R Cd/dt ) (V A-V ) +V /A -RC(+/A)dV /d t , V
-RC(+/A)dV /dt V A -RCdV /dt f A 1 and f V A « RCdV /dt
»
2
5-33 ( 5 14) PULSE-COUNTING CIRCUIT ) uing puse, the vtge css C is ppxitey equ t Vp nd Q C Vp Afte the fist puse, the vtge css C 2 is C Vp / C 2 The p ces s epets itsef . The vtge css C2 inceses by C Vp / C 2 t ech puse
CTER 6 6-1 ( 6 1 ) THE IPOLE MONT p ) p P N 10 - / 6 02 x 10 2 3 3 5 / 2 ) 1 0 6 5 x -3 C -3 -19 - 19 b) s p/Q 5 x 10 6 x 1 6 x 10 5 9 x 10 The diete f n t is f te de f 10 -10
6-2 ( 6 2 )
THE VOL N SURFACE BOUN CHARGE ENS IT IE S
T PdT
o
+ b d S
6-3 ( 6 2 ) BON CHARGE ENSIT AT N INTERFACE In the figue, we hve shwn he tw edi septed f city On the fce f 1, b
C
On 2, b 2
-P 2 (-n)
6-4 ( 6 4) COAXIAL LINE Cnside ve f dieectic hving the shpe shwn in the figue
T
28
VT EdT Ea
T
S hee S s the surce b oundng The surces A nd B re the ony ones here Ed s ot ze ro . Then, ther rd re rA nd rB ,
Ed
S
-(/2 o rA) rAL
V·T dT
o
( /2 rB ) rB L
o
nd, snce rA , rB nd e rb trry ,
V· E
.
6 5 ( 6 . 7 ) COAXIL LINE ) Ner the nner conductor, E /2 r o I s the chre er met er nd C the cc tnce er met er , C V [ 2 r o /n(2 / ) V Thus E
5 10 6
=
l
V/R n R /R ) l 2 l
500/ n( 5 10 -3 /R ) , R 1 . 77 2 10 -5 m b) One shoud use No 34 re. 3393 F/m c) C' 2 2 . . 5 -12 /n(5/. 6 )
6 - 7 ( 6 . )
CHAGED I EEDDED IN DIELECTIC THE FREE ND BOND CHAGES ) Insde the de ectrc, D / 2 r , E /2 r r On the nner surce o the de ectrc, On the outer surce,
2
6- ( 6 . ) AALLELLATE CACIT ) e cn tret ths ro em s e hd to cctors n seres C
l
C
=
E A/ ( s- t) , C z o
=
= C C / ( C C ) = r A t(s-t)
s-t
G
5
E E A/ t r o
oA
t
r A t (s- t) r
oA
- ( t / s ) ( / )
6-9 ( 6 . 7 ) ) Snce the ony ree chrge s Q, E. 6-7 ges us tht Z -9 Z - Z D = Q/4r 0 /4r 7 . 9 6 /r C /m oth nsde nd outsde the de ectrc. Insde the de ectrc, E D / r o Q / 4 r o r Z - / r 7 . 95 0 . 65 0 - /r
Outsde the de ectrc, E o D/o Q/4 o r . 9 4 / r To nd V, e set V 0 t nnty. utsde the shere, V o Q/4 o r 9 . 0 0 / r V
At the surce o the shere, V 450 Insde the shere, V = 450 (Q/4 r r )dr r
0.1
00 00 /r
c) Let us y Gusss to sm e ement o ome t the surce Te od cre o te e ement o re s
2) ( - ) , ) (Q/4R -E ( r \ o s eviou s y . The discontinuity in E is due to the bound surce chrge.
0 l
6-1 0 ( 6 . 7 ) CHARGED DIELECTRIC SHE Outside the shere, E (4/)R /4 o r 2 At r R, V (4/)R /4 oR Inside the shre , D (4/)r /4r 2 V
At the center,
R2
R / o + ( r / o
R P / 3E r f o
2
2
R P / 3 E R f o
=
R P / 3 E O
r / , E
r / r o
dr
6-11 ( 6 7) ASURING SURFACE CHARGE DENSIT IES ON DIELECTRI CS 4 12 ( 1 9 6 9 ) . Reerence: Journ o hysics E
6- 12 ( 6 . 7 )
VARIABLE CAACITOR UTILI ING A RINTED CIRCUIT BOARD
Y
C
In cses, dC y )
y
r oydz/t, dC/dz
10 - dC 10 9 dC 26.55 dz x 8.85 x 10 -12 10 9 10 -9 7 . 7 26.55
r oy / t
1
b)
y
=
0 9 0 - 2 z 2655
075z m
6-1 ( 6) EQUI OTENTIAL SUFACES ) No. b ) = D . The chrge dens ty s os t e on the sd e here the ect or D onts y rom the shee t . 6-14 ( 6)
NON-HOMOGENEOUS DIELE CTICS r) = 0 r o = o r 0 , E 0 nd b 0
vn = V(E E E (E v· + ·VE Snce VEr" b E v (E r 'VE r
6-15 ( 6 9) FIELD OF A SHEET OF ELE CTONS TED IN LUCITE ) The tot ree chrge s -10 - 7 C - -4 -7 - 10 25 2 -2 000 10-2 / m
b) From Guss's , D on ech sde, n the neutr regon, s one h the ree chrge er sure meter -4 Dn - ( 1 / 2 ) 1 0 - 7 /2 5 10 2 - 2 . 0 0 -5 Cm -5 C/m 2 1 . 7 5 ) D n n r
(E
En
(Dn - n ) Dn r o -7 . 062 10 5 V m
Snce E n -dVn d, nd snce Vn 0 t = 6 10-3 5 Vn 7 0 62 - 4 , 2 7 V c) At both surces, ont nrd, ke E nd D, nd b n - 1 . 7 5 1 0-5 C /m 2
2
d) Inside the chagd egin,
c
dD c /dx
2 0 00 x l - 2 C /m 3 , D
2 0 0 0 x l 2x C /m2
The cns tant in tega tin is ze b ecaus e D changes si gn at x S D o at x A l s , c d c /dx ( + b ) / 7 0 6 2 x l V /m 2 , c 7 0 6 2 x l x V /m The cnst ant int eg ati n is aga in ze , the same ea sn Fm issn's equatin, 2V c d 2 V c /dx 2 ( b ) 7 062 x l V/m2
2 dVc /dx 7 062 x l x V/m The cnstant integatin is ze, since dV c /dx V c 3 3l x l0x 2 3 4 V
c
The cnstant integatin is nw chsen t make V c
e) See cuves
l 3 ) The s ted eneg is 2 x ( /2 ) V c 2 5 x l 4 dx
o
g) N 616 ( 6 10)
Thus 10 3
Vn at x
1 3 x 10 4
SHT LCT
O
-
t
T E�
P ! E = C b
/
E o: P o
6 7 ( 6 7 ) LATION BTWN R D C FOR AIR OF LCTRODS Let th e aea ne plate be S and the spa cin s Then /
c c
3
Q=0
CHTER 7 7 - 7 CONTNUTY CONDTONS AT NTERFACE D /r both inside nd outside the dieletric Ei / r o r inside
o
E = A /2� r out s i de .
is continuous t the surce but its slope dV/dr is smller inside thn outside. 7- 7 CONTNUTY CONDTONS AT NTERFACE D Q/4r both ins id e nd outside the dielectric Q/4 r r insie E E o Q/4 o r outside Thus t the sur c e E o E i Q/4 R l-l/ r V is continuous t the surce bu t its s lope is smller inside the dielectric
7- 7 ENERGY STOAGE N CAPACTORS W QV/ CV / 0 -6 x 10 6 / 05 mgh 0 . 5 h 0 5 / l x 9 8 5
=
ENERGY STOGE N CPACTORS 7-4 Fo Mylr 5 x l 8 / W . x 8 8 5 x l 5 x l 0 /m One would use te geometry shown in the igure We need n b so lute mnmum o one kilowt t-hour Then we need 600 x 00 0 or . 6 x 0 6 . Assuming 00 eiciency which is unrelistic the ctul overll eiciency might be sy 5 the cpcitor wuld hve volume o 11 m . The density o Mylr being pproximtely eul to tht o wter the cpc itor would hve mss o 1 tons which is bs urd
%
%
4
7- 6 7
BOD S RFE HARGE DENSTY
7-7 ( 7 ELE OF LARE ELETR ORE Te orce per square meter is r o E / 5 x 8 8 5 x l - ( 4 x l 7 2 / 5 x l 05 a Te orce is 5 atmosperes 7-8 ( 7 PEET-MOTON HNE We ave our seets o carge as n te igure Seets a and are - O b coincdent and are situated n te ields o c and d oosing te rigt-and direction as positve te ield at te position o a an b is E a , b / - b / (/
- ( l- l / r
/ r
+ C
Ten te orc es per unit area on a and b are 2 - l- l / E ) ( /2E E Fa (2 /2E r E o' Fb = -((b /2E r E r r Similarly, E c /2E o - b /2E o + /2E O = ( 1 /2E o ) �2 - ( l- l / E r ]
( / r b / o
Fc
=
( 1 + 1 / E ) / 2E o r
( l l / r (l-l/ r / o
/ o
Fnally F F F F b a c d
0
7-9 ( 7 SELF-ANG APAITOR F ( o E /S ( / S (S / (V/S (S/ (V / ( r o 8 r SV /t = x 8 . 85 x l - x 4 8 x l - x 6 x l / 4 x ( 7 6 x l -4 x l N Tis is a very large orce t is approximately te wegt o a mass o one ton
7-
7 . 3) ELECTROS TTIC CLS a) /Z) r Z o /d) Z Z x 5 P a d
5 m b) Th E in th Myar i 3 / . 5 x 5 /m. Thn th E in th air i 5 8 3 . Z x 3 x / . 5 6 . 4 x /m c) /Z) r o /d Z Z x 5 Pa d 8.4 m Rrnc: Static Ectriication 975 . Z5.
CALCULATIG ELECTRIC FORCE BY TE TOD OF IRTUAL WO Lt th orc b F . Am a vir ta diacmnt d . Thn th ork don by th battry i qa to th mchanica ork do n th incra in th tord nrgy th to qantiti bing qa. Th F dQ/Z) d Z C/Z) Z dC/Z Z / Z ) r o d Z F r o Z 3 x 8 . 85 x Z x . x 6 /Z x -3 . 33 x -3 7- 7.3)
7-Z 7 . 4) ELECTRIC FORCE E Z / x ) E Z / y ) EZ
i
j / z ) E
i
ZEE/x
...
ZEE
7-3 7 . 4) ELECTRIC FORCE S Pro 7-. Th mchanica ork don i qa to th incra in ctric nrgy. Both nri ar id by th battry dx d Z C/ Z) Z dC/Z Z /Z ) r -l) o dx/ F Z / Z ) r- ) o / 6 / Z ) 3 - ) 8 . 8 5 x -Z x . / 3 8 . 85 x 4 7- 5 74) ELEC TRIC FORCE O A DIELEC TRIC / Z o ) / r From Ga a E From Prob . 46 / Z r o ) n RZ /R ) Th E rnRZ /R ) d E Z /d r /nRZ /R ) ] Z -Z /r 3 ) Th orc i dirctd inward. Dirgarding th ign Z Z 3 8 . 8 x 2 x . 5 x 62 5 x 6 / n 2 5 x r 3 ( E -l) E V / 2n (R /R ) r F' r o Z l
x 3 r 3 /m3 3
a h i c od o dcc o , F ' 3 2 - 39 3 2 m3 ravia ia ioa ioa forc r ci ci c mr i 9 8 3 m3 So Th rav Ecic foc foc Gavia avia ioa for c 3 . 2 9 8 3 3 3 POARATO URET URET - . DIPAET D POARATO Fom Fom S c . . 4 voag o caa caa ci o i R ] , E - R
D oE r o - - R] R] dDd o R R -R - R P - oE - o - -R] dPd - o R ] - R R OERIO - . DIT EER OERIO 8 -2 2 2 -4 oA 8 Q 34 - W
e
Q V /2 - Q V / 2 = 0 . 2 4 8 ( 3 50 0 - 7 0 0 )/2 l 2 1 1
34F
34
W 2 9 x 2 - 4 3 3 4 4 4 W Wh 34 34 4 4 . 2 Th fficicy i oy 2 rc L oo , Dic Dic Egy Egy ov ovri ri o , 84 ; Poc Poc EEE , , 838 93) .
TER 8 8- 8 . ET ET I IDUTIO O TE TE AXIS AXIS OF A IRULAR IRULAR LOOP fo I 2 z 2 32 4 - x 0 12 2 . z2 32
}
2 - 32
3
+0 ·3 ·3
LO O P CUNT LOO SQUARE SQUARE CUNT
( 8 . 1)
B 8 ( �oI/4
o
d
a ( �oI / 4 a d/(a 2 + 2 3/2 o 21 � o I/a
20
8- 3 (.l FIED OF A HARGED ROTATING DISK
a) E
=
/ ' b ) O
a
v
20
wr
=
A rng o radus r and dth dr ats as a urrent loop. So rom Se .1.2 R B � (wrdr/2r o o d E 1 0- 6 / 85 x 10 - 1 2 1.13xl0 5V/m B 0 5 x 4 x l - 7 x l 0 3 x l x 10 - 6 6 . 2 x l - l l T - 4 ( l SUNSPOTS a The urrent loop eteen rnd r + dr arres a urrent rdr rdr (w/2 (w/ 2 wrdr At the enter R B ( � /2 wrdr/r = �o wR/2 o 2B/�o wR = 2x.4/4xl - 7 x 3 x l - 2 x l 0 7 20/3 The The eletron ele tron densty dens ty s (20 (2 0 /3 /3 / 1 6 x 10 - 1 9 10 19 m- 2 The urrent s the total harge dvded y the perod: /w 3 3 x 10 12 A I [ x 10 1 4 x (20/3 ] / ( 2 /w The negate harge o the eletrons s neutralzed y quas statonary postve ons. - 5 ( l HEOT OIS B = 2� oNI} /2( /2 (+} +} /4 3/2 0.8 3 / 2 � NI/a = . 992 x 10- 7NI/a Reerenes: Durand Magntostatque pp 44270; ODell The Eletro dynams o Magneto -Eletr Phenomena Appendx 4; Ruens Rev S 8
sr 243 945 8 - 6 (8.1) HEOT COIS a) In he norhern hemisphee he magneic ield poins donard. In a N- S plane looking W he coils are oriened as in he igure.
\
c) 8 . 992 x l - 7 NI/a xl - 5 NI xl - /8.992xl- 7 55.6 A. d) Try a curren o 2 amperes so s o make he numer o urns as small as possile pos sile . Then Then e need leas 282 8 urns urns in each coil coi l . Then Then R 2 8 x 2 x l x 2 . 7 x l - 3 3.82$ V 7 . 64V P 4 x 3.3 . 82 15.3 W or each coil. No cooling is required or his size o coil. INEAR DISPACE DIS PACENT NT TRSDUCER TRSDUCER 8- 7 ( 8 . 1) INEAR 3/ 2 B ( oa 2 /2) l[a l [a2+ z- a) 2 J 3/2 _ 1 / [ a2 + ( z+ a ) 2 J 3 / 2 (oa2 /2) [l/(z 2 - 2az+2a 2) 3/2 _ 1/(z 2 +2az+2 +2az+ 2 a2 ) 3 / 2 J ( o I/2a)[1/(z 2 - 2z+2) 3/2
2 3/2 J - 1 / ( z, + 2 z ' + 2 )
z '
z / a
THE SPACE DERIVA DERIVATIVES TIVES O B A STATIC IED 8 - 8 ( . 2 ) THE \ B x/+ By/ Y + B z/z 0 B y / Y is posi po siiv ivee . By syery syery B x /x is also posiive. Then B z / is necessarily negaive. MONOPOES 7 - 9 ( 3) METIC 1 5 ( 10/4 /41x9 10- ) x 0 . 6 . 0 5 x 0 - 8 J Q*H£ . 2 7 x 10- x (10 1 . 05 x 0- /6 x 10 - 66 Gev
3
(8.4 GNETI FIED OF A HARGED ROTATING SPHERE a Q/4r 2 V Q/4 R o V/R v ( oV/RwR sn V sn B o ( wV sn Rd (R sn 2 /2R3 (2/3 o wV o d B (2/3 8 . 85 x l- 12 x 4 x l x 2 x (10 4 /60 x l 4 7 . 75xl - 11 T The eld s parallel to the axs o rotaton. e m (1/2 [ ( o V/R (2R n RdJ (wR sn R sn R3 o wV sn 3 d (4/3R 3 wV m (4/3 10- 3 x 8 .85 x 10- 12 x (l4 /6010 4 3 .882 x 10- 7 2 g (/410 I 3 . 882 x 10 - 7 I 4.943 x 10- 5A
\
-7
-2
HAPTER 9
9 -1 (9 .1 DEFINITION OF
9 - 2 (9 . 1 GNETI FIED OF A URENT- ARRING TUBE a B I/2r A s parallel to the tue and n the same dreton as the urrent B nsde s zero + d +A s as aove. lux lnkage. I the urve s entrely stuated ns de the tu e here B 0 the ntegral +s zero and A must e unorm. Its vaue s o no nterest sne B . 9 - 3 (9.1 onsder the dashed urve ,
I t is not zero .
gives the
Fo r any curve C ,
M G NETI C FIED COSE TO A CURENT HEET 1-
-
B
-
-
- - - -
01
40
I
- 4 . 1 V DE AF HIGH-VOLTAGE ENERATOR a 2 oE 2 x 8 . 8 x 1 0- 12 x 2 x 1 0 3 /10- 3 3.4x10 - C/m 2 3.336x10 - 4A. I 3.4x10 - x . x x.1x60 b B 4 x 10- 7 x(3.336x10- 4 /0./2 4 x3.336x10- 11 4.192 x 10 9 - (9 . 1 SHORT SOLENOID +L B (o a2 /2 (N/Ldz/[a 2+( z - 2 ] 3/2 -L (oNa 2 /2L +Ldz - /[a2 +(z - 2 ] 3/2 -L z 2 2 2 z a {a + - -L
�
L- Z
[a +(L -
=
2
2 3/2
]
+
L+ Z
[a +(L+ 2
2
]
9 - 6 (9.1 FIELD AT THE CENTER OF A COIL R2 L /2 2 2 2 3/2 a) B (onI/2 x dxdz/(x+z R1 -L/2 (onI/2 2 (x + z 2 ) R -L / 2 1 L R 2 (onI/2 2 (L 2/4Ldx+ x 2 R1 2 2 )2 } R2
( nIL / 2) n{x + (L / 4+x o
R1
2
2 2 +( (o nIL/2 n )2 Note: Integratng rst th respet to x ould be muh more d ult. l+ ( l+ S
41
) The numer o turns s LRZ -R)n and the average length o one turn s Z(Rl+RZ / Z . Thus the lengt o e re s L(RZ -Rl )n(Rl+RZ ) R Z -R/ Ln Vn here V Zs the volume3 o the ndng. Also Z 3 ! Z( -) (L/ZRl )Rl n Zn( - l)Rl
URRENT DISTRIBUTION GIVING A UNIFO B The eld nsde the hole s the sae as one had to superposed urrent dstrutons: a unorm urrent densty throughout the ross - seton plus a urrent n the opposteZ dreton n the hole. Z Z The urrent n the ull ylnder s I a IR /(R a ) . Z From peres la B ax - � o I a y/ZRZ B ay � I ax/2R The urrent n the small ylnder s I Bx �o y/Za Z � o aY /ZR Z By - �o Ia (x-)/ZR Z Bx B ax + Bx By B ay + By � o I a /ZR Z �o I/Z(RZ - aZ ) The eld s thereore unorm nsde the hole. Note that B s proportonal to . Thus B o hen and B hanges sgn th . Also hen a + R + and B . 9-8 9 . ) SADDLE OILS Ths urrent dstruton s otaned y superposng to ull ylnders o urrent long n opposte dretons. Let I e the urrent long through the ol. Then the ur rent I that ould lo through one omplete rle s related to I as ollos: I = l - (Z/) os - a/R) - (a/R) (l-} /R Z ) Inse the let -hand rle at the radus r the B due to that sde s B � o (I/Zr)r Z/R Z) (� oI /Z) r/RZ . So 9-7 9.)
4
2 2 2 2 + y 2 ] ! R , B 2 = ( o I I 12'H (a-x) + y ] ! R ly [ (a+x) 2 + y 2 ] ! + B 2 y [ ( ax) 2 + y 2 F 2 B l c o s 8 l + B 2 c o s 8 2 = B l ( a+x) / [ ] + B 2 (ax) / [ ] o I a / 'R
Bx
=
By
o
1
S o B is unifo rm and parallel to the yaxs . TOROIDAL COIL
9 -9 ( 9 . 1 )
a)
<
J
B . da =
rt 0
( NI / 2 ') 0
0
'
NI 2'(R+cos8) dd8 0
0
'
d8
R+ cos8 d
0
The integration with respect to 8 mus t be done with care , taking into account the two branches of the curve . We integrate from ' /2 to +'/ 2 , where cos 8 is positive , and then from '/ 2 to 3' /2 , where cos 8 is negative . Then
J
o
2'
'/2
-'/2
+
3'/2 d8
' /2
R+ cos8
2 !
(R -p )
43
' /2
'/2
d8
R+ c o s 8 +
-'/2
'/2
' /2
d8
Rcos8
/2 d Rcos /2 2 _ 4 a rc tan 2 ! 2 arc tan 2_ 2 ) 2 R+cos _ (R (R _ p ) (R o Since arc tan a arc tan b = arc tan (ab)ab) , 2_ 2 ! d } 4(/2) 4 p ) 2R/ (R = a rc tan 2 2 ) 2 2 Rcos l(R )/(R 0 Thus ( �oNI/2T) f r 2 2T 2 pdp � oNI[R(R2 r2 2 ]
1
t
o
22
(R _ p l !
(R _ p )
The integration is more difficult ith Cartesian coordinates b) B = � oNI R(R2r 2 ) ! /r 2 at the radius Set R cos Then � oN/2 �NI[R(R 2r 2 ) ]/r 2 r2 / 2[R ( R2 r 2 ) ! ] r 2R[1(1r 2 / R2 ) ! ] For r2 «R 2 , r2 /2R(r 2 /2R2 R
CHAPTER 10 There is an interesting article on the crossedfield mass spec trometer in The ournal of Physics E, Scientific Instruments, Volume 10 (1977) page 458. 101 (10 .1) THE CCLOTRON FREQUENC 2 T cen trp oe beig BQv, BQv mv /R. b) Then W vR = BQ/m-1 9 c) BQ/2m = lx16xl /2x2x1 7 xl 27 = 7 5 megahertz . 102 (10.1) MOTION OF A CHARGED PARTICLE N A UNIFOM B The velocity component paralle to B is affected The component normal to -B gives a circular motion as in the preceding probem. 44
10 - 3 10 1 MGNETIC MIRORS The fgre shos part of a helcal orbt for a postve partcle The partcle drfts to the rght The magnetc force ponts to the left fter a hle, the drft veloct ll also pont to the left There s a good artcle on the magnetosphere n Contemporar Phscs, 165 1977) . 10- 4 10 1) HI ENERG ELECTRONS IN THE CB NEBULA a) W 2 x l 0 14 x 1 6 x l - 19 3.2xl - 5 J b) m 3 2 x l - 5 /9xl0 16 3 6 x l - 22 kg, m/mo c) R d) 2x3.4xl0 13 /3xl )/24x3600) 2 das
2 2 mc /m c
3 . 6 x lO -
22
/ 9 . l x lO -
3l
=
4 x lO
S
10 - 5 10 1) MGNETIC FOCUSING a) A electron goes throgh one fll crcle n a tme T 2/ Drng that tme t travels a dstance L vx T So B 2 3/2 m/e) ! /L L 2eV/m)2m/Be) 2 3/2 mV/e)/B, b) B 2 3/2 9lxl - 3l x 10 4 /1 6 x 10- 19 ) ! /05 424xl- 3 T IN' B/� o 3373 10- 6 101) DESTER SS SPECTROTER a) mv2 /R BQv, mv BQR, 2m1/2)mv2 ) BQR 2mQV) BQR, m 1 QR2 B 2 /2V b) B 2mV/Q) /R - 2 7 2 x 1 7 x l x l/1 6 x l- 19 ) ! /0 06 77xl - 2T HI 011 T 0 094 T 45
Fo r
+
'
B l
Noe ha m/m 2B/B Thus, or large m's, here m/m becomes small rom one isoope o he nex, B/B becomes even smaller c B 2mV/Q /R 2xl 7xl27 l/l6xl 19 ! /006 3 4xl 5 /1 6 !! /0 06 7 68xl 2 ! here is he aomic eigh This value o m is approximae 10 7 101 M PECTROTER mv2 /R BQv, R mv/BQ z x 2R 2/Bm/Qv The ime o ligh rom o he arge is R/v /Bm/Q During ha ime he ion ris hrough a isance y 1/2 QE/m / BQ 2 2E/2B 2 m/Q Reerence Rev ci Insr 819 1974 108 10 1 HIHTEETURE PLM a mv2 /R BQv, R m/BQ 2v2 /2 /BQ 0 225 m b ion has he same velociy, bu hal he mass, so R 113 m Reerence: lassone an Loveberg, Conrolle Thermonuclear Reacions pages 156 an 395 109 10 HIH TEERTURE PL a By sery, B can only be azimuhal Bu he line inegal o B· over a circle perpenicu lar o he paper an ih is cener on he axis o s ery mus be zero, since he ne cur ren is zero. Then B < ® imilarly, B 0 insie he inner cyliner b ee igure c I bens oars
46
It bens upars e They also return to the scharge. Reference G1asstone an Loveberg, Controlle Thermonuclea Reacton p 278 1010 (10 . 1 ION BE DIVERGENCE a I v, I/v b QE Q (/2 oR QI/2 Rv c QvB Q(o I/2R QI o/2R QEvB (QI/2R(1 ov o 1011 (101 ION THRUSTER See the soluton to Pob . 21. Here, the foce exete on the ejecte fuel, n the eference frame of the vehcle, BIs, o m'v. I 2 R I 2 (s/A , hee A s the aea of oe of the electoes , C o D. l/(lP D /P G 1 D A BIsv / ( 1 2BIs /B 2 w) 1/(12m'/B 2 T Also, 2I/v 2/Bv 2E/Bv 2E/Bv, 1/(1 2E/Bv . As v ncreases , l, an l for v » 2E/B 1012 (10.3 G (1 2 ! 1/01, 2 11/1, S 1013 (10.5 ENCE 1 Y 1/(11/ 1.155 1.15 (11. 5x10 8x1 1. 732 x 10 8 m
/
=
Y
1
= 1 m, z
2
= z
l
= 1 m
1.15 s. 7
(10. 5)
REENCE FRAS
1.155 as aboe 8 1.155 (1l.5xl xl) 1. 732 x 10 8m, Y l z l 1m, t l 1.155 s 1015 (10.8) HALL EET a) M(E+x) here i k x B x Y z 0 0 B x + Y + zk M Ex+EYj+YB xB , vx M(E x+ YB) , vY M(EY x B) z 2BE M(E x+E ) +M x b) x M[Ex M(EY x B)B l+M 2B 2 1+M2B 2 ) x2 E y Y M y (E l+M B 2 2 2 E M M2 2 (Ey+Ex) M Ey + M B x neM(Ex Y B)/(l+M 2B 2) y neM(Ey +x B)/(l+M2 B2 ) c) V y (b/a)MVxB (10 /5xl 3) 7 x 1 x 10 4 1. 4 x 10 V d) When E Y 0, x ne x /(l+M2B 2 ) , I x bcneM(Vx /a)/l+M2 B2 ), R Vx Ix a(1+ B 2 )/bcneM, /Ro M2B 2 Let us caculate the mobility in copper . 28E, (/)E. 19 7 Thus the mob ilitY : s a/p a /n 5 . 8 x l O / 8 5 x 10 x 1 . 6 x 10 3 = 43xl . Reerences HH Wieder, Hall Geerators and Magnetoresi stors ;
]
� ) ] f
4
3
=
48
H . Weiss S tructure and Appl icat ions of Ga1vanomagneti c Devices .
( 8) V
ELECTROMGNETI C FLOTERS
vBa
CHTER 11
( 11 . 1) a) Bv
BOAT TESTING TANK
2 x 10
-5
x 20 x 3
=
1 . 2 mV
b ) Zero . 11-2 ( 11 1) a) I
=
EXDING LOOP
Bvs/R
2 (Bvs) /R b ) ( B s ) v 2 2 c) I R = (Bvs ) /R . The p ower exp ended to move the bar app ears as
heat in the resistance R. 11-3 ( 11 . 2)
INDUCED URNTS
t
Reference: Rev . S c i . I n s t rum . ,
�, 49
1581 (19 77) .
4 (11 .) NDUCED CUENTS a) Counterclockse b ) Counrclockse c) Snce the flux-lnkae s con stant, and snce vxB 0, the nduced electromotance s zero (11.2) NDUCED ELECTROOTANCE d/dt NA(dB/dt) 100 x 10 - 2 x 102 x x 60 sn( x 60 t) sn(2 x 60 t)V We have dsrearded the sn. 116 (112) ELECTRONETIC PROSPECTION
a) The nduced electromotance s azmuhal . Over a crcle of radus r, 2 2rE r B owsnwt, E (rB w/2)snwt The nduced current dnsty s E and s also azmuthal J (rBo w/2)snwt. b) Wth our sn conventon, a postve J ves a postve B At t 0 , dB/dt 0 and J . Then, as B dcreases, J ncrases as pr Lenz's la, etc. 1 7 (12) NDUCTON HEATIN a) d/dt -r2 (d/dt)( oN'I coswt) b ) The lenth of the conductor s 2r and ts crosssecton is Ldr 50
Hence R 2r/L r o o2w2N 2 2 /4 Lr r c) P a 2x2/Lr The average va 1ue f n 2 wt b eng equa1 to 0 5 R ) P av ( )L r 3 r ( )LR4 /4 (4x7 x2x60x5000x2) 2 o (x 5 16)(6x ) x 5 71 W Note The poer issipate in the inding is I 2R 7here R is its resistance The conductiity of copper being 5 8 x 10 sieens per eter if 2there are n ayers the crosssection 7of the ire2 is 2 (n/5000) /4 an R 2 x 6 x x5000/58x0 (n/5000) /4J I 2 R 103 I2 /n2 If n 10 I 2 R 40 W . Reference Stanar Hanbook for Eectrica Engineers p 2228 an fooing z 118 (112) INDUCED ELECTROMOTNCE y t (w/2)t wt = 2Y / Y/4 B o sin(2Y /)sinwt(/4)dy y /4 y B o sinwt(/4) ( /2) [cos (2Y /) ] y 2 /8)B o sinwtcos(2Y //2) cos(2Y /) J ( 2 /8)B o (sin2 wt sinwtcoswt)
2 2 ( 1 U WN ' I )
2 4
=
(B /4) (sin2wt cos2wt)
1110 (114) THE TOLMN AND ARNETT EFFECTS In the reference frae of the conuctor the force on a partice of charge e an ass is p
Lanau an Lifshitz Eectroynaics of Continuous Meia 51
(5 ELETRI ONDUITS Fro he definiion of (Eq 88), A is parallel o he ire Then A/ is also parallel o he ire and, if here is a single ire, here is a longidinal indced elecrooance in he ire Sandard Handook for Elecrical Engineers, Sec 7 (5) THE POTENTIALS V and A Since x (A ) xA, B is no affeced lso (V + /) (/)( ) v / and is no affeced eiher HAPTER 2 2 (2) [H] [o ][L], [o ] [H]/[] 22 (2 ) TUAL INDUTNE Asse a crren I in he ire The flx liage hrogh he oroidal coil is (o IN/2) nl+/a), M (oN/2)n(+/a)
23 (2) TUAL INDUTNE a) Fro Sec 82, coil a prodces a a agneic indcion B = oNa I a a2 /2 + z 2 ) 3/2 {JoNaa2 /2(a 2+ z 2) 3/2}N 2 oN aN a2 2 /2(a 2 + z2) 3/2 So " ab / I a ) M varies as he cosine of he anglar displaceen c) No 24 (22) OUTSIDE A SOLENOID The agneic flx inside he solenoid is R2 oN'I Then he elecro oance indced in a loop of radis r> R2 coaxial ih he solenoid is 2 R oN'dI/d 2rdA/d, A = ( o /2r)N'R I M
=
=
52
(22) NSDE SOLENOD Te magnetc fux nsde a oop of radus r
-3
-7
1210 (23 COXL LNES rom Ampere's crcuta a, te B n te annuar regon beteen e conductors s te same at a frequences nsde te conductors ere s more fed at oer frequences Hence Wm s arger at o frequences and L £f Lf 53
12 1 1 125 ONG OENOID T ENTER TP 2 2 2 2 L N / 2 ) rR , ( � N / ) rR , L L o BC AB o AC Our frmua fr a ng send s based n he assumpn ha s � N I nsde, and zer usde h hs assumpn, he cupng cefcen k zer, and M s zer 121 127 OTGE URGE ON INDUTOR
=
M
O.
a (R Rs I dI/d R s I dI/d, 100 Rs I 10 4 The v1 ages acrss Rs and are b h abu 10 4 b nnec he dde R/ as n gure b Upn penng he sch, he curren s I /Re 0e
1214 127 TRNIENT IN R IRUIT In crcu a, / s b , 10 4 x 106 /4 2 5 x 1 0 H , In d 2 Q/d 2 RdQ/d Q/ = s s n Prb 1215, he parcuar un s Q s 2 + l / C 0 , n - R/2L) ± (R2 / 4 2 - 1 C) , n + R/2 100 /5 x 10 2 x 10 4 1/ Thus he vaues n are eua and Q expR/2 s 0, Q 0 and s s, I dQ/d expR/2 [ R/2 0, I = 0 and R/2s 54
R
=
=
Fnally Q -Vs C(1Rt/2 exp(Rt/2 + Vs C nce R/2 2/RC, Q Vs C[1- (1+2t/RCexp( - 2t/RCJ, v Q/C, V/Vs 1(1+2t/RCexp(2t/RC umarzng, V/V s lexp( - t/RC, for crcut a V/V s 1(1 +2t/RCexp(2t/RC, for crcut ettng t/RC t ' , V/Vs l - exp( - t', for a, V/Vs 1 - (1+2t'exp( - 2t, for b The fgure shos Q/V s C as a functon of t for the to crcuts The charges are the same at t 1 26 RC ' The crcut th the nucto s sloe at frst, an then faster The nuctor s not useful 12 - 15 127 TNIENT IN RC CIRCUIT a I/t + RI V, I Aexp( -Rt/ V/R nce I 0 at t 0, I (V/R [l-exp( -Rt/J l[ l- exp (lt J b ) I/t + RI + Q/C V, 2 Q/t 2 + RQ/t + Q/C V The partcular soluton s Q vc 10 4 The complementary functon s Q ent , n [ -R±(R3 2- 4 / C ! J 3/ 2 - 5 4±10 J Q exp(5t (Bcosl t + Dsnl 4t + 10 - nce Q 0 at t 4 0, B 3 - 10 , 3 4 Q exp(5t ( - 10 cos t +-4Dsnl 3 t + 110 , 3 3 I Q/t exp( - 5t (5 x 10 cos t snl t - 5Dsnl 3t +10 3Dcosl t 7 lso, at t 4 Then D 5xl - cos 3 t 5 x 10 7 snl 3 t + 10 - 4 10 4exp (5t cos 3t Q exp (5t ( 10 + 1044 I 10 exp( - 5t ( - 10 3 snl 3 t +45 x 10 4 exp(5t cosl 3t 0exp( - 5tsnl \ + 5 x 10 exp( - 5tcosl \ 0lexp( - 5tsnl \
3
55
CHAPTER 13 131 (13. 1) - 2GNETIC FORCE2 BIL = 5 x 10 x 400 x 5 x 10 32 (13.1) GNETIC FORCE a) Let the ire have a rosssetion a, and let the urrent density be . For a length of one meter, Ba ag, 8 g/B = 8 . 9 x l 0 3 x 9 . 8 x l - 4 8 . 7 x l 0 8/m2 I 8.7xl0 a b ) R /5. 8 x 108 7 a. "/m p ' ( 8 7 x l 0 a) 2 /5.8xl0 7 a .3xl l a W/m If a 10 - 8 , then p ' 130 W/m. The ire ill beome very hot. Convetion ill spoil the measurement. ) In the Northern hemisphere there is a South magneti pole. The lines of B point South. The urrent must point West. d) t the poles the lines of B are vertial and the magneti fore on a horiontal ire is horiontal and perpendiular to the ire. 133 (13 .) GNETIC FORCE F 50 x 00 x 0.5 x 10 x sin 70 0. 235 N 134 (13.1) GNETIC FORCE F I a d a x B I aB x d a . 13- 5 ( 3 . 1) ELECTROGNETIC PUS y Consider an element of volume dxdyd, as in the figure, ith the urrent floing along the axis The urrent is dxdy. Both and B are uniform inside the infini tesimal element of volume The d fore per unit volume is -
-4
56
HOMOPOR GENRATOR D HOMOPOLAR MOTOR
13-6 ( 1 3 . 1 )
v
BR 2 /2
x ( 3 2 /6 ) . 2 5 /2
3 9 2 7
13-7 13 . 1 HOMOPOLAR MOTOR The curr ent has a radi al component po intin towards the axis . The azimuthal component of the current ies a B pontin to the riht. The wheel turns counterclockwse 13-8 13 . 2)
METIC PS SU
.
-
.
a) We have prec isely the si tuation des cribed in Se c 1 3 . 2 . b ) Inside the inner solenoid there is zero manetic field Between the two solenoids the field is B The manetic pre ssure B 2 / 2�o pushes inward on the inner so lenoid . 13-9 ( 13 . 2 ) MGNETI P SS URE a)B 2 /2� B 2 /2 4 x l- 7 Pa B 2 /2 x 4 x l- 7 ) 1 -5 atmospheres o 4 B 2 atmospheres .
b
57
c) ( i) The pr es sur e is always equal to the enery dens ity (ii) The electric "pressure" we have considered is associated with the fact that lines of force are uner tension This "pressure " is always at trac tive (We have not considered the repulsion betweeen electric lines of force, which ives a positive pressure of oE 2 / 2 For example, if we have two electric chares of the same sin, one can find the correct force of repulsion by interatin oE 2 /2 over the p lane ha lf-way be tween the chare s where te lines o o rc e clas ) .
E
E
(iii) The manetic pressre we are concerned with here is associated with the lateral repulsion between lines of force This pressure is repuls ive In P rob 15-6 we are concerned with the tension in the lines of force, which ives an attractive "pressure" of B 2 / 2 o ( iv) In prat ice , the ele ctr ic "p res sure" is nearly always neli ible while manetic pressure is often lare. For example , a lare E of 1 6 V/m ives an elec tric "p ress ure" o f 5P , while a lare B of 1 T ives a manetic pres sure of 4 x 1 5 P 13-1 13 2) ETIC PRESS U a) B = o I/2R otside, B inside, from pere's circuital law. Thus m ( 1 / 2 o ) ( o I/2R) 2 = ( o /8 2 ) (I /R) 2 b) m ( 4 x l -7 / 8 2 ) ( 9 x 1 8 / 25 x l -8 ) 5 . 7 3 x 5 73 atmospheres Reference: J Phys D. App1d Phys 2187 (1973)
13-11
(1 3 2) GNETIC PS SU a) P 1/ 8 x 1 - 7 4 x 1 5 Pa 4 atmosphe res b) The pressure would be unchaned, since B is uniform inside a lon solenoid. 13-12 (1 3 3) ENER STORAGE 2 /2 8 . 8 5 x lO -12 x 10 12 / 2 4 . 4 3 J m 3 E o B 2 / 2 o 1 /8 x 1 - 7 3 9 8 x l 5 J /m 3
E
=
13-1 3 1 3 . 4) GNETIC PRES SURE a) The manetic force is 2Rpm It acts throuh a distance dR Then the work done by the manetic force is 2Rp mdR 5
b) The mechanical work done is Z /Z dR Z R(B Z /Z o ) dR Z R 0NI / 0
f
J
0
c) I (Nd INd ( RZ o NI/) 13-14 ( 1 4)
FLUX COSSI ON
a) As the tub e shr ins in di meter , an azimuthal current is induced that maintains the enclosed ma net ic f lux approximately constant Hence B � B o (R 0 /R) Z
l u
o , 1 3 / (4 1-7 ) 1 9 A/m 1 3 T c) B 1 ( 1 / 1 )
b) B
(1) (Z)
d) The solenoid maintains a constant B o in its interior The current in the tube incre as es the induct ion insi de the tube to B Thus the increase in manetic enery is m
(
Z Z 'R L B -B o
Jj
Z)
o
Z = 'R L
(3)
x ( 1 -4 / ) x Z ( 1 4_ l ) Z /8x l - 7
6 x l 6 J .
(4)
The source supplies an extra enery W s N I 0 (d/dt)dt NI 0 NI 0 RZ ( B-B 0 ) , Z Z Z -1 B , Z Z Z ' R (B L / ) R NI R R , R - 1 B R o 0/ 0 0 0 0 0 ( / o ) LB o Z (Ro Z -R2 ) / 4 x l 7 ) Z x l 2 ( 1 -Z - 1 -4 ) / 4 1 3 x l 5 J .
(
1
[ ' )
=
Te explose supplies an enery R R 4 4 - ZrL ( B Z - o Z / Z o dr - ( L/ o ) R0 r l Ro Ro
[ /
J
[
L / o ) B 0 Z R0 4 /ZRZ + RZ / Z R0 Note that
m
z
6 x l 6 J .
h0
Z rdr,
(5) (6)
(7) (8)
(9)
s + exp 1 Note also that, althouh the manetic field jus inside the solenoid 5
is unaected by the current I in te tube the current I prduces a A/ t in the solenoi d that makes W s The explos ie supp lies mos t o the enery We hae nelected the mecanical enery required to crush the tube acoustic enery etc
13-15 (13 4) 2
PULSED GNETIC FIELDS a) Wm = (B /2 o V = 4 x 1 6 J Cot $ 8 x 1 6 b) Wm 4 x 1 6 J 4 x 1 6 /3 6 x 16 1 kwh Cost 2 to 1 cents , dependin on preailin rates c) B 2 /2 o 4 x 1 9 Pa 4 x 1 4 atmospheres (1 3 5) a) Wm
GNETIC ENERGY
I a /2 Ib b /2
I a ( aa
a ) /2 Ib ( ab b ) /2
b) Wm 13-17 ( 13 5 ENER STOAGE a) Wm LI 2 / 2 / 2 I/ 2 b) Wm I N ¢ / 2 INR2 0 NI/2£
=
1
13-18 ( 13 5) a)
ENER S TOGE
= V I
i) LdI/dt
(V/L)t V2 /2L) t 2
ii) (1/2) (Vt/L Vt 2 / 2 L(Vt/L) 2 /2
=
b)
i) V
o R2 N2 I2 /2£
Q/C
=
V2 /2L t 2
(I/C)t
ii) ( 1 / 2 ) [ ( I t / C ) I t ( 2 /2C) t 2 iii) CV 2 /2 C ( I t / C ) 2 /2 (I 2 / 2 C ) t 2
=
13-19 (13 4) a) r 0E 2 / 2 B 2 /2 o
ENERGY S TOAGE 3 2 x 8 85 x 1 12 x 1 16 /2 = 1 4 x 1 5 J /m 3 6 4 / 2 x 4 x 1 - 7 = 2 5 x 1 7 J /m 3
b ) ( D 2 / 4) L x 2 5 x 1 7
( D 2 /4 ) 2 D x 2 5 x 1 7 1 1 x 3 , 6 3 6 x 1 13
D = 45 m, L
9
25 atmospheres A DL = 4 5 9 = 1 . 2 7 1 5 m 2 Foner and Schwartz , Superc onduc in Machines and Devices , p 41. SUPERONDUTIN POWER TRNSMISSION LINE 2 /2 D = 4 1 -7 ( 1 11 /2 1 5 ) 2 / 2 5 1 2 I I/2D)I a) F o o 6 1 N /m 2 o I/D 8 1 - 7 5 1 5 /5 1 -2 8 T . b) B 2 o I / 2 ( D / 2 ) B 2 /2 o = 6 4 /4 1 -7 5 1 7 J /m 3 c ) W ( 1 / 2 ) 2 ( 1/ 2) ( 6 . 6 1 -7 1 6 ) ( 5 1 5 ) 2 8 2 5 1 1 J ost = 5 1 -3 ( 8 . 2 5 1 1 / 3 . 6 1 6 ) = $ 115 . 13-2 ( 1 3 6 )
Reference: Proc I . E E E , April 1967 pae 57 13-21 (13.8) ELETRI MOTORS ND MOVIN-OIL TERS a) F 1 NIBb , perpendicular to both 1 and B F 2 NIBa, in the vertical direction
3
F 3 NIBb , p erpendicular to both 3 and B
F 4 NIBa, in the vertical direction b) T = 2NIBb ( a sin / 2) NIBab sin 13-22 (1 3 . 8) METI TORQUE a) T I / I ( B S c o s / ISB sin
m B
e
e
b) T
61
e
132 3 (1 3 ) ATTITUE CONTROL FOR SATELLITES a) NBA cos T I ( / ) (NBA cos ) NIBA s S ee fure for Prob 13- 22 2 3 b ) IN T/BA s 1 / x 1 -5 x ( / 4) 1 . 1 4 x . 8 7 3 13 24 ( 13 9)
s
28 At
CNICAL FORCES ON IS OLATED CIRCUIT
I(d/dt)dt
I
2 Wm ( L I 2 / 2 ) ( 1 / 2 ) I L ( 1 / 2 ) I 2 ( I ( 1 / 2 ) I Thus the mechacal work s Wm s equal to Wm
CTER 14 141 ( 14 . 2 ) ETIC FIELD OF THE EARTH e M s
N
If the sphere carred a surface chare des ad rotated at a aular velocty we would have R s R M v
s
142 ( 14 . 3) EQUIVALENT CURNTS The surface curret dest s the same as f a torodal col wre woud o the torus ad e M 143 (1 4 4 EQUVALENT CURNTS The equvalet currets are equal ad oppos te drec tos Thus B s de the tub e .
M 62
4-4 4 4)
DL CTRCS D GNT C TRS COD
( b)
( oj
a) b) D is unaec ted is reduced by r c) Te eey is mmum
See Fis a and b
0 0 0 0 0 0
0 0 0 0
0 0 00 0
0 0 0 0 0
(C)
d
d) e) H is unae cted , B is increased by r ) The enery is minimm Se e Fi e The loop is in stable equilibrium
See Fis cd
(e)
14-5 ( 14 4) TIC TORQU The manet acts like a solenoid.
See Probs 3 -22 and 3-2 3
46 4 4) ASUNT OF M B in cos 0 n 2 45 derees The ield is larest at
63
147 ( 14 4) a) V
MICROTEORITE DETECTOR
(d /dt) (Mb ) a a 2b 2 Ib (d/dt) 2 3/ 2 2( a + z )
b) See the iure Rev Sci Instr 42 663 (1971) 148 (1 4 4)
CHICAL DSPACENT TRSDUCER
a) See curve b ) z / ( x2 + z2 ) 5 / 2
95xl 4z, l / ( l + z 2 ) 5/2 9 5 x 1 3 , z 14 4 Reerence: H Wieder, Hall enera tors and Manetoresis tors , p 9 5
=
149 ( 14 5) MAGNETED DISK Outside, we can use the ield o a current loop: B ex oMt /2 ( + z 2 ) 3 /2 , H ex Mt /2 ( + z 2 ) 3 /2 Inside , we have the s ame value o B , with z 2 « a2 :
} }
} }
141 ( 1 4 6 ) TOROIDAL COIL WITH MGNETIC CORE Let N ' be the nb er o turns p r meter in bo th ca ses . From Eq 1415 , H N in bo t case s With the air core , B N With the manetic core, B is larer by a actor r he equialent curr ent low in the same dire ct ion as I
1411 1 4 6
EQUVALENT CURENTS
64
b) On the inner urfce,
ae
M
B /o - H
r l) H
mH mI / 2 b ,
in the sme direcion s te current On the outer surfce, e mI/2, in the opposite direction
a
c) B
o I / 2 r , i the iron were bsent
14-12 14 . 7 )
V B V · H+ ;
TH DIVRGN OF H
= V r . + r o .
) r o if
Vr ;
nd i
V r
+
is not perpendiculr to H
14-13 14 . 8) TH TIATION URV Interpoltin lorithmclly between the points mrked 2 x 1 3 nd 2 x 1 4 , 4 3 r 6 l x x l 1 . 2 2 x l . 14-14 14 . 8) ROWND RING ) H 5 x 2 4 /2 x 2 1 A/m, B . 5 T b ) Nd/dt NSdB/dt 1 x l -4 x . 5 1 /2 4 )
6 6 mV
14-15 14 . 9 ) TH WBR -TUN A weber is unit of mnetic f lux , nd d /d t is volte . Tu weber is vol t second . Te num er of turns is pure numbe r . So weber][mpere] [volt second][mpere] [wtt][second] [joule]
14-17 14 . 9 ) TRNSFOR HUM The hum is due to mnetostriction Reference: S tndrd Hndb ook for lec tricl nineers , Se c 11-9 6 nd followin. 14-18 14 . 9) POWR OS S DU TO HYS TRS IS The re of the loo is pproxmtely 2 . 8 x 16 , or 45 W /m3 cycle
5
THE FLUXGATE METOTER ND THE PEING STRIP
14-19 149
Reference: H. ijlstra Experimental Methods in Manetism Vol 2 p 37 Brandt Introduction to te Solar Wind p 14 ; M . S tanley Liins ton and John P . B lewett Par ticle Accelerators p 2 76 .
CHTER 1 11 (1 . 2 Wm
(1/2 LI 2
1-2 ( 1 . 2 ) L
RELUCTNCE
N
1- 3 ( 1 . 3) a) < = B A
(1 /2 ) LI) I
RELUCTNCE N 2
( 1 / 2) N I
( 1 / 2 ) < 2 /NI
G
CLIP-ON T ER I 2R-L / r o A L / oA
0 IA/L B 0 I/L
ithout the iron core B is o I/2R and is much smaller. b The po s it ion o the wire is unimpor tant . 1-4 For r
ETIC CIRCUIT
B = 2R/ NI r 0 + / 0
43
foI
2R/ r
4 x 1 - 7 x x 2 . 4 2 x . 2 /+ 1-3
T. 66
1 8 x 1- 3 1 . 2 5 7 / + 1 -3
Ths B s too l o � 5 B . 5 8 x l -3 44 T B 3 .257/525+
32 T
Ty �
525
Ths B s n too l; fo � 525, B 38 T . Ty � 55 B 49 T, instd of . 5 on th p Ths s stisfctoy 5-5 ( 5 3)
ETORES IS TE MULTIPLIER
5-7 ( 5 4) RELAY F (B 2 / 2 � o ) A ( A/ 2 � ) ( � N I / ) 2
0 0
( � A/2 ) (NI / ) 2
0
( 4 x l - 7 x l -4 / 2) 4 x l -2 / 2 x l -3 ) 2
6 N
5-8 (5 3 ETI FLUIDS ) Th mntc flx s concnttd in t ld, wh B is l noh to i n ppcibl antc pss b S th li tt pblishd y Fofldcs op
HAPTER 6 6- 6 .
RETIFIER IRUITS OARED
b
67
R
16-2 (162)
!
( l /T)
V
T
VALUE OF A SE WVE
16-4 (16.2) R VALUE a) V b ) 4V 2 /2 2 V
[
c)
i r (l/T)
[
2 2 V coS Wtdt 0
}
V (1-2t/T) 2 dt
V
2 0
2
/WT
Jf :
2 os ad a
( V /T )
r
y
(1-2t/T) 2 t
,
d) For the complete sine wave, V r V /2 , so that the mean square value is V 2 / 2 For the half s ine wave , the mean square value is V 2 /4, and V rms V / 2 .
16-5 2 (24)
THE-WE SNGLE-PHASE CUNT 33 V
16-6 (6 . 4) THREE-PHASE CURENT cost + cost (2/3) - sintsn 2/3) + cos( 4/3) - sintsin( 4/3) c os t [ 1 + co s ( 2 / 3 ) c os ( 4 / 3 ) ] - s in t [ s in ( 2 / 3) in ( 4 / 3 )] cost[ l+cos 12 cos 24 ° ] - sint sin 12 sin 24 ° ] cost[l cos 12 cos(-12 )]- sint[ in 12 sn(-12 )] c o s t ( 1 - . 5 - . 5 ) - s in t ( s in 1 2 - s in 1 2 )
16- 7 (1 6 4) ROTATNG GNETC FELD a Us in trionometric funct ions , Bx B o [ c o s t - 5 c os t 2 / 3 ) - . 5 co s t + 4 / 3 ) ]
B o [ c o s t - . 5 c o s tc o s2 / 3 - s in t s in 2 / 3 ) - 5 costc4 /3 - sintsin4 3 ]
B ° [ co s t - 5 - 5 c os t - . 5 sint) 5 ( 5cost 5 sint) ] 1 5 B° co st 8
B
B 0 . c ( Wt +2 1 / 3 ) - 0 . c o ( W t4 1 / 3 ) ]
y
0
O.S!B o cosWtc 21 /3 - sinwtsin21/3 - coswtcos41/3+sinwtsin41/3) O . S IB o ( -0 . S c os w t - O . S snwt + O . S coswt - O . Ssinwt) -2 ( 0 . S ) 2 B o sinwt -l.SB 0sinwt
7
o B
7
7
1 . S B c o sw ti - 1 . SB s in wt j , B O O
1.S B ' O
=
Using exponential func tions , B x B o [expjwt - O . Sexpj (Wt2 13) - O . Sexpj (Wt41 3) J , B o expjwt(-0.Sexpj21/3-0 Sexpj41/3) , B o exp j wt - O . S ( -O . O . S j ) - 0 . S ( -0 . S - 0 . S /j ) J ,
b)
By
. S B o expjwt, B 0 [ 0 . S exp j wt2 1 / 3) - 0 . S e x p j ( w t 4 1 / 3 ) J , O . S B o expj wt expj 2 1 3-expj 41 3 ) , O . S B o exp j w t ( - 0 . S0 . j 0 . S0 . S j , 2 2 ( 0 . S ) B o jexpjwt, . SB 0 expj Wt1 2 ) •
Thus B 6-8
. SB o coswt! - s inw tj as previously.
( 16 . 4 )
DIRECT- CUENT HI VOLTAGE TNSMISSION LINES
1
a) 2 V / 2 I
o
.
Sp
b ) 3 ( V / 2 I o TP
2 V I ' I o DC Sp
2 DC
V I ' I o DC TP
( 2 / 3 ) I
.
69
DC
0.94I
DC
169 (16 .5 ) ELECTROGNET OPERATING ON ALTENATING CUNT (NV rm L / Vrm i N , from Prob 152 . q rm NI rm
jQ=
Q
( 16 . 6) CLEX NERS a) 1 + 2 j 2 . 2 36 1 . 1 7 ; 1 + 2j 1 2 j 2 . 2 3 6 4 . 2 49 ; 1 2j
22362.34; 2 . 2 3 6 1 . 1 7
b) (1+2j ) (12j) 1 + 4 5 ; ( 1+2j) 2 1 4 + 4j 3 + 4j 2 1 / ( 1+2 j ) 1/ 3+4j) ( 3 4 j ) / 3+4j) (34j) ( 3 4j ) / ( 9 + 1 6)
. 12 . 1 6j
( 1+ 2 j ) / ( 1 2 j )
( 1+2 j ) 1+ 2 j ) / 1+4)
. 6 + 8j
COLEX NUERS
2.236 1.17 1612 ( 16 . 6) expjx
COLEX NERS 1 + j x x 2 /2! _ j x 3 / 3 + x 4 / 4 ! + j x 5 /5!
cox
1
jinx
+ jx
J X 3 /3'
1613 ( 16 . 6) COLEX NUERS expj + jin 1 16 1 4 (1 6 . 6) COLEX NUERS a) Multiplication by j increae he arumen b ) e increae by . c ) e decreae by /2.
e
by /2 radian
1615 ( 16 . 7 ) THREE-PHASE ATEATING URRENT co t co t+12 ) cot coco12 o + intin12 o cos wt ( l+ c o s6 0 ) + sin wsin6 0
: [o!
�
F
=
3 /2) c t + 2 int 3 co(t3 3 co(/6)
/ 2 ) c swt + / 2 ) s inwt
!)
7
(c3 o ct + in3 in)
1- 1 (1. 7) CCUTING AN ERGE PWER WITH PHR a) P av (/T) TV coswtI cos(wt+)dt o o I /T) T coswt(coswtcos - siwtsi)dt o (V /T) c 2 w - coww C o I /T)cos(T/2) (Vo I /2)cos The itegral of siwt over oe period is zero b) P av (1/2)R c exp(jwt)I exp - j(wt+) (1/)V I cos
[ !
{
!
CHAPTER 17 17 - 1 (171) IEDANCE R(l R a ) R + wL + R/jwC + + L w 2 2 + / j wC R w C2 + 1 R + R2 w2 RC2 + 1 + L - R2 Rw22 CwC2 + 1 2R at f 0 ad + jwL at f + b ) R+R/(R2 w 2 C 2 +1) 10+10/(100x42x10 x25x10- 18 +1) 20 wL - R2 wC/ (R2 w2 C2 +1 2x10 3x5x10 -3 - 100x2x10 3x5x10-9 /1 31 . 4 1 1 (20 2 +31. 4 2 ) ! 37.2 arc ta (31. 4/20) 575 c) / 2 9 x 10 - 2 , - 575 d) P 20 2 0.20 W e) No f X is iducive (positive for L > R2 C/R2 w2 C2 +1) , hich is aays true O ) X is zero at f
1
.[
1
( 17 1)
INDUCTORS
R+ j L+1/jC
(R+jL) [ (1 2 LC) (l 2LC) 2 + R2 2 C 2
R+jL RC 2LC+1
2 2 . R+[L (l LC) R ( l 2 LC) 2 + R 2 2 C 2
R( l2 LC) + R 2LC + j [L ( l2 LC) R2 ( l 2 LC) 2 + R2 2 C2 17 3 (1 7 3)
COENSATED VOLTAGE DIVIDER
R2 /jC 2 R2 + 1/jC 2 R1 /jC 1 R2 /jC 2 R2 + 1/jC 2 + R1 +1 / j C 1
V /Vi
174 ( 17 . 3)
R1 R2 R2 j C 2 + 1 + R1 j C 1 + 1
RC FILTER
.
/
a) Cal l V ' the po tetia l at the coect io be twee A ad C o /V' D / ( C +D) , B(C+D) B (C+D) + AB+C+D) , V /V o i
b) A V /V i
=
( V /V ' ) ( V ' /V ) i o
R, B
l/jC, C
BD B ( C +D) + A ( B + C+D )
=
l/jC, D
R/jC + (R+1/jC)
2
=
BD + (A+B ) ( C+D)
R
R 2 R + R j C + l/j C + 2R 72
3 +j (1 l/RC)
I vo /V I is maximum at R = 1 . 1 When I o / = ( 1 / 3 ) / 2 R
Ten V
0 v 1 / 3 .
. 33 or 3 3 .
The pass-band is very
broad.
ASURING N IEDE WITH A PHASESENSITIVE VOLTTER
17-5 (17.3) r
=
Z
=
V /V 2 1
=
R'r/(l-r)
R ' ab j ) / l- a-b j )
(abj ) ( l-abj ) ( -a) 2 b 2
=
Settin R jX , a _ a2 _ b 2 R 1 a2 b 2 _ 2a
=
a(-a) - b 2 2 2 ( -a) b
1 r 2
I I
x
-
, " =
2a
b 1 r 2 2a
I I
2 r sin r cos r X R o 1 r 2 - 2r cos 1 r o 2 2r cos o Ref er en ce : Electronics, Jly 25, 1974 p 117
0
17- 6 ( 1 7 . 3)
0
=
IEDNE BRIDGES
0
0
THE WIEN BRIDGE
73
Fr R R2 / 2 , R3 R , C 3 C the first equatin is stisfied and the send ne yields R3 C 3 7-7 ( 7 . 3)
LOW-PS RC FILTER
l/C ' 0 l R + l/ RC + l V0 /Vl I / RC f RC b ) V /V l l/(R 2 2 C 2 +) /(0 8 2 xf 2 x0 - 8 +) / ( 2 f 2 +) ) db 2 0 l g V l I ) V /V
•
7- 8 ( 7 . 3)
PHASE SHIFTER
wC
V
- R+/C
l
V /V
R-/ C i R+/C V i
R+ /C = R- /C
ex 2 r tg l / C)
7- 9 ( 7 . 3)
ASURING SURFACE PETS
) V
d E
c
=
E + E r o d o a
=
d E ' d d
E + ( d E /d ) o a r o a d
( / ) / ( l r d /d ) d
E
b)
E
/ ( l r d /d d )
c) V
d E
( d / ) / l r d d ) d
d) IR
C IR
0
(dQ/dt)R V(dC/dt)R ( 10 13 /2 ) ( lex t ) d C / dt ( 10 -13 /2) ( ex t) RV
( 10 -13 /2) ex t
1975 ges 173 182; Ctlg f
References Sttic nre Elec trnics
17 -1 0 (1 7 3) FERENCE TEERATUES NER S OLUTE ERO Let I nd I s be the currents in the riries nd in the se cndri es rese ctiely . At blnce V . Then 1 I Reference
R I ' j wM2 I p = R I Ml /M2 2 s 1 S
=
l /R2
Re Sci Instr. 1537 (1973) .
1 7 1 1 ( 1 7 3 ) EOTE-READING ERCURY THEOTER ) Since C l » C 2 the ccitnce f C l nd C 2 in series y be set equl t C 2 nd V ' /Vs
R R l / C 2
RC 2 /RC 2 l)
where C 2 ries linerly with the teer ture b) Since V is t ry linerly with C 2 we ust he tht C 2 « 1 nd then V ' « V c) C is cylindricl cci t with n utsi de rdius f s y 2 The ercury cln hs rdus f s y 0 05 Then C 2 2 rL / n 2 / 0 0 5 2 rL/n 2x885xl -12 x3/n 0 L 5 L F e he set r 3 Here L is the ength f the ercury c lun ins ide the electrde C Setting L 100 if l » ls R2 » 2s
75
d) C 2 5 F , R / C2 /2 x 0 7 x 5 x 0 2 3 0 0 0 Setting R 30, RC2 /00 , V' /Vs /00 , V' 0 lt , 9 5 ( 9 7 6 ) Referene Reiew f Sientifi Instruents
72 ( 7 3) WAER S ine L , we ay set the ltage arss the lad equal t tat at the sure Als sine R , we ay set the urrent thrugh the lad equal t the urrent su lied by the sure he il rdues a B that is r r tinal t , and in hase with , the urrent thrugh . he ltage arss R2 i s R2 / (Rl R 2 ) ties the ltage arss the sure hen
v a
00
KV I s/2)
KVrs Irs s
7-3 ( 7 3) NSIEN SUES SOR FOR INDUCOR We an alulate te ltage arss the ndutr, after the swith is ened , in anther way. We nsd er a l kis e esh urrent in L and C a) First, we ind Q and as funtins f t, wh the swith en Fr Kirhff's ltage law, LdI/ dt + 2RI + Q/ C C , Ld 2Q /dt 2 2RdQ/dt Q/ C ry a slutin f the f Q Q ex nt hen Ln2 2n l /C 0 n -R/L (R2 /L2 -l/LC) R/L, sine R 2 L/ C he tw r ts are equal hen Q (ABt) ex (Rt/L) . Set Q Q at t O hen A Q ex Rt/L) - (R/L) (ABt)B Nw I dQ/dt at t O hen B Set I RQ /L, Q ex (Rt/L) (Q tRQ t / L) , I ex -Rt/L - (R/L) (Q tRQ tL) RQ /L , ex (Rt/L ) I R /L ) ( I RQ /L ) t
±
0
10
10
0 0 0 0
0
10
b) Nw let us find a reatin between Q and Set I L and I C the urrents thrugh the indutr and aaitr befre t 0 in the 76
directions shon, so as to give a clockise current in the right hand mesh hen the sitch i s open . Th en 1
V -
=
2
2
( R +w )
2
cos [ wt- tan
-1
( w /R) ] ,
V
o cos [w tHan - 1 ( l/C ) ] 2 2 2 ( R + l/w C ) d Q /dt,
=
V
Q
I
1 o sinw t + tan- l/ Rw C ] 2 2 2 W ( R + l/w C )
=
2 2 2 - Vo R / ( R +w ) ,
o V
o
2 2 � s in [ tan W( R + l/w C ) 2
I Q
o
2
2 2 2 R (R w C +1) 2 2 2 C (R + w )
-
o
-1
-R /
( l /RwC) ]
�
-1 / ( C)
oul be - V R , Q ou d o o l ould again be -l / RC, or - 1 / ( C )
If the source supplie DC, instea of AC , I
e VC , an I /Q o o Since 1 0 -RQ o / , Q = Q o exp (-Rt /) , I = I o exp (-Rt /)
c The ol tage acros s the inductor , a te r th si tch has been opene , is , RI + I / t RI o exp ) + (-R/ ) exp ( ) despite the fact that I decreases exponential y ith tie
Reference : R eference Data for Radio Engineers , p 6 - 12 SE RIES RESONCE Z R + j ( w-l/wC 2 Z - _ j for w - 0 Z R fo r w C = 1 , 1 7 - 14 ( 1 7 . 3 )
y
Z
oj for w
R
77
X
75 73) E RESONNCE t ° Q
O
-II
(Rjw/jwC a _ Rjw/jwC R wCw 2C 2C) R2 CJ R j (w (R jw)2 (w2 2CRjwC) w[ w C R2 w2 C2 (w2 C) 2 R2 w2 C 2 Rea part f (w 2 C) 2(RR 2w 2C 2) \ 4 2 f 29 ) _ 4 J Imaginary part f 2f[ ( ) 2 4 2 f 2 gnitude f )
hase f = arc tan w J /R b) = hen [ J r hen (w 2C) R2 C w 2C R2 C/ w 2 (R2 C/) /C f (R2 C/)/C/2 = ( / 3 )/9/2 = 4 77 kHz c) w 2C 4 2 x64x 6 x3 x 6 4 2 x64x 3 527 j (28x 3 ) 3 (527) 2 x 6 8 78j
Z
2 . 3 3 2 + 10
=
d)
2
x 4
2
x 6 4 x l0
6
x 10
-12
=
2 584
R' /jwC' R' 3869 C' 629 F hiips Technica Revie 3 N 4 (9 7)
78
116 ( 14 STARDELTA TANSFORTON
J
- ,o � 863 X X
11 ( 14 STARDELTA TNSFOTON
11 14 B RD GEDT R + , R -, 1/ R -Y Zj C wZ C Z ( r+j wL) Zj wC + w Z C Z r + j 3 LC Z The w Z C Z rR 1, w Z LC Z
MTUAL INDUCTNCE
1 7- 20 ( 17 . 5 )
Tr ans f orm th e c i r cu i t in t o
the
<
one
shon in the figure of te preceding page. 10 1 - 2 - ! ) 102x103 x1.9 j _ 2 x 10 3 x 0 . 9 5 2 = 5 / 5 5100j = 5 / 10314j 5 /( 10 2 314 2 ) 1 . 5 9 2 x 1 0 - 2A 15 . 9 2
U� �
CHTER 18 18- 1 ( 18.1) DECT - CUNT MOTO RS a ) R eplace V' by a resistance R ' V ' / Then the poer supplied by the source is V ( R R ' ) The first term represents the arious losses and V' is the useful poer. c ) At no load , V ' V , = (V- V ' ) /R , B . Since VV'wB , w . d ) V ' increases . The motor sl os don and V ' decrease s , so increases fater, R ' V ' / decreases and the efficiency decreases. 18-2 ( 1 8 . 2 ) OWE RFACTOR CO RCTON a I z l 600/100 6.00 Q R 6 x 0. 65 3.90 $ x 6 x sin arc cos 0 . 65 ) 4.55 Q
b)
I
vi z
=
V(R-j x) / l z I
2
600 /36) ( . 90-j 4 . 5 5) The in-phase component is 600/3)3.90 6 5 . 0 A . Te quadrature component is 76.0 A , lagging . Check: 65 . 0 2 76 .0 2 100 2 •
8
c VwC 6 , C 76/600x260 366 This capacitor oul cost about $ 400.00 R eferee: Staar Habook for Electrical Egieers, 5 - 98 a 16 - 185 183 1 8 2 OWE R- ACTO R CORECTON WTH LUORESCENT LS Te ipase compoet of the curret is 80/120 0 6 6 7 A Sice cos 0 5 , 60 egrees The curret is 2 x 0 66 7 1 33 A The reactive curret is 133 si 60 116 A Te VwC 1 16 C 116/260x120 20 Heerso a Marse, Lamps a Lightig, p 325 Staar Haook for Electical Egieers, 19 - 33
18 - 4 183 ENE RG TRNSE R TO A LO AD
R
L
c
I t
= R ' '/C,
rjw Rj w ' /C Q' r/ R, xr/ RC V Y Vt oe ccle oe ccle 0 r
81
Rj w '
c For a resstor, V Vo coswt, I ( V o /Rcoswt dQ / dt, ( V /wRsnwt Q See Fg R For a capactor, V V cos w t , Q CV cosw t See Fg C For an nductor, V Vo coswt, Ld /dt Vo coswt, Ld 2 Q / dt 2 Vo coswt, Q - (Vo / 2 Lcoswt See Fg L v For a resstor n seres th an nductor, th R jwL, j Q I Vo exp jwt/ (Rj wL (V Rexp j wt / (1j ( V /2 Rexp j (w t -/4 , o Q ( V o /2wRexp j wt-/4-/ 2 See the fourth fgure Reference Rev Sc nstr � 109 ( 19 71 =
0
0
0
l - 5 ( l3 ENER TRNSFE R T A L AD Let the voltage across G at a gven nstant , be V The voltage at y s then approxately equal to V Let the currnt through , at a gven nstant, be , and the pulse duraton b T Then the energ dsspated n durng a pulse s ae T VdQ W Vdt one cycle o The voltage at x s Q /C Then the spo t on the os cllosc ope s creen descrbes a curve as n the fgure The area under ths curve s proportonal to the above nteral Reference Rev Sc nstr 1004 ( 1974
2
A postve 2 s eqvaent to a negatve n the prmary - 7 ( 4) ASUT OF TH COFFCNT OF COLN k th the secondary open, jL 1 th the secondary short - crcted, 2 j w2 /jwL 2 j wL - j wL j L 2 /j L2 / 1 2 - ( 4) FLCTD DNC a ) n R + j wL + W2 M2 / (R2 j WLZ ) ( 1+j R2 ) R2 / R2 2 +1 b) =
[
- 9 4) ASURN TH A UNDR A CURV Dra a ne arond the perphery o the gre th condctng nk Then measre the votae ndced hen the Hehotz cos are ed, say at 1 kohertz The system can be cbrated th a crce or h a rectange o knon area Reerence Rev Sc nstr 41, 1663 1970) ( 4 SOLD RN UNS a) R /A 10 1 /4 x - 6 x x 7 4 3 x - 4 > sec 10 0 /4 3 x -4 ) ! 40 A , V se c 4 0 x 4 3 x - 4 0 2 vot b ) pr 100/120 A
=
=
3
111 . 4 CUT TSFO Disregarding e sign, te induced elecromoance is d/dt, it ba o /2r 2adr ba v o a/ n [ b a / b - ad/d. eerence ev Sci. nstr. 324 1975
I
1 - 12 1 . 4 NDUCD CUTS / A 2a/b The tube is a single turn solenoid . Hence L 113 1 . 5 DDCUT LOSSS See the standard andbook or lectrical ngineers, Sec. 2 -74. 114 1.5 DDCUNT LOSSS For a solid core, he poer loss is 2. 2 / d/dt 2 /4a/b bL/4a e core is spli ino n laminaions, insulaed one rom e othe n n [d /n /d 2 / 4an/bL l /n 2 . eerence Standard Handbook or lectrical ngineers, Sec. 274, 91, 92, 93 and olloing.
1 . 5 HSTSS LOSSS lace he laminations inside a sole R/f noid and measure the resistve part o the impedance o the solenoid as a unction o the requency. Then A B C 2 , here A is the DC resisance. Then A / B C . A plot o R-A / as a unction o gives bo B and C.
4
8- 8 ) P-N T ER NAB/t NAB NA� �0 /2 a 0 0 4 x 0 -4 ) 2 x 0) 4 x 0 -7 x 0 4 ) / 2 x x 0 -2
2
oop te we cayn te unnown cuent seveal tes aoun te coe
HTER 9 9-2 9 )
E ' S EQUATNS
- -
-
Dvn y � o an cance ln x M on o t se s - -Jf + D/- t fo Eq - x M f + � o x H + M) - o � o E / t
� o J f+ / t + x M f o E q 4- 20
9- 9 ) E' S EQUATNS Use te equatons o te p evous poles s et tn 0 / t = j
B
V V a
9-4 9 )
V
n
0
E
E ' S EUATNS
Vj
x - o 0 /t 0 Tan te veence of ot ses an eeen tat x fo any vecto B
-
-
8
0
1 . 3 ET MS D ME S ETS a Takng te dvergence of te euaton forx E and rememering tat the divergence of a curl s alas eual to zero, ( 33t(V B . * , .* 3 * / 3 t . ) rom the euation for te curl of , * . V x E · da c s
f -!*. �
16 1 1W 1 1V 1 1S 1F 1W 1T 1H
1.3 1.m 1/s ls l/ 1V/ 1 -1
1 kgm/s2 m 1 kg m 2 / s 3
1 kg m 2 / s 2
1 kg m2 /s 3
lkg m 2 /s 2 /s 2 2 3 1 kg m /A s
12 s 3 /kg m2 1/V ls/kg m 2 /s 3 1Vs* 1 kg m2 /s 2 1W/m2 1 kg/s 2 1W/** 1 kg m 2 / 2 s
*From the act that , in a changing magnetic feld , the induced voltage is eual to the rate of change of the magnetic flux ** From / et us chec 2 2 1 kg m k m , orrect V glves s - A a) j wLI s s The energ stored in a caactor is V 2 /2. Ten =
2 kg m
2 4 A s
s
kg m
=
orrect
6
2 c T tod n a dcto L / 2 . Tn 2 2 k _ k A2 Correct - 2 2 s2 A s d) Te powr loss n a resstor s R Tn k 2 A2 2 Correct A2 s 3 s
lS
w 2 C a pure nuer Ten 2 2 4 - l Correct 2 2 2 2 e
A s k etc
CHAPTER
20
20 - 1 ( 204 PNE WAVE N FE SPACE E E o exp j w t- z H H o exp j (w t-z/ , wre E o and H o are ndependent of x y z t and ave no z-coponent a) Ten fro ·E 0 0 0 ( / y E ( / x ox exp j oy exp j
Tese
(
�
are denttes
) We ave slar equatons for H c) Fro
x
- -j k0
0
/ z
- � o / t
-�
°
/ t
- -j -k 0
0 -j / Ey 0
-j � o-H
Ex Y 0 Ts -j / x -j �H x E H Ex
d
E
- - - cH0 ° - Fro x H E 0 / t x E ° cE
0 - 2 2 0 4 P NTENNA o o ax 10d/dt ax 10 cos 0 dB/dt) ax 10 cos 0 dE/dt) ax /c l c os 30 o 2 x 3 x 10 0 1 /3 x 10 8 5 4 4 V
=
8
(2 0 . 6)
PONTING VECTOR 2 3 . 8 x l 26 / 4 x 49 x l 16 2 . 6 5 x l -3 E rms '
EE
'
E
= 1 . 5 3 x lO rms
5
V im
2 2 E E2 /E S2 (l/RS E ) /(l/Rs ) EE /E S RS /RS _ E ( 7 x 10 8 / 1 . 5 x 10 11 ) 1 . 5 3 x 1 05 700 Vm
=
2 c) E 2 . 6 5 x l - 3 x (700) 1 . 3 x l 0 3 W/m2 6 0 x 1 . 3 x l 3 ( cal /4 . l9 ) / ( 10 4 cm2 ) 1 . 86 calorie /minute centimeter 2 This quantity is called the solar constant We have neglected absorption in the atmosphere The avera ge daily lux at the ground in the United S tate s is abo ut 0 . 4 calorie /minute centimeter 2 . Reerence: erican Institute of Physics Handbook 3rd ed p 21 43 .
20 4 (2 0 . 6) SOLAR ENERGY At the surace o the arth
1 . 3 x 10 3 W /m2 rom Prob . 203 . 5 x l 0 7 / 1 3 x l 0 3 4 l 0 4 m2 P A/50 A 5 0 P or a square 200 meter s on the s ide .
�
20-5 ( 20 . 6) POYNTING VECTOR o 2 3 x 10 8 x 8.85 x 10 -12 x 20 2 1.06 W/m 2 In one second the energy absorbed by one square meter of the coper sheet is 1 . 06 J This energy will increase the tmperature of one kilogram o copper by 1.06/400 elin. In one s econd the tempe rature of the shee t rise s by 100 x 06 /40 0 1 . 0 6 / 4 0.265 kelvin.
E
20-6 (2 0 . 6)
POYNTING VECTOR
a)
G C oAt Or
c-- - 8
