FAADAY'S LAW 31.1 Faaday's Law of Induction In the peious chapte we leaned that electic cuent poduces agnetic field. Afte this ipotant discoey, scientists wondeed: if electic cuent poduces agnetic field, is it possible that agnetic field can poduce an electic cuent? Conside loop of wie that is connected to a galanoete and a agnet is oed in the icinity of the loop. It was obseed that if both the loop and the agnet held stationay elatie to each othe thee will be no deflection in the galanoete. If the loop is held stationay while the agnet is oed towad the loop, the galanoete will deflect in one diection. If the agnet is oed away fo the loop, the galanoete will deflect in the opposite diection. The sae obseations is occued when the loop is oed while the agnet is held stationay. Fo these obseations one concludes that a cuent is poduced in the loop as long as thee is elatie otion between the loop and the agnet. Such a cuent is called the induced cuent, and it souce is called the induced ef. The phenoenon itself (the poduction of electic cuent fo changing agnetic field) is called the electoagnetic induction. Michael Faaday studied these obseations quantitatiely and put the in a atheatical foula consideed as one of the fundaental laws in electoagnetic theoy. He found that the induced ef is popotional to the ate of change of the agnetic flux, i.e., = N 31.1 N S N S N S (a) (b) (c) Figue 31.1 (a) A ba agnet is stationay elatie to the loop and thee is no deflection in the galanoete. (b) The ba agnet is oing towad the loop and the galanoete deflects in one diection. (c) The ba agnet is oing away fo the loop and the galanoete deflects in the opposite diection. Whee N is the nube of tuns in the loop. Fo a unifo agnetic field the agnetic flux becoes Φ = BAcosθ
With θ is the angle between the agnetic field and the aea ( The diection of the aea of a plane is noal to the plane). The SI unit of the agnetic flux is Webe (Wb) with 1 Wb equals to 1 T.. Faaday's law now eads ( BAcosθ ) d N 31. Fo this expession we conclude that an induced ef can be ceated if eithe B, A, θ, o a cobination of the ay with tie. The inus sign in Faaday's law is a consequence of the law of conseation of enegy. Exaple 31.1 A coil consists of 00 tuns of wie. Each tun is a squae of side 18 c, and a unifo. field pependicula to the plane of the oil is tuned on. If B = 0.5T in 0. 8 s, what is the induced ef in the coil. Solution Knowing that θ = 0, and the aea A is constant we hae ( B) d db 0.5 NAcos θ NA = 00(0.18) = 4.1 V 0.8 Exaple 31. A loop of wie of aea A is placed in a. field pependicula to the plane of the loop. The agnitude of B aies with tie accoding to B = B ax e, what is the induced ef in the loop. Solution Again θ = 0, and the aea A is constant we hae αt NAcos θ ( B) d db NA = αabax e αt 31. MOTIONAL EMF To undestand how the induced ef is oiginated we now study in details the natue of the induced ef. Conside a od of length l oing with speed in a unifo agnetic field B diected into the page, as shown in Figue 31.. The fee electons inside the od will expeience a agnetic foce F = eb which is diected downwad. The electons then will accuulate at the lowe end of the od leaing a net positie at its uppe end. As a esult of this chage sepaation, a net electic field E will be set up inside the od. Theefoe, fee electons now will be affected by an upwad electic foce F e = ee in addition to the agnetic field. Chages continue to build up at the ends of the od until the two foces balanced. At this point otion of chages ceases leaing the od with two opposite polaities at its end, that is an ef is poduced acoss the od. To calculate this ef we hae, fo the equilibiu condition F = o eb= ee so E = B F e
B l Figue 31. (a) A conducting od oing in a unifo agnetic field into the page. The agnetic foce akes electons to accuulate at the lowe end of the od, leaing the uppe end with positie chages. (b) A conducting od slides along conducting ails in a unifo agnetic field into the page. A cuent will be induced in the loop. l B x Since the electic field in unifo inside the od, the potential diffeence acoss the od is elated to this electic field accoding to = V = El = lb 31.3 This potential diffeence is aintained acoss the ends of the od as long as it is oing in the field and is called the otional ef. If the od is a pat of closed loop, as shown in Figue 31.(b), a cuent will flow in the loop fo the positie end to the negatie end (counteclockwise). Let us now poe Equation 31.3 using Faaday's law. Conside again Figue 14.(b) whee the od is sliding along conducting ails in the agnetic field such that it fos a closed loop. The ate of change of the agnetic flux though the loop is now popotional to the change in the aea of the loop. As the aea of the loop at soe instant is A= lx, the agnetic flux though the loop is Φ = Blx Whee x, the wih of the loop is changing with tie as the od oes. Using Faaday's law, we find that the induced ef in the loop is d Φ d = = ( Blx) = dx Bl But ( dx ) epesents the speed of the od, so we obtain
lb 31.4 Which is the sae esult of Equation 31.3 except of the inus sign. Exaple 31.3 A conducting ba of length l otates with a constant angula speed ω about one end. A unifo. B is pependicula to the plane of the otation. What is the potential diffeence induced between the ends of the ba. Solution It is clea that is not constant along the length of the ba We hae to diide the ba into sall eleents each of length d. Noe the ef acoss one of these e eleents is dv = Bd = ωbd l Integate to find the ef acoss the ba L V = ω B d = 31.3 Lenz's Law 0 1 Bωl It tells us that the induced cuent ust be in a diection such that it poduces a agnetic field to oppose the change in the agnetic flux. Lenz's law can be explained by the following two ules: (1) If the agnetic flux though the loop is inceasing, the diection of the induced cuent is such that it poduces a agnetic field opposite to the souce agnetic field, () If the agnetic flux though the loop is deceasing, the diection of the induced cuent is such that it poduces a agnetic filed in the sae diection as that of the souce agnetic field.
Exaple 9.3 A squae loop of side L and esistance oes with constant speed though a egion of wih 3L in which a unifo agnetic field B diected out of the page as shown. Plot the flux and the induced ef in the loop as a function of x, the position of the ight side of the loop. Solution The agnetic flux is zeo befoe the loop entes the field. As the loop is enteing the field, Φ =Blx, that is, the flux inceases linealy with x, eaches its axiu alue, Bl, when the loop is entiely in the field. Finally, as the loop is leaing the field Φ =Bl(4L-x), that is the flux deceases linealy with x, eaches to zeo when the loop is entiely outside the field. Φ 3L x Now dx dx dx F Noting that /dx is the slope of the cue in the fist gaph, (Φ s. x). While the loop is enteing the filed the flux is inceasing and, accoding to Lenz s ule, the agnetic field set up by the induced cuent is into the page (opposite to the oiginal field). Hence the induced ef is clockwise. While the loop is leaing the field, Φ is deceasing and the agnetic field set up by the induced cuent in, this case, is out of the page (siila to the oiginal field). This eans that the induced ef is counteclockwise. To find the foce on the loop it clea that while the loop is enteing the only side that cause the net foce is the ight side. Now F = Il B = ILB ( j k) = ILB( iˆ ) While the loop leaing the left side is the only side that cause the foce. Again
F = Il L B = ILB ( j k) = ILB( iˆ ) 9.3 INDUCED ELECTIC FIELD A changing agnetic flux ceates an induced ef and thus an induced cuent in a conducting loop. Theefoe, an electic field ust be pesent along the loop. This field, which is ceated by changing agnetic flux, is called induced electic field and gien by. in = E ds 31.5 Using Equation 31.1, Faaday s law can be ewitten as E ds 31.6 It should be noted that this esult is also alid fo any hypothetical closed path. The induced electic field gien by Equation 31.6 is quite diffeent fo the electostatic field (poduced by static chages). The foal one is a non-conseatie field poduced by a changing agnetic flux. Hence no electic potential can be associated by the induced electic field. The potential diffeence between two points i and f is V f V i f E ds i which would be zeo fo a closed path, contay to Equation 31.6 The diection of the induced e.f. is deteined by Lenz's ule Exaple 31.8 A long solenoid of adius has n tuns of wie pe unit length and caies a cuent gien by I = Iax cosωt, with B o and ω ae constant and the tie t is in seconds. Calculate the induced electic field inside and outside the Solenoid. Solution Φ The. flux though the aea enclosed by the closed loop is = BAcosθ = µ o ni ( π ) = µ nπ I cosωt o ax Now applying
E ds = µ nπ ωi o ax By syety, the agnitude of E is constant aound the path and tangent to it ( ax E π) = µ onπ ωi Fo which we find that µ onωiax E = To calculate the electic field outside the sphee, the closed path has now a adius >. Since the agnetic field is confined only to the egion <, the agnetic flux though the path is Φ ( π ) = µ nπ I ωt = µ oni o ax cos The electic field is now ( ax E π ) = µ onπ ωi Fo which we find that µ n I E o ω = ax