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1. CLASS – XII SUBJECT- PHYSICS CHAPTER- MOVING CHARGES AND MAGNETISM

2. M A G NE TIC E F F E C T OF C U R R E N T - I M a gn e t ic Eff ec t of C u rre nt – O ers t e d’s E x p er i me nt A m p ere ’s S w i mm ing R ule M a x w e l l ’s Cor k S cr e w Ru le R ight Ha nd T hu m b R ule B iot – S a v a r t ’s L a w M a gn e t ic Fi el d due t o I n f i ni t e ly Long St ra ight C u rre nt – car r y ing C ondu c t or M a gn e t ic Fi el d due t o a C i r c ul a r Loop car r y ing c u rre nt M a gn e t ic Fi el d due t o a S ole noid

3. N Magn e t i c E f f ec t o f C ur re n t : A n e l ec t r ic c u rre nt ( i. e . f l o w of e l ec t r ic c h ar g e ) p r odu ce s ma gn e t ic e ff ec t in t he s p ac e ar o und t he c on d u c t or ca ll e d s t re ng t h of M ag n e t ic f i e ld or s i m ply M a gn e t ic f i e l d . O ers t e d’s E x p er i me n t : Wh e n c u rre nt w a s a l l o w e d t o f l o w t h ro u g h a w ir e pl ace d para ll e l t o t he a x is of a ma g ne t ic n ee dle ke pt d irec t l y b e l o w t he w ire , t he nee dle w a s f o und t o d e f l ec t f r om i t s no rma l po si t ion. E I K N K I E Wh e n c u rre nt w a s r e v e rse d t h r ough t he w i re , t he n ee dle w a s f ound t o d e f l ec t in t he oppo s i t e di rec t ion t o t he ear li e r case .

4. B B N Ru l e s to d e t e r m i n e th e d ir ec t i o n o f ma g n e tic f i e ld : A mpere ’ s S w im mi n g R u l e : I ma gini n g a m a n w h o s w im s in t he di rec t ion of c u rre nt f r om s ou t h t o n o r t h f ac ing a ma g ne t ic n ee dle ke pt u n d e r him s u c h t h a t c u rre nt e n t er s his f ee t t h e n t he N o r t h p o le of t he n ee dle w il l d e f l ec t t o w ard s his l e f t h and , i. e . t o w ar ds W es t . Ma x w e l l’ s Cor k Scr e w R u l e o r R i g h t H an d Scr e w R u l e : If t he f o r w a r d m o t ion of a n ima gin a r y r ight h an d e d scr e w is in t h e di rec t ion of t he c u rre nt t h r ough a lin ea r c ondu c t o r , t h e n t he d i rec t ion of r o t a t i o n of t he scr e w g i v e s t he di rec t ion o f t he ma gn e t ic lin e s of f o rc e ar ound t he c ondu c t o r . S N W I I I

5. x P R i gh t Han d Thum b R u l e o r C ur l R u l e : If a c u rre nt car r y ing c ondu c t or is i ma gin e d t o be h e ld i n t he rig ht h a nd s u c h t h a t t he t hu m b p o in t s in t he di rec t i o n of t he c u rre n t , t h e n t he t i ps of t he f ing er s e n c i rc ling t he c o nd u c t or w il l g i v e t he di rec t ion of t h e ma gn e t ic li ne s of f o rce . I B i o t – Sa v ar t ’ s L a w : The s t re ng t h of ma gn e t ic f ie ld dB due t o a sma ll c u rre nt e l em e nt dl car r y ing a c u rre nt I a t a point P di s t a nt r f ro m t he e l eme nt is di rec t ly p r opo r t i o n a l t o I , d l , s in θ a nd i n v e r se l y p r opo r t i o n a l t o t he s qu ar e of t he d i s t a n c e ( r 2 ) w her e θ is t he a ngle b e t w ee n dl a nd r . θ dl r dB α I dB α dl dB α s in θ dB α 1 / r 2 dB α I dl s in θ r 2 dB = µ I dl si n θ 4 π r 2 P ’ B I

6. B i o t – Sa v ar t ’ s L a w i n v ec t o r f o r m : dB = µ I dl x r 4 π r 2 dB = µ I dl x r 4 π r 3 V a lue of µ = 4 π x 10 - 7 Tm A - 1 or Wb m - 1 A - 1 D i rec t ion of d B is sam e a s t h a t of d i rec t ion of dl x r w hic h ca n be d e t erm in e d by R ight Ha nd S cr e w R ul e . It is emer ging a t P ’ a nd e n t er ing a t P in t o t he p l a ne of t he di agram . C u rre nt e l em e nt is a v ec t or qu a n t i t y w h o s e ma gni t ude i s t he v e c t or p r odu c t of c u rre nt a nd l e ng t h of sma ll e l e me nt h a v ing t he di rec t ion o f t he f l o w o f c u rre n t . ( I dl ) x

7. M a gn e tic F i e ld du e t o a S tr a i g h t W ire ca r r y i n g c u rr e n t: P θ r a I Ф 2 Ф 1 Ф l A cc o r ding t o B iot – S a v a r t ’s l a w dB = µ I dl si n θ 4 π r 2 s in θ = a / r = c os Ф or r = a / c os Ф t a n Ф = l / a or l = a t a n Ф dl = a sec 2 Ф d Ф S ub s t i t u t i ng f or r a nd d l i n d B , dB = µ I c os Ф d Ф 4 π a M a gn e t ic f i e ld due t o w ho le c ondu c t or is ob t a in e d by in t e g ra t ing w i t h li m i t s - Ф 1 t o Ф 2 . ( Ф 1 is t ake n n e g a t i v e s in c e it is a n t i c lo c k w i se ) Ф 2 µ I c os Ф d Ф - Ф 1 4 π a B = ∫ dB = ∫ µ I ( s in Ф 1 + s in Ф 2 ) B = 4 π a dl x B

8. If t he s t ra ight w ir e is in f ini t e ly long, t h e n Ф 1 = Ф 2 = π / 2 µ 2 I B = 4 π a µ I B = 2 π a or B a a B B D i rec t ion of B is sam e a s t h a t of di rec t ion of d l x r w h i c h ca n be d e t erm in e d by R ight Ha nd S cr e w R ul e . It is p er p e n d i c ul a r t o t he pl a ne of t he di agra m a nd e n t e r ing in t o t he pl a ne a t P . M a gn e t ic Fi el d L i n es : I I

9. M a gn e t ic F i eld d u e t o a C i r c u lar Lo o p ca r r y i n g c urr e n t : 1 ) A t a point on t he ax i a l l i n e : O a r dB dB dB c o s Ф dB s in Ф dB s in Ф I I dl C X Y dl The pl a ne of t he c oil is c o ns id ere d p er p e ndi c ul a r t o t h e pl a ne of t he di a g ra m s u c h t h a t t he di rec t ion of ma gn e t ic f i e ld ca n be v i s u a li z e d on t he pl a ne of t he di a g ram . A t C a nd D c u rre nt e l eme n t s X Y a nd X ’ Y ’ ar e c on s id er e d s u c h t h a t c u rre nt a t C e mer g e s out a nd a t D e n t er s in t o t he p l a ne of t he di a g ram . X ’ Y ’ D 90 ° Ф Ф Ф Ф x P dB c o s Ф

10. dB = µ I dl si n θ 4 π r 2 dB = µ I dl 4 π r 2 µ I dl si n Ф 4 π r 2 B = ∫ dB s in Ф = ∫ or B = µ I ( 2 π a ) a 4 π ( a 2 + x 2 ) ( a 2 + x 2 ) ½ µ I a 2 B = 2 ( a 2 + x 2 ) 3/2 ( µ , I , a , s i n Ф a r e con s ta n ts , ∫ d l = 2 π a a n d r & s i n Ф a re r eplace d w it h m easu r abl e a n d c onsta n t v alue s. ) or The a ngle θ be t w ee n d l a nd r is 90 ° b eca u s e t he ra dius of t he l oop is v er y sma ll a nd s i nc e s in 90 ° = 1 The sem i - v e r t i ca l a ngle ma de by r t o t he l oop is Ф a nd t he a ngle b e t w ee n r a nd dB is 90 ° . Th ere f o re , t h e a ngle b e t w e e n v e r t i ca l ax i s a nd dB is als o Ф . dB is res o l v e d in t o c o m p o n e n t s dB c o s Ф a nd dB sin Ф . D ue t o di ame t r i ca lly oppo s i t e c u rre nt e l e me n t s , c o s Ф c o m pon e n t s ar e a l w a y s oppo s i t e t o eac h o t h e r a nd h enc e t h e y ca n ce l out eac h o t h er . S in Ф c o m p one n t s due t o al l c u rre nt e l e m e n t s dl g e t ad d e d up a long t he sam e di rec t ion ( i n t he di rec t ion a w a y f r om t he lo o p ) .

11. I I I D i ff ere nt v i e w s of di rec t ion of c u rre nt a nd ma gn e t ic f i el d due t o c i rcu l a r loop of a c oil: I B x x ii) If t he ob se r v a t ion po i nt i s f a r a w a y f r om t he c oil, t h e n a << x . S o, a 2 ca n be n e gl ec t e d in c o m p ar i so n w i t h x 2 . µ I a 2 B = 2 x 3 S p ec i a l Case s : i) A t t he ce n t r e O, x = . µ I B = 2 a B B B

12. dB dB = µ I dl si n θ 4 π a 2 µ I dl 4 π a 2 I dB = µ I dl 4 π a 2 The a ngle θ be t w ee n d l a nd a is 90 ° b eca u s e t he ra dius of t he loop is v e r y s ma ll a nd s in c e s in 90 ° = 1 B = ∫ dB = ∫ ( µ , I , a ar e c on s t a n t s a nd ∫ dl = 2 π a ) a x O 2 ) B a t t he ce n t r e of t he l o o p: The pl a ne of t he c oil is l y ing on t he p l a ne of t he d i a g ra m a nd t he d i rec t ion of c u r r e nt is c lo c k w i s e s u c h t h a t t he d i rec t ion of ma gn e t ic f i e ld is p er p e ndi cu l a r a nd in t o t he pl a n e . I dl 90 ° µ I B = 2 a B a

13. M a gn e t ic F i eld d u e t o a So le n o i d : I I x x x x x x x TI P : Wh e n w e look a t a ny e nd of t he c oil carr y i ng c u rre n t , if t he c u rre nt is i n a n t i - c lo c k w i s e di rec t ion t he n t h a t e nd of c oil b e h a v e s li k e N o r t h P o l e a nd if t he c u rre nt is in c lo ck w is e di rec t ion t h e n t h a t en d of t he co il b e h a v e s li k e S ou t h P ol e . B

14. M A G NE TIC E F F E C T OF C U R R E N T - I I Lo re n t z M a g ne t ic Fo rc e Fl em ing’s L e f t Ha nd R ule Fo rc e on a m o v ing c h ar ge in u n i f o r m E l e c t r ic a nd M a gn e t ic f i e lds Fo rc e on a curre nt ca rr y ing c ondu c t or i n a uni f o r m M a gn e t ic Fi el d Fo rc e b e t w e e n t w o in f ini t el y long p ara ll e l c u rre n t- ca rr y i ng c ondu c t o r s De f ini t ion of a m p er e Re p re s e n t a t ion of f i e lds d ue t o p ara ll e l curre n t s To r que ex p er i e n ce d by a c u rre n t- c a r r y ing c oil in a uni f or m M a gn e t ic Fi el d M o v ing C oil G a l v a nom e t e r C o n v e rs ion of G a l v a no me t e r in t o A m me t e r a nd V ol t me t e r D i ff ere n c e s b e t w ee n A m m e t e r a nd V ol t m e t e r

15. Lor e n t z M a gn e t ic F or c e: A c u r re nt ca rr y ing c ondu c t or pl ace d in a ma gn e t ic f i e ld ex p er i e n ce s a f o rc e w hic h mea ns t h a t a m o v ing c h ar ge in a ma gn e t ic f i e ld e x p er i e n ce s f o rce . F F m = q ( v x B ) or F m = ( q v B s in θ ) n w her e θ is t he a ngle b e t w ee n v a nd B S p ec i a l Case s : q + B v I θ q - B v F θ If t he c h ar ge i s a t res t , i. e . v = , t h e n F m = . S o , a s t a t i on ar y c h a r ge in a ma gn e t ic f i e ld d o e s not ex p er i e n c e a ny f o r c e . If θ = ° or 180 ° i. e . if t he c h a r ge m o v e s p a ra ll e l or a n t i - p ara ll e l t o t he di rec t i on of t he mag n e t ic f i e ld, t h e n F m = . iii) If θ = 90 ° i. e . if t he c h ar ge m o v e s p er p e ndi c ul a r t o t he ma gn e t ic f i e ld, t h e n t he f o rc e is max i m u m . F m ( m ax) = q v B I

16. F lemi ng ’s L e f t H a n d R u le: F o rc e ( F) M a gn e t ic Fi e ld ( B ) E l ec t r ic C u rre nt ( I) If t he c e n t ra l f ing er , f o r e f in g e r a nd t hu m b of l e f t h a nd ar e s t re t c h e d m u t u a lly p er p e ndi c ul a r t o eac h o t h e r a nd t he ce n t ra l f ing e r poin t s t o c u rre n t , f o r e f ing e r p o in t s t o ma gn e t ic f i e ld , t h e n t hu m b p oin t s in t he di rec t i o n of m o t ion ( f o rce ) on t he c u rre nt ca rr y i ng c o n du c t o r . TI P : Rem e m b e r t he ph ras e ‘ e m f ’ t o re p rese nt e l ec t r ic c u rre n t , ma gn e t ic f i e ld a nd f o rc e in a n t i cl o ck w is e di rec t ion of t he f ing er s of l e f t h a nd. For ce o n a m o vi n g c h a rg e in un i f o r m E lec tr ic a n d M a gn et ic F iel d s : Wh e n a c h arg e q m o v e s w i t h v e lo c i t y v in re gion in w hic h bo t h e l ec t r ic f i e ld E an d ma gn e t ic f i e ld B ex i s t , t h e n t he Lo re n t z f o rc e is F = qE + q ( v x B ) or F = q ( E + v x B )

17. For ce o n a c urr e n t - ca rr y i n g c o nd u c t o r in a u n i for m M a gn e t ic F i el d : θ v d dl F A I I B l Fo rc e ex p er i e n ce d by eac h e l ec t r on in t he c ondu c t or is f = - e ( v d x B ) If n be t he num b e r d e n s i t y of e l ec t r on s , A be t he ar e a of cr o s s sec t i on of t he c ondu c t o r , t he n no. of e l ec t r ons in t he e l eme nt dl is n A dl . w her e I = n e A v d a nd - v e s i g n re p rese n t s t h a t t he di rec t ion of dl is o ppo si t e t o t h a t of v d ) or F = I l B s i n θ - Fo rc e ex p er i e n ce d by t he e l ec t r ons in dl i s dF = n A dl [ - e ( v d x B ) ] = - n e A v d ( dl X B ) = I ( dl x B ) F = ∫ dF = ∫ I ( d l x B ) F = I ( l x B )

18. Fo rce s b e t w e e n t w o p ara ll e l in f ini t e ly lo n g c u rre n t- car r y ing c ondu c t o rs : r F 1 2 F 2 1 1 2 π r I 1 B 2 P Q I 2 x B 1 S R B 1 = µ I 1 2 π r M a gn e t ic Fi el d on R S due t o c u rre nt in P Q is Fo rc e ac t ing on R S d ue t o c u rre nt I 2 t h ro ugh it is F 2 1 = µ I 1 2 π r I l s in 90 ˚ 2 B 1 act s pe r pe n dicula r an d i n t o t h e p lan e o f t h e diag r a m b y Rig h t Han d T hu m b R ul e . S o , t h e angl e b e t w e e n l an d B 1 i s 9 ˚ . l is l en g t h of t he c ondu c t o r . M a gn e t ic Fi el d on P Q due t o c u rre nt in R S is or F 2 1 = µ I 1 I 2 l 2 π r B 2 = µ I 2 2 π r Fo rc e ac t ing on P Q due t o c u rre nt I 1 t h ro ugh it is µ I 1 I 2 l F 1 2 = µ I 2 2 π r I l s in 90 ˚ F 1 2 = (Th e an g l e b e t w ee n l an d B 2 i s 9 ˚ an d B 2 I s e m e r g i n g o ut ) F 1 2 = F 2 1 µ I 1 I 2 l = F = 2 π r µ I 1 I 2 F / l = 2 π r or Fo rc e p e r unit l e ng t h of t he c ondu c t or is N / m ( i n m ag n i tud e ) ( i n m ag n i tud e )

19. r F F I 1 P Q I 2 x S R r I 2 F x S R I 1 F P Q x B y Fl em ing’s L e f t Ha nd R u le , t he c ondu c t o r s ex p er i e n c e f o rc e t o w ar ds eac h o t h e r a nd h e n c e a tt rac t eac h o t h er . B y Fl em ing’s L e f t Ha nd R u le , t he c ondu c t o r s ex p er i e n c e f o rc e a w a y f r om eac h o t h e r a nd h e n c e re p e l eac h o t h er .

20. D e fi n i t i o n o f A m p e r e : Fo rc e p e r unit l e ng t h of t he c ondu c t or is F / l = µ I 1 I 2 2 π r N / m Wh e n I 1 = I 2 = 1 A m p er e a nd r = 1 m , t h e n F = 2 x 10 - 7 N/m . One am p er e is t h a t c u rre nt w hic h, if p asse d in eac h of t w o p ara ll e l c ondu c t o r s of in f i ni t e l en g t h a nd pl ace d 1 m a p ar t in v ac uum ca u s e s eac h c ondu c t or t o ex p er i e n c e a f o rc e of 2 x 10 - 7 N e w t on p e r me t r e of l e ng t h of t he c ondu c t o r . R e p r e s e n t a ti o n o f F i e ld d u e to P a r a ll e l C u rr e n t s : I 1 I 1 I 2 B I 2 B N

21. B To r qu e ex p e ri e n ce d b y a Cu r r e n t L oo p ( R ec t a ngu l a r) in a un i fo rm M a gn e tic F i e l d : P Q R S x θ l I | F PQ | = I l B si n 90 ° = I l B F R S = I ( l x B ) | F R s | = I l B si n 90 ° = I l B Fo r ce s F P Q an d F R S b e in g equa l i n m a gn i t u d e b u t oppos i t e i n d i r e cti o n canc e l o u t eac h o the r an d d o n o t p r oduc e a n y t r a nsla t ion a l m ot i on . B ut t h e y ac t a long di ff ere nt lin e s of ac t i o n a nd h e n c e p r odu c e t o r q u e a bout t he a x is of t he co il. F QR θ F R S Le t θ b e t h e a ng l e b e t w ee n th e plan e o f th e lo o p an d th e di r e c t io n o f th e m a g ne t i c f i e ld . Th e axi s o f t h e coi l i s pe r pe n d i cula r t o th e m a gne t i c fi e l d . b I F SP = I ( b x B ) | F S P | = I b B s in θ F QR = I ( b x B ) | F QR | = I b B s in θ Fo r ce s F S P an d F QR a re equ a l i n m a g n i tud e b u t oppos i t e i n d i r e cti o n a n d th e y c ance l o u t eac h othe r . M o r e o v e r t h e y a c t a lon g th e s a m e lin e o f a c t i o n (axi s ) an d henc e d o n o t p r o d uc e to r q ue . F PQ = I ( l x B ) F SP F PQ

22. P Q R x S b θ θ N To r que ex p er i e n ce d by t he c oil is ז = F PQ x P N ז = I l B ( b c os θ ) ז = I lb B c os θ ז = I A B c os θ ז = N I A B c o s θ ( in ma gni t ud e ) ( A = lb) ( w h er e N is t he n o . of t u r n s ) I f Φ i s t h e a n gl e bet w ee n th e n o r m a l t o th e c o i l an d th e di r e c t io n o f th e m a g ne t i c f i e ld , the n Φ + θ = 90 ° i .e . θ = 90 ° - Φ S o, ז = I A B c os ( 90 ° - Φ ) ז = N I A B s in Φ N OT E : One m u s t be v er y c are f ul in u s ing t he f o r m ula in t erm s of c os or s in s in c e it d e p en ds on t he a ng l e t ake n w h e t he r w i t h t he pl a ne of t he co il or t he no rma l of t he c o i l. Φ Φ B B F PQ F R S n n I I

23. ( s in c e M = I A is t he M a gn e t i c D ipole M o m e n t ) N o t e : The c oil w il l r o t a t e in t he an t i c lo c k w i s e d irec t ion (f r om t he t op v i e w , acc o r ding t o t he f igu re ) a b o ut t he ax is of t he c oil sh o w n by t he do tt e d lin e . The t o r que a c t s in t he u p w a r d di rec t ion al ong t he do tt e d l i ne ( acc o r ding t o M a x w el l’s S cr e w R ul e ) . 3 ) If Φ = ° , t h e n ז = . 4 ) 5 ) 6 ) If Φ = 90 ° , t h e n ז is max i m u m . i. e . ז m a x = N I A B U ni t s : B in T es l a , I i n A m p e re , A in m 2 an d ז in Nm . The a b o v e f o rm ul a e f or t o rq ue ca n be u se d f or a ny loop i rres p ec t i v e o f i t s s h a p e . or ז = N I ( A x B ) ז = N ( M x B ) Torqu e i n V e c t o r f o r m : ז = N I A B s in Φ ז = ( N I A B s i n Φ ) n ( w her e n is u nit v ec t or no rma l t o t he pl a ne of t h e lo op )

24. P B W P T – T o r si o n He a d , T S – Te r m i n a l sc r e w , M – M i rr o r , N, S – P ole s piece s o f a m a g ne t , L S – L e v e l l i n g S c r e w s , P Q R S – Rectan g ula r c o il , P B W – P ho s pho r B r o n ze W i re LS F R S S Q R Mo v i n g C o i l o r Suspend e d C o i l o r D ’ A rson v a l T y p e G a l v an o me t er : N S x T E LS B To r que ex p er i e n ce d by t he c oil is ז = N I A B s in Φ Res t o r ing t o r que in t he c oil is ז = k α ( w he re k i s r esto r in g t o r q u e pe r u n i t angula r t w ist , α i s th e angula r t w is t i n th e w i r e ) A t e quilib r iu m , N I A B s in Φ = k α I = k N A B s in Φ α The f ac t or s in Φ ca n be e li m in a t e d by c hoo s ing Ra di a l M a gn e t ic Fi e l d . M Hai r S p r i n g TS F PQ

25. La m p S cal e R ad i a l Mag n e t i c F i e l d : The (t op v i e w P S o f ) p l a ne o f t he c oil P Q R S li e s a long t he ma gn e t ic lin e s of f o rc e in w h i c h e v e r po s i t i o n t he c oil c o me s t o r es t in e qu i lib ri u m . S o, t he an gle b e t w ee n t he p l a ne of t he coi l a nd t he ma gn e t ic f i e ld is ° . or t he a ngle be t w ee n t he n orma l t o t he p la ne of t he c oil an d t h e ma gn e t ic f i e ld is 90 ° . N S B P S i. e . s in Φ = si n 90 ° = 1 I = k N A B α or I = G α k w her e G = N A B is ca ll e d G a l v a no me t e r c on s t a nt Curr e n t S e n si t iv it y o f G alva n o m e t e r : I t i s t h e d e f e c t i o n o f g a l v anome t e r p e r un i t cu r re n t. N A B k α I = Vo l t a g e S e n si t iv it y o f G alva n o m e t e r : I t i s t h e d e f e c t i o n o f g a l v anome t e r p e r un i t v o l t age . N A B k R α V = M i rr o r 2 α

26. Con ve r s i o n o f G alv a n o me t er t o A mme t e r : G a l v a no me t e r ca n be c o n v e r t e d in t o amme t e r by s hun t ing it w i t h a v e r y s ma ll res i s t a n c e . P o t e n t i a l d i ff ere n c e a c r o s s t he g a l v a no m e t e r a nd s hunt res i s t a n c e ar e e q u a l. ( I – I g ) S = I g G S = I g G I – I g Con ve r s i o n o f G alv a n o me t er t o V o l t me t e r : G a l v a no me t e r ca n be c o n v e r t e d in t o v ol t me t e r by c onn ec t ing it w i t h a v er y high res i s t a n c e . P o t e n t i a l d i ff ere n c e a c r o s s t he g i v e n lo a d res i s t a n c e is t he s um of p .d acr o s s g a l v a no me t e r a nd p . d. acr o s s t he hi g h res i s t a n c e . V = I g ( G + R ) G I I g I s = I - I g S or R = V I g - G I g G R V or

27. D iff e r e n c e b e t w ee n A mme t e r a n d Vol t me t e r: S . N o. A mm e t e r V ol t me t e r 1 It is a l o w resis t a n c e in s t r u me n t . It is a h igh res i s t a n c e in s t r u me n t . 2 Res i s t a n c e is GS / ( G + S ) Res i s t a n c e is G + R 3 S hunt Res i s t a n c e is ( GI g ) / ( I – I g ) a nd is v er y s m a ll. S er i e s R e s i s t a n c e is ( V / I g ) - G an d is v e r y high. 4 It is a l w a y s co nn ec t e d in ser i es . It is a l w a y s co nn ec t e d in p ara ll e l. 5 Res i s t a n c e of a n id ea l amme t e r is z e r o. Res i s t a n c e of a n id ea l v ol t m e t e r is in f i ni t y . 6 I t s res i s t a n c e is l es s t h a n t ha t of t he g a l v a n o me t er . I t s res i s t a n c e is g rea t e r t h a n t h a t of t he v ol t me t er . 7 It is n ot p o ss ible t o d ecrea s e t he ra nge of t he g i v e n amme t er . It is p o ss ible t o d ecreas e t h e ra nge of t he g i v e n v ol t me t er .

28. M A G NE TIC E F F E C T OF C U R R E N T - I II C y c lo t r on A m p ere ’s C i rc ui t a l L a w M a gn e t ic Fi el d due t o a St r a ight S o l e no i d M a gn e t ic Fi el d due t o a T o r oid a l S ol e n oi d

29. N W S D 1 D 2 + B C y clotro n : D 1 , D 2 – Dee s W – W ind o w N , S – M a g n e ti c P ol e P iece s B - M ag n eti c Fi e l d H F O s c i l la t o r D 2 D 1 Wo rk ing: I ma gin i ng D 1 is p o s i t i v e a nd D 2 is n e g a t i v e , t he + v e l y c h ar g e d p ar t i c le ke pt a t t he ce n t r e a nd i n t he g a p b e t w ee n t he dee s g e t ac c e l e ra t e d t o w ar ds D 2 . D ue t o p er p e n d i c ul a r ma gn e t ic f i e ld a nd acc o r ding t o F l em ing’s L e f t Ha nd R u l e t he c h ar ge g e t s d e f l ec t e d a nd d escr ib e s sem i - c i rc ul a r p a t h. Wh e n it is a b o ut t o l e a v e D 2 , D 2 b ec o me s + v e a nd D 1 bec o me s – v e . Th ere f o r e t he p ar t i c le is a g a in acce l era t e d in t o D 1 w her e it c on t inu e s t o d escr ibe t he sem i - c i rc ul a r p a t h. T he p r o c es s c on t inu e s t ill t he c h ar ge t r a v e rs e s t h r ough t he w ho le s p ac e in t he d ee s a nd f in al ly it c o me s out w i t h v er y high s p ee d t h r ou g h t h e w i n d o w . W B

30. Theo r y : The ma gn e t ic f o rc e ex p er i e n ce d by t he c h ar ge p r o v id e s ce n t r ip e t a l f o rc e re qui re d t o d e scr ibe c i rc ul a r p a t h. m v 2 / r = q v B s in 90 ° ( w he re m – m a s s o f th e c ha r g e d pa r ti c le , q – cha r ge , v – v el oc i t y o n th e pat h o f r adiu s – r , B i s m agn e ti c fi e l d a n d 90 ° i s t h e angl e b/ n v a n d B ) v = B q r m If t is t he t i m e t ake n by t he c h ar ge t o d escr ibe t he sem i - c i rc ul a r p a t h in s ide t he d ee , t h e n Ti m e take n in si d e t h e de e dep e nd s o n l y o n t = π r v or t = π m B q th e m a g ne t i c f i e l d a n d m / q r a t i o an d n o t o n th e s pee d o f th e cha r g e o r th e r adiu s o f th e pat h . If T is t he t i m e p er iod of t he high f re qu e n c y o sc ill a t o r , t h e n f or res on a n ce , T = 2 t or T = 2 π m B q If f is t he f req u e n c y of t he high f re qu e n c y o sc ill a t or ( Cyc lo t r on F re q ue n c y ) , t h e n B q f = 2 π m

31. M ax i m um E n er gy of t he P ar t i c l e : K in e t ic E n er gy of t he c h ar g e d p ar t i c le is 2 2 2 K . E . = ½ m v 2 = ½ m ( B q r ) 2 m m B q r = ½ M ax i m um K in e t ic E n er gy of t he c h ar g e d p ar t i c le is w h e n r = R ( r adiu s o f th e D ’s ) . B 2 q 2 R 2 K . E . m a x = ½ m The ex p ress ions f or Ti m e per iod a nd C y c lo t r on f re qu en c y only w h e n m rema ins c o n s t a n t . ( O t h e r qu a n t i t i e s ar e a l rea d y c on s t a n t .) m = m [ 1 – ( v 2 / c 2 )] ½ If f re qu e n c y is v a r i e d in s y n c h r oni sa t ion w i t h t he v ar i a t ion of mas s of t he c h ar g e d p ar t i c le ( by ma in t a ini n g B a s co n s t a n t ) t o h a v e res on a n ce , t h e n t he c y c lo t r on is ca ll e d s y n c h r o – c y c lo t r on . If ma gn e t ic f i e ld is v a r i e d in s y n c h r oni sa t ion w i t h t he v a r i a t ion of mas s of t he c h ar g e d par t i c le ( by m a in t a ini n g f a s c on s t a n t ) t o h a v e res on a n ce , t h e n t he cyc lo t r on is ca ll e d i s o c h r onous – c y c lo t r on . N O T E : C y c l o t r o n ca n n o t b e u se d f o r acc e le r a t i n g ne u t r a l pa r t icles . E le c t r o n s ca n no t b e acc e le r a t e d becaus e t he y g ai n spee d v er y q u ick l y du e t o thei r l ig h te r m as s an d g o o u t o f p has e w it h a l te r n ati n g e . m . f . an d ge t lo s t w ithi n t h e dees . B ut m v a r i e s w i t h v a c c o r ding t o E in s t e in’s Rela t i v i s t ic P r in c i ple a s p e r

32. A m p ere’s C irc u ital L a w : The l i ne in t egra l ∫ B . d l f or a c lo se d c u r v e is e qu a l t o µ t i me s t he n e t c u rre nt I t h rea ding t h r o u gh t he are a bou n d e d by t he c u r v e . ∫ B . dl = µ I ∫ B . dl = ∫ B . dl c os ° = ∫ B . dl = B ∫ dl = B ( 2 π r ) = ( µ I / 2 π r ) x 2 π r ∫ B . dl = µ I I B B r O dl I C u rre nt is e m er ging out a nd t he m a gn e t ic f i e ld is a n t i c l oc k w i se . P r oo f :

33. M a gn e tic F i e ld a t t h e ce n tre o f a S tr a i g h t So l e n o i d : I I x x x x x x x P Q S R ∫ B . dl = µ I ( w he re I i s t h e ne t c u rr e n t th r ea d in g th r o u g h th e s ole n oi d ) ∫ B . dl = ∫ B . dl + PQ QR ∫ B . dl + ∫ B . dl + R S ∫ B . dl c os 90 ° + ∫ B . dl SP ∫ . dl c os ° + B = ∫ B . dl c os ° + = B ∫ dl = B .a ∫ B . dl c os 90 ° a nd µ I = µ n a I ( w he re n i s no . o f t u r n s pe r u n i t len g t h , a i s t h e len g t h o f t h e p at h a n d I i s th e c u rr e n t passin g t h r ou g h th e lea d o f t h e sole n oi d ) a a B = µ n I

34. M a gn e tic F i e ld du e t o To r o i d a l So l e no id (To r o i d ): I dl B O B = B = Q P B ≠ ∫ B . dl = µ I ∫ B . dl c os ° = B ∫ dl = B ( 2 π r ) A n d µ I = µ n ( 2 π r ) I r B = µ n I ∫ B . dl = N OT E : The ma gn e t ic f i e ld ex i s t s o n ly in t he t ubul a r are a bound by t he c oil an d it do e s not ex i s t in t h e are a in s ide a nd ou t si de t he t o r oid. i. e . B is z er o a t O a nd Q an d non -z er o a t P .