- If one knows that a magnetic field has a symmetry, one may calculate the magnitude of B by use of Ampere s law: The integral of scalar product

Transcription

1 11.1 APPCATON OF AMPEE S AW N SYMMETC MAGNETC FEDS - f one knows ha a magneic field has a symmery, one may calculae he magniude of by use of Ampere s law: The inegral of scalar produc Closed _ pah * d l ne * dl dl cos over a closed pah is o (1) ne is he ne curren flowing hrough he area inside closed pah. Noe ha he field is due o all currens around; no only o he conribuion of ne. The righ hand rule fixes he posiive sense of circulaion on he closed pah as follows; if he humb poins along ne curren ( ne ) sense, he curled fingers indicae he posiive sense of circulaion on he closed pah. Examples. a) Find he magniude of field due o a curren flowing in a sraigh long wire (fig.1). We Fig.1 know ha, from symmery poin of view, has he same magniude a all poins a disance from he wire and i is angen o he circle passing by hese poins; he righ hand rule gives he same direcion for "+" sense of circulaion and (fig.1).we selec a circular pah and apply he expr. (1) * d l dl * () circle _ circle _ b) Calculae he magneic field of an ideal infinie solenoid ha has n urns/m and carries he curren. n ideal infinie long solenoids he field ouside he solenoid is zero (Fig. ) and inside i is prey much uniform (same everywhere). s direcion (see righ side of fig.) is found by he righ hand rule. Fig. There are nl wires in lengh l; so he ne curren hrough he square pah is nl. Then, from (1) we ge l ne * l n* l * and n (3) * Noe ha, in he express. (1), vecor d l has he same direcion as vecor. One may selec a closed square (side l ) pah like ha shown in fig. ("+" sense is fixed by rule of Ampere law) and apply expr. (1). * d l dl along axis, inside solenoid * d l ouside and along verical sides c) Calculae he magneic field of a oroidal coil wih N urns carrying he curren. l Fig.3 A quick observaion shows ha he magneic field lines are circles passing perpendicular o he oroid secions. We selec a circular pah wih radius r as shown in fig.3 and apply he Ampere s law. As he field magniude is he same around he circui, we find: N - inside wired secion * r N (4) r - ouside oroid = because he ne curren is N* + N* = - inside oroid = because ne curren is zero. ( = ) 1

2 11. NDUCTANCE - A carrying curren solenoid ( ) is he bes ool o build, change and conrol a magneic field. The main propery of a solenoid is he reacion versus he change of magneic flux passing hrough i. ind Fig.4 ε i Fig.4 presens wha happens a a solenoid a few momens afer he swich is urned on. The curren due o source ε creaes inside solenoid he field direced o he righ. This means an increase of magneic flux (i was ) inside he solenoid. The enz law ells ha he " solenoid will reac " by inducing a curren " i " such ha he relaed magneic field ind be opposie o. The induced curren i flows in he opposie sense wih respec o. The reacion of solenoid does no allow insananeous se of curren value in circui as given by Ohm s law ( = ε / sol ). This kind of reacion happens when he swich is urned off, oo, bu he direcion of induced curren "i" is he same as ha of and ind has he same direcion as ; he circui reacion does no wan o leave he flux decrease. n he firs case, one observes a gradual increase (Fig. 5a) of ne curren and in he second case one observes a gradual decrease (Fig5.b) of curren in circui. max = ε/ sol Fig5.a Swich urn on 5.b Swich urn off - The phenomena presened in he upper paragraph, is a known as self-inducion; he E.M inducion is relaed o he flux buil by he solenoid and passing hrough iself. This is a phenomenon ha happens in all circuis; he presence of a solenoid ino he circui jus makes i more pronounced. f here is a magneic flux Φ hrough a single urn of a solenoid(or coil), hen, he oal flux hrough a solenoid wih N urns is N*Φ. The quaniy N*Φ[Weber] is known as flux linkage hrough he coil. -n many real siuaions (ex. ransformers), wo coils are arranged in such way ha he magneic flux from coil 1 hrough coil be maximal (Fig.6). Noe ha each coil is par of a differen circui and carries a differen curren ( 1, ). n his siuaion he flux hrough each urn of a coil ( le's refer o coil 1 ) is consiued by wo componens; Φ 1 due o curren 1 and Φ 1M due o curren in second circui. So, (5) 1 1 1M and he ne flux linkage hrough coil_1 is N N ) (6) 1 1 1( 1 1M 1 Then, he expression for he efm induced in coil_1 is Fig.6 d NET (7) d( N d N d N 1 1) ( 1 1 ) ( 1 1M ) M

3 - Till now one uses several parameers (loop area, coil lengh, curren and number of urns) o express he inducion behaviour of coils. Nex, one measures he coil s behaviour versus E.M. inducion by a single parameer. Consider firs is self-inducion par presened by facor "N 1* Φ 1 " a expression (6). The magniude of curren 1 depends on circui " 1" where coil_1 is conneced. The field " 1 " and is flux hrough a urn "Φ 1 " are proporional o 1. All oher conribuions o "N 1* Φ 1 " flux depend only on he coil_1 geomery. So, one pus hem ogeher inside a single parameer known as coil self-inducance (). This way, he self-induced flux linkage in coil " 1 " can expressed as N1 * 1 * 1 (8) [H-Henry] is a parameer ha depends on coil geomery (lengh, diameer, number of urns) and d( N1 1 ) d( 1) d defines he "self-induced emf " by relaion 1 (9) The direcion of his induced emf is such ha he relaed curren opposes he curren changes in circui. i i Fig. 7.a Fig. 7.b - The "muual inducance M" helps o calculae he par of flux in firs coil due o curren in second coil. So, N1 * 1M M * (1) This parameer M is "muual" and depends only on he geomery of he whole se of wo coils. has he same value when calculaing he effec of curren in coil 1 on flux hrough coil, i.e. N * M M * 1 One may figure ou easily ha M is bigger when he coils are closer o each oher and when hey have he same cenral axe. ased on (1) one ges ha emf in coil 1 due o change of flux originaed from coil is 11.3 CCUTS d( N1 1M ) d M M (11) Fig.8 -A real inducor has always a resisance; in he following model one assumes an ideal inducor (zero resisance) and includes he inducor resisance ino exernal resisor. To find he evoluion of curren ino a circui, one may apply he Kirchhoff rule o he circui in Fig. 8 a a momen when here is a self-induced emf ε wih magniude d/ acing in circui (his is no a seady say siuaion). i a) Once he swich is urned on, he curren sars flowing along he direcion shown and builds a magneic field in he coil. This increases he magneic flux hrough coil and an emf is induced in i. This emf gives rise o curren i direced opposie sense o. Consequenly, only a moderae increasing curren "" happens in circui. Kirchhoff s rule gives: d d (1) To solve equaion (1) one uses a new variable y (13) Noe: The inducion effec is couned by ε (i.e don include i in calculaion) 3

4 Time derivaive of (13) gives ges y dy ( ) dy y * dy d (14) y subsiuing (13) and (14) ino equaion (1) one y dy y ln y ln y ln and y y y y y e (15) A = _ and _ y /. So, afer noing ℇ / = * * * he expression (15) gives y e e (1 e ) (16) Fig.9 Afer defining he ime consan as (17) he expression (16) ransforms o (1 e ) (18) For he ime inerval = τ he curren increases from o.63. The same funcion shows he charge Q evoluion during charging a a C circui bu for ha circui he ime consan is τ = C. - Afer he curren ges o he value o here is no induced efm in he circui because he flux remains consan. The flux hrough he coil changes anew if he source ε is swiched off. Noe ha one needs a closed circui o observe he curren evoluion. Tha s why we refer o he scheme presened in Fig.1a where he swich S is urned on as he swich S 1 is urned off. n his siuaion, he induced emf i i Fig.1.a Fig.1.b opposes he decrease of magneic flux hrough coil by producing a curren i along he same sense as ( ε = - *d/ is posiive because d/ < ). The Kirchhoff rule for his siuaion gives ln Afer noing * e d For τ = / _ and _ d and d ( ) * * e * / e (19) The graph in figure 1.b presens he decrease of curren following expression (19). For = τ = /, he curren falls down by 63% of is iniial value. 4

5 11.4 ENEGY STOED NSDE AN NDUCTO -e s consider anew he circui in fig.8 a few momens afer he swich is urned on. The Kirchhoff rule gives d( d( ( ( () Noe ha ( is smaller han he curren value a seady sae ( = ε /). y muliplying by ( boh sides of () one ges d( ( ( ( (1) f considering an ideal emf source he produc (*ε would presen he power supplied by his source ino he circui a he momen. The facor (* is he power dissipaed hermally ino he resisor a d( his momen. Then, he erm ( would presen he power being supplied o he inducor a momen. eferring o he direcion of curren in scheme given in Fig.8, one migh figure ou ha he source "ε" is supplying energy in circui and he "induced emf source ε " is soring energy inside he inducor as magneic field energy U (. So, he increase rae of magneic energy inside he inducor (i.e. du /d is equal o he power delivered by he source ε o he "ε ". This way, one ges he relaion du( d( ( du ( d( du ( d U ( A he end of ransiory period he curren is o ( = ε /) and 1 U () This energy is sored inside he inducor. is due o he presence of magneic field inside he inducor and i remains he same during all he seady sae in circui. emember ha he elecric field energy sored inside a capacior a he end of he ransiory period is 1 Q 1 U C or UC CV and one uses he leer U o indicae ha his is a ype of poenial energy. C - e s see he case of a solenoid wih n [urns/m], lengh l [m], cross secion area A and curren [A]. The fields inside solenoid (ideal model, see relaion 3 ) is n (3) The magneic flux hrough i is N ( A* ) ( n* l)*( A* ) n* l * A* n * n * A* l * * (4) So, he self inducance for a solenoid is * n * A* l (5) 5

6 y subsiuing (5) a relaion () and isolaing from (3) as, one ges n 1 1 U n A* l A* l (6) n This energy is sored inside he whole volume (A*l) of solenoid. So, he energy sored inside he uni volume or he densiy of magneic energy is u 1 * (7) - is ineresed o menion ha his energy appears as an elecric spark a he swich when i is urned off. Exemple: The igniion coil in an auomobile makes use of his effec o fire he spark plug. mporan Noes: - The expression (7) is valid for any magneic field. - is very similar o he densiy of elecric energy (any elecric field u E * E ) - As expeced (from wave naure of fields) hese energy expressions are ~ o he square of field srengh. 6