IX INDUCTNCE ND MGNETIC FIELDS 9. Field in a solenoid vaying cuent in a conducto will poduce a vaying magnetic field. nd this vaying magnetic field then has the capability of inducing an EMF o voltage in the conducto. The combination of these is the phenomenon of inductance: a changing cuent in a conducto will cause an EMF. The effect is enhanced consideably if the conducto is in the fom of a coil o solenoid. Fistly the field poduced is popotional to the numbe of tuns, since what mattes is the total cuent flowing though an aea. nd secondly, the voltage is induced in each loop so that the d contibutions fom each tun add togethe. We stat with the fist pat of the poblem: calculating the magnetic field poduced by a cuent flowing in a solenoid. The pictue shows a coil of n tuns pe unit length, caying a cuent of I ampees. lthough we do not need it immediately, we shall specify the coss section aea of the coil as a. Calculation of the magnetic field of the solenoid may be done using mpèe s law (exploiting the symmety of the system). Thus we use closed loop B. dl = I with an appopiately chosen closed path of integation. We fist show that the magnetic field outside a vey long solenoid is zeo (except, obviously, at the ends). Conside the loop labelled. By the symmety of the system, and since the magnetic field must be pependicula to the diection of flow of the cuent, the magnetic field must point paallel to the axis of the coil. Then only the two hoizontal elements of the loop make a contibution to B. Thee can be no contibution fom the uppe element since it may be placed as fa away as we like. So the only contibution comes fom the lowe element. The field along this element multiplied by its length is popotional to the cuent theading the loop. This is zeo as the loop does not intesect the coil. Since the length of the element is abitay, it follows that the field must be zeo. Thus thee can be no magnetic field outside a long solenoid. Tuning to the field inside the coil, conside loop in the figue. We have just established that the field outside the solenoid is zeo. Thus only the line integal along the intenal hoizontal element of the loop contibutes to the mpèe s law integal. If the length of the element is d and if thee ae N tuns contained within the loop then mpèe s law gives Bd = NI. But since thee ae n tuns pe unit length, the numbe of tuns in the loop of length d is nd. Then PH4 / B Cowan 9.
o Bd = n B = ni. (9.) This is the expession fo the field inside a long solenoid. The field is independent of the position along the coil. This is because we ae consideing a vey long coil, which esults in tanslational invaiance. Note also, that the field is independent of the adial position in the coil; anothe consequence of symmety. We thus conclude that the field in a long solenoid is unifom o homogeneous. This is a good way of poducing a unifom magnetic field. 9. Magnetic vecto potential We have seen that we can obtain the electic field E fom the scala potential V as E = gadv in the electostatic case. Recall that the possibility fo doing this elied on the fact that in electostatics the cul of E is zeo. Note, howeve, that in the pesence of vaying magnetic fields the above equation does not hold since then cule is no longe zeo. In the pesence of souces of B, that is, electic cuents, we cannot expess B as the gadient of a scala potential since the cul of B is then non-zeo: culb = j. The asymmety between the teatment of electic and magnetic fields is not so supising since the (static) E field is iotational wheeas the B field is solenoidal. While the B field cannot, in geneal, be expessed as the gadient of a scala potential, it can be witten as the cul of a vecto potential: B = cul. (9.) Hee is called the magnetic vecto potential. Thee is no immediately obvious advantage of this quantity since we have simply eplaced one vecto by anothe. Howeve we will see that the use of the vecto potential will help with a numbe of issues. lthough not of diect elevance to this couse, the magnetic vecto potential goes togethe with the scala electic potential to make a elativistic fou-vecto. nd using the magnetic vecto potential will facilitate a coection to E = gadv, to make it geneally tue. We shall stat this discussion of the magnetic vecto potential by showing that the magnetic field can be expessed as the cul of anothe field and we will then investigate some of the consequences and elated mattes. In the inteests of simplicity we shall conside the field poduced by an element of a conducting wie. The field of the entie wie can then be obtained by integation ove its path. The magnetic field at a point due to a cuent I flowing in an element dl of a wie can be witten as PH4 / BPC 9.
dl I P I dl db =. 3 We now use the esult = gad 3 to ewite the magnetic field as I db= dl gad. Next we make use of the vecto calculus identity to wite culab= aculb b gad a cul d l = cul dl dl gad. Now cul dl = since this is like diffeentiating x with espect to x. Thus the expession fo db takes the fom I dl db = cul. This means that we can wite B as the cul of a vecto : B = cul whee the contibution to this vecto fom cuent flowing in the element dl of the wie is given by I dl d =. (9.3) The vecto is known as the magnetic vecto potential. To check fo consistency, we can take the divegence of B expessed in tems of divb = div cul = since div cul is identically zeo. So we see that expessing B in this fom automatically ensues that divb =. In electostatics we saw that the elation cule = implied that E could be expessed as the gadient of a scala potential since cul gad is zeo. nd in the magnetic case we now see that the elation divb = implies that B can be expessed as the cul of a vecto potential since div cul is zeo. Thee is an impotant connection between the magnetic vecto potential and magnetic flux. Recall the definition of cul, embodied in Stokes s theoem PH4 / BPC 9.3
aea cul. da=. d peimete whee we have used the vecto agument. Since B = cul, the left hand side of this equation is simply the magnetic flux though the aea. Thus Φ=. d peimete ; (9.4) the line integal of aound a closed loop gives the magnetic flux theading the loop. In egions of space whee thee ae no cuents pesent it is possible to expess B as the gadient of a scala potential and that potential will obey a Laplace equation. Then it will be easie to use this potential as an intemediay in the calculation of B. The inteio of a solenoid would be one such example. 9.3 Fields fom potentials Now let us look at the poblem of incopoating electomagnetic induction into the expession E = gadv. s it stands, this expession is incompatible with since cul gad of any scala is zeo. cule B = t Let us expess the cul E equation in tems of the magnetic vecto potential cule= cul. t This can be witten as F I HG K J = cul E + t. Now we know that any vecto whose cul vanishes can be witten as the gadient of a scala since cul gad is identically zeo. Fom this it follows that E + t = gad of a scala. This equation is clealy tue in the time-independent case, whee we identify the scala as minus the electic potential V E = gadv. Thus in the geneal case we now have E + t = gad V, o PH4 / BPC 9.4
E = gadv. (9.5) t This is the geneal esult, valid in dynamic as well as static cases. Thus we have succeeded in ehabilitating the deficient E = gadv. We conclude that the electic and magnetic fields may be witten in tems of the scala and vecto potentials as E = gadv t B = cul. U V W (9.6) 9.4 Self inductance We now poceed to the discussion of inductance. s we stated above, a changing cuent will poduce a vaying magnetic field, and the vaying magnetic field will induce a voltage. Equation (9.) tells us how a cuent in a solenoid will poduce a magnetic field. nd since the field was seen to be unifom ove the aea a of the coil, the magnetic flux though a single tun is nia. If the length of the coil is l, then the numbe of tuns is given by nl. nd then the total magnetic flux Φ linking the solenoid is Φ = n Ial. We know fom mpèe s law that the flux (linkage) must be popotional to the cuent. The constant of popotionality depends, essentially, on geometical popeties of the conducto. Let us denote this quantity by L, so that fo the long solenoid L = n al (9.7) nd in geneal Φ = L I. (9.8) We shall see that L is what we know as the inductance of the solenoid. Fom Faaday s law we know that a vaying flux will cause a voltage: V dφ =, ignoing the minus sign since we may choose the diection in which to measue the potential. Theefoe if the cuent in the coil is changing, thee will be a voltage acoss the coil given by V = L. (9.9) This is the conventional expession fo (self) inductance. Recall its magnitude fo a long coil: L = n la. PH4 / BPC 9.5
Obseve that L is popotional to the volume la of the coil and the squae of the numbe of tuns, as hinted at the stat of Section (9.). s you should know, the unit of inductance is the heny. If the solenoid is not infinitely long then thee is not total flux linkage with all the tuns. Then the inductance is somewhat less than that of Equation (9.7). In such cases a fudge facto α (known as Nagaoka s facto) is intoduced. We then wite L = n laα (9.) whee the facto α is a function only of the aspect atio of the coil: its atio of adius to length. If the aspect atio of the solenoid is x then a good appoximation to α is /( +.9x). 9.5 Mutual inductance By an extension of the aguments of the pevious section, it follows that if two coils ae in close poximity then a vaying cuent in one coil will induce a voltage I in the othe and vice vesa. This effect is known as mutual inductance. V nd such a coupled assembly of coils is known as a tansfome. coil I coil Let us fistly assume that the two coils occupy the same space so that all the flux which links one coil also links the othe. We ae thus assuming hee that the lengths and aeas ae the same fo both coils. The only diffeence we pemit at this stage is in the numbe of tuns pe unit length. We shall specify that coil has n tuns pe unit length and coil has n tuns pe unit length. If thee is a cuent I flowing in coil then this will poduce a magnetic field B = I n. Denoting by l the length, and a the aea of the coils, the numbe of tuns in coil is given by n l. Then the magnetic flux linking the tuns of coil is given by and the voltage acoss the second coil is then Φ = n lba = n n la I dφ V = = n nla. (9.) The constant of popotionality between the flux in one coil and the cuent in the othe, o the voltage in one coil and the ate of change of cuent in the othe is known as the mutual inductance, denoted by M. Thus we wite Φ = M I (9.) = V M d t whee, in this idealised case M is given by PH4 / BPC 9.6
M = n n la. (9.3) This expession is symmetic in the indices and, demonstated fo the idealised case whee the coils occupy the same space. This implies that if the cuent wee to flow in coil then the same voltage would be induced in coil. This is a vey geneal popety, dl dl I loop loop independent of the shape and elative locations of the coils, as we shall now show. We calculate the vecto potential at loop due to the cuent I flowing in loop. The contibution to the vecto potential at the position of dl due to the cuent in the element dl of loop is, Equation (9.3) I dl d =. Then at this point the total vecto potential is given by integating the expession aound the whole of loop : = I d l. The magnetic flux theading loop is then found by taking the line integal of aound loop Φ =. d l so in this case loop loop I π dl. dl Φ =. 4 loop loop This gives a geneal expession fo calculating the mutual inductance M Φ I = = dl. dl loop loop (9.4) but this is seen to be symmetic in the indices and ; the flux in loop due to a cuent in loop is equal to the flux in loop due to the same cuent in loop. nd thus we conclude that M = M. (9.5) 9.6 Coupling coefficient and matix epesentation We see that fo two coils, wound in the same place (including the fudge facto α), we have: L L = n laα = n laα. If we calculate the mutual inductance between the coils, then clealy the same fudge facto would apply. In othe wods we would have M = nn laα. PH4 / BPC 9.7
The mutual inductance is thus seen to be equal to the geometic mean of the self inductance of the individual coils. M = LL. If thee is not complete coupling between the coils then M would be less than this value. Fo widely sepaated coils, clealy the mutual inductance would be zeo. To teat the geneal case we intoduce a coupling coefficient K, and wite M = K LL. (9.6) We finish this section by witing the equations fo the cuents and voltages in a tansfome in the geneal case. If thee is a cuent in both coils then the voltage in a given coil depends on the vaying cuent in that coil and the vaying cuent in the othe coil: V = L + M V = M + L (9.7) which may be conveniently expessed in matix fom as V L M d I V = M L I. (9.8) 9.7 Enegy of a magnetic field When we evaluated the enegy of an electic field we did it in an indiect manne, by calculating the wok needed to establish a given potential acoss a capacito. Then the potential was elated to the electic field. The meit of using a capacito was that within its inteio the E field is unifom. We now evaluate the enegy of a magnetic field in a paallel fashion. We will calculate the wok needed to establish a given cuent in a long inducto. The meit of using a long inducto is that within its inteio the B field is unifom. The defining popety of an inducto is Equation (9.9): V = L. So in a small time the cuent will change by a small amount : V = L. If we multiply both sides of this equation by the cuent I, giving IV = L I then on the left hand side IV gives the powe, so multiplying this by the time inteval gives the wok done duing this inteval, which we denote by dw. Thus we have PH4 / BPC 9.8
dw = L I, which we may integate up fom an initial cuent of zeo to a final cuent I: W I = L idi = LI. (9.9) This gives the enegy stoed in an inducto of inductance L caying a cuent of I ampees. Now the cuent is elated to the magnetic field though Equation (9.), which we wite as B I = n. nd since L is elated to the vaious geometic factos by Equation (9.): L = n la, these may be combined to give W = B la (9.) whee we see that the n cancels out. Obseve that la is the volume of the egion containing the magnetic field, which tells us that thee is a magnetic enegy density U B given by U B = B (9.) So in a combined electic and magnetic field, adding the two contibutions to the field enegy, we have ε U = E + B. (9.) This is the geneal expession fo the enegy density of an electomagnetic field. The field enegy contained in a egion of space is then found by integating the field enegy density ove the volume of the egion. Recall that in Section (7.6) we consideed the question of enegy consevation in an electomagnetic field, whee we intoduced the Poynting vecto. PH4 / BPC 9.9
9.8 Finding the potentials How ae the potentials detemined by the souces? We know that in the electostatic case the electic potential obeys the Poisson equation V = ρ ε. We now ask how the electic scala potential V and the magnetic vecto potential ae elated to the electic and magnetic souces ρ and j in the geneal dynamic case. to give In the electic case we combine while in the magnetic case we combine to give (ecall cul cul = gad div ) dive= ρ/ ε and E= gadv t ρ ε V = div (9.3) t E culb = j + and B= cul c t = j+ gad div + c t c V t. (9.4) Equations (9.3) and (9.4) ae diffeential equations fom which the potentials V and may be detemined fom the souces ρ and j (and the appopiate bounday conditions). Howeve the equations do look athe complicated. nd simplification is possible. It is only the gadient of V that is impotant so one can add an abitay constant to V without changing the obsevable electic field; this we know aleady. But similaly, since cul gad, we can add the gadient of an abitay scala field to without changing the obsevable magnetic field. It then follows that we can simplify Equations (9.3) and (9.4) though the imposition of supplementay estictions on V and which have no effect on the obsevable E and B. Helmholtz s theoem tells us that a vecto field is detemined once its cul and its divegence ae specified. We have no choice with the cul of ; this must give the coect value fo the B field. But the divegence of is anothe matte; we can choose this to be whateve we like; it can even depend on time if we wish. The choice in the pecise specification of, fo instance stating what its divegence is, is called the choice of gauge. n elegant choice of gauge is to make the backet in the ight hand side of Equation (9.4) to be zeo. That is, we choose div to be V div =. (9.5) c t Not only does it emove the gad tem in Equation (9.4), it also convets the PH4 / BPC 9.
div t tem of Equation (9.3) to ( c ) V t This sepaates electic and magnetic effects; V is given solely in tems of ρ and is given solely in tems of j. We have two inhomogeneous wave equations; electic and magnetic effects ae untangled and the vecto equation fo is equivalent to thee scala equations fo the its thee components (so long as we ae using ectangula Catesian coodinates). V = ρ ε =. j (9.6) This is called the Loentz gauge. ctually this is named afte the wong Loentz; it was intoduced by L. Loenz, not H.. Loentz! special case of the Loentz gauge occus in the static case: when time deivatives ae zeo. Then div = ; we have the Coulomb gauge. nd we then ecove the conventional Poisson s equations V = = ρ ε. j (9.7) When the magnetic field is unifom thee is a special gauge that is paticulaly convenient to use. The Landau gauge is a special case of the Coulomb gauge (which is a special case of the Loentz gauge.) In the Landau gauge the vecto potential is specified by The B field is found by taking the cul of : = Byx. ˆ (9.8) z xˆ yˆ zˆ B= cul = x y z = B zˆ. z By z In othe wods the vecto potential given by the Loentz gauge Equation (9.8) esults in a unifom magnetic field pointing in the z diection. This poves useful in many pactical applications; it is the simplest way of witing a vecto potential which gives a unifom magnetic field. The solution of Equations (9.7) ae given by: V ( ) ( ) ( ) ( ) = ε volume = volume ρ j d v. dv (9.9) PH4 / BPC 9.
We shall not conside the solutions in the dynamic case; these ae two complex fo this couse. The components of ae expessed in tems of the components of j just at the value of V is expessed in tems of the value of ρ (to within a constant facto). This shows that just as V augments to ceate a fou-vecto, so the chage density augments the cuent density to give a fou-vecto. It is also wothwhile to point out that while the equation elating and j is cetainly a vecto equation, the stuctue of the equation is such that x is detemined by j x alone, y by j y and z by j z. In othe wods this vecto equation is equivalent to thee independent scala equations (so long as we ae using ectangula Catesian coodinates). Fo completeness we give the expessions fo E and B in tems of thei souces in the static case ρ ( ) E ( ) = ˆ dv ε volume (9.3) j ( ) B ( ) = ˆ d v. volume Obseve that the equations fo V and ae much simple. 9.9 Summay of magnetostatic esults The magnetostatic elations between the thee quantities B, and j ae summaised in the following diagam, boowed fom Intoduction to Electodynamics by D. J. Giffiths. This paallels the simila diagam fo the electostatic case given in Section (4.6). j 4 π = j volume d v j = cul ; div 4 π B = j volume = = B j B ˆ d v B = cul B PH4 / BPC 9.
When you have completed this chapte you should: be able to calculate B inside a long solenoid; be familia with the concept of magnetic vecto potential; be able to calculate fom an abitay cuent distibution; undestand the connection between and magnetic flux; be able to genealise E = gadv to the case of electomagnetic induction, using ; know the meaning of self inductance and be able to calculate L fo a long solenoid; undestand that L is popotional to the volume of a solenoid and the squae of the numbe of tuns; know the meaning of mutual inductance and be able to calculate M fo co-positioned long solenoids; undestand why and be able to demonstate that M = M ; be familia with the idea of coupling coefficient; be able to wite the V I elation fo a tansfome in matix fom; intepet the wok done in establishing a magnetic field in an inducto in tems of magnetic field enegy; be able to calculate the E and B fields fom V and in the geneal case; undestand the choice of gauge in the specification of and be familia with the Loentz gauge and the Coulomb gauge. PH4 / BPC 9.3