Time-varying electromagnetic field

Transcription

1 Chapter 5 Time-varying electromagnetic field In this chapter we take up the time-varying electromagnetic field. By this study, we will be able to understand in a systematic way that actually the electric and the magnetic fields induce each other and such laws will complete the system of Maxwell equations. The most dramatic consequence of this mutual induction will be the existence and propagation of the electromagnetic waves. 5.1 Time-variation of the magnetic field Faraday s law of electromagnetic induction Discovery of Faraday 1831, stimulated by the works of Oersted and Ampère, Faraday conducted an experiment of varying the magnetic flux which goes through a circuit loop and discovered the following phenomemon: Φ Φ + Φ Φ = magnetic flux = B ˆndS induced current 1

2 Increase of the magnetic flux Φ Generation of the electromotive force in such a direction that the induced current tends to decrease the magnetic flux through the loop ( Lentz law ) Here the electromotive force is defined to be the work received by a unit charge as it goes around the closed circuit once. The precise relation Faraday found was E(electromotive force) = dφ dt This is called the integrated form of the Faraday s law. There are two apparently different origins for the time-variation of the magnetic flux: A B The circuit moves in a non-uniform magnetic field The circuit stands still. B varies in time. This is the same as A as seen in the rest frame of the circuit. B( x) In the first paper on special relativity, Einstein starts by asking how we should understand the above two apparently different but obviously relative descriptions of the induction of the electric field. This was one of the strong motivations of Einstein to develop the theory of special relativity. B(t) Motion of circuit in non-uniform magnetic field In this case, together with the circuit the electrons in the circuit move in the magnetic field with the velocity v. Then the Lorentz force acts on them and due to the non-uniformity of the magnetic field, unbalance of the force is generated and it acts as the source of the electromotive force. 2

3 A simple model : Let us check this mechanism quantitatively. For this purpose, consider a simple model where a rectanglular loop moves in the x direction where there is a magnetic field B( x) pointing in z-direction with non-uniform strength along the x-direction. z y F a 1 x v The Lorentz force acting on a unit charge in the circuit is given by F = v B The work done by this force as the charge goes once around the loop is E = F d l line integral is counterclockwise Along the edges 2 and 4, F is perpendicular to d l and hence no work is done. Along the edge 3 (resp. edge 1), F and d l are parallel(anti-parallel). Therefore the total work done is E = avb(1) + avb(3) On the other hand, from the figure, the change in t of the magnetic flux penetrating the loop is Φ = B(1) av t B(3) av t = t ( avb(1) + avb(3)) v t Therefore, evidently we have the relation E = Φ t = dφ dt a 3 1 which is precisely the Faraday s law. Area= av t 3

4 More general case : One can derive the Farady s law for a loop of general shape moving in an arbitrary direction as well. However since the mathematics is more involved we will not describe it in this course. From Faraday s law to the Maxwell s equation : We shall now rewrite the Faraday s law into the 3rd equation of Maxwell. Generation of the electromotive force means that an electric field is induced. So what we should do is to express the law above as a relation between the electric field and the time-variation of the magnetic field. Recall that the electromotive force is defined as the work done by the induced electric field to a unit charge as it goes around the loop. Therefore, making use of the Stokes theorem, we can write E = E d l = ( E) ˆndS Stokes On the other hand, the Faraday s law says that this is equal to dφ/dt, which can be expressed in terms of the magnetic flux density as dφ = d B ˆndS dt dt Since the equality holds for an arbitrary loop, we can remove the integral and obtain a local equation E = B This is so-called the 3rd Maxwell equation or differential form of the Faraday s law. 4

5 5.1.3 Case where B varies in time with circuit at rest Clearly, this case is equivalent to the situation where one moves with the circuit in the previous case. So it must simply be a different view of the same phenomenon. Therefore there must be a force of the same magnitude as v B on a unit charge in the circuit. However, in the rest frame of the circuit the charges are also at rest and hence cannot feel the force from the magnetic field! This means that there must somehow be an electric field of the form E = v B present. Now in the rest frame of the circuit, magnetic field is varying in time. This variation in an infinitesimal interval dt is given by d B = B( x + vdt) B( x) = dt( v ) B + O(v 2 ) B = ( v ) B To show that the Faraday s law holds, we need to check that this equals E, where E = v B. Using the formulas of vector analysis, we get E = ( v B) = v( B) ( v ) B = ( v ) B B = 0 = B So indeed the same form of the Faraday s law is valid in this case as well. Actually, in special relativity, one can show that E = v B holds due to the Lorentz transformation of electromagnetic fields Example of the calculation of the induced electric field Electric field produced by the time-variation of the magenetic field of the solenoid : The form of the Faraday s law is very similar to that of the Ampère s law for the static magnetic field. In fact if we make the replacement E B B/ µ 0 j, the Faraday s law beomes the Ampère s law. 5

6 E = B B = µ0 j Therefore the problem at hand is completely parallel with that of the generation of the magnetic field due to a straight wire, which was treated before. (The left figure is for the case Ḃ > 0) B a E = B r E d l Lentz j B = µ0 j B So just as we did before, we will make use of the integrated form of the Faraday s law derived using the Stokes theorem: dsˆn ( E) = E d l = 2πrE(r) ( sign is because d l is in direction opposite to E.) On the other hand, dsˆn B = dsḃz = dsḃ Therefore we get E(r) = 1 2πr dsḃ We know that the magnetic field is constant both inside and outside the solenoid. In particular, it vanishes outside. Taking this into account, the 6

7 RHS can be computed as for r a for r a dsḃ = πr2 Ḃ E = r 2Ḃ dsḃ = πa2 Ḃ E = a2 2r Ḃ Exercise 5.1 Using the correspondence between the Farday s law and the Ampère s law discussed above, show that the familiar result for the magnetic field produced by a straight line current can be obtained from E = a2 Ḃ, the form valid for r a. 2r B a S Exercise 5.2 Consider the situation where there is a disc of radius a and a uniform magnetic field of magnitude B is applied in the direction perpendicular to the disc. When we rotate this disc at the angular velocity of ω, an electromotive force is induced between the center and the points on the perimeter. Find its magnitude. 5.2 Time-variation of electric field and Maxwell s displacement current The Faraday s law, which connects the magnetic field with the electric field, tells us that when the magnetic field changes in time, in general the electric field must also change in time. Since we have so far been dealing with the situation where E is timeindependent, we must reconsider all the laws we have derived and see if they hold also for the time-dependent E. In other words, we have to check if the Maxwell s equations are mutually consistent for general situation. 7

8 (1) First let us see if the Faraday s law is consistent with the non-existence of the magnetic charges. If we take the divergence of the both sides of the Faraday s law, we find E = B ( E) = 0 = B The LHS is identically zero, while the RHS vanishes due to the absence of the magnetic charge. So the Faraday s law is quite consistent in this regard. (2) Next let us see what becomes of the Ampère s law when we allow time dependence. For the static magnetic field, it reads Taking the divergence of both sides, B = µ 0 j ( B) = 0 = µ 0 j Again the LHS identically vanishes, while the RHS is zero due to the charge conservation for steady current. However, in the general time-dependent situation, the charge conservation takes the from j + ( ρ/) = 0 and this apparently leads to inconsistency. How can we remedy this problem? The hint for the appropriate modification is obtained by looking at the Maxwell s first equation coming from the Coulomb s law. If we tentatively assume that this law is valid even for time-varying field, i.e. ɛ 0 E = ρ upon taking the time derivative we get ρ = ɛ 0 E (5.1) 8

9 Therefore the charge conservatino equation j +( ρ/) = 0 can be written as ( j + ɛ ) E 0 = 0 (5.2) So, although j is no longer divergenceless, the combination j + ɛ 0 E/ is still divergenceless. Therefore if we use this quantitiy in place of j in the Ampère s law, we can avoid the inconsistency. That is ( B = µ 0 j B = µ 0 j + ɛ ) E 0 This shows that when the electric field depends on time, the there is an additional source of the magnetic field in the Ampère s law, namely j d ɛ 0 E This term was discovered by Maxwell and is called Maxwell s displacement current. It will play a crucial role in the existence of the electromagnetic wave, as we shall see later. Also, this shows that even in the region where there is no real flow of electrons, there occurs and effective current flow if there is a time-varying electric field. A typical example is provided by the time-variation of the charges stored in a condenser. One can indeed check that in such a situation a current flow is observed across the empty space between the two plates of the condenser. So by addition of the term expressing the displacement current in the Ampère s law, the system of Maxwell equations become logically consistent and also experimentally consistent. Historically, the observation of the electromagnetic wave constituted the crucial evidence for the existence of the displacement current. 5.3 Summary of the Maxwell equations Differential form 9

10 E = ρ ɛ 0 B = 0 E = B B E = µ 0 j + µ 0 ɛ 0 Integral form With the aid of the Gauss and Stokes theorems, we obtained ɛ 0 dse ˆn = d 3 xρ = Q V V dsb ˆn = 0 E d l = dsb Σ ˆn = Φ Σ ( B d l = ds µ 0 j + µ 0 ɛ ) E 0 ˆn Σ Σ Together with the Lorentz force law below, we have the complete system of equations of electromagnetism: ( ) F = q E + v B 10

11 Appendix: fields : Comparison of the laws for electric and magnetic Recall the formula for the electric field due to charge density and the formula for the magnetic field due to current density. E = 1 d 3 x ρ( x )( x x ) (5.3) 4πɛ 0 x x 3 B = µ 0 d 3 x j( x ) ( x x ) (5.4) 4π x x 3 Their forms are notably different. This is primarily because there are no magnetic charges as the source of magnetic field. However, actually we can reveal a hidden beautiful correspondence between the electric and magnetic fields. To see this, recall the formula which expresses the electric field in terms of the electrostatic potential: E = φ (5.5) Now a natural question is whether there is a corresponding potential for magnetic field. As we already know, the Lorentz force due to magnetic field is perpendicular to the velocity of the particle and hence there is no work performed by such a force. This means that unlike the case of the electric field, there is no potential energy stored due to the magnetic field. However, from the basic equation B = 0, we can write B in the following way: B = A (5.6) In fact ( A) = 0 holds as an identity and such a vectorial quantity A exists. This is called the vector potential. Although we shall not prove it here, A is determined from j in the following way: A( x) = µ 0 4π d 3 x j( x ) x x (5.7) (In other words, by taking the rotation of the RHS, we get the expression (5.4).) This is evidently in parallel with the formula for the Coulomb potential φ( x) = 1 4πɛ 0 11 d 3 x ρ( x ) x x (5.8)

12 Behind all this, there is actually a beautiful special relativistic structure. This unfortunately is beyond the level of this course and hence we will not describe it. 5.4 Maxwell s equations in the vacuum and electromagnetic wave The most striking phenomenon that proved the correctness of the Maxwell s theory was the prediction and the verification of the existence of the electromagnetic wave (H. Herz, 1888). In this section, we shall study the Maxwell s equations in the vacuum and see that the electromagnetic wave arises as a solution to such equations Maxwell s equations in the vacuum Let us recall the Maxwell s equations: E = ρ ɛ 0 B = 0 E = B B E = µ 0 j + µ 0 ɛ 0 In the vacuum, i.e. in the region where no charges nor currents are present, these equations take simpler forms which are more or less symmetric with respect to the interchage of the electric and the magnetic fields: E = 0 B = 0 E = B B = 1 c 2 E ( ) 1 c = ɛ 0µ

13 5.4.2 Electromagnetic wave equations in the vacuum Let us first eliminate the magnetic field by judicious combinations of these equations and derive an equation satisfied by the electric field E. Specifically, take the rotation of the 3rd equation and use the 1st and the 4th equations. With the use of an identity, we obtain Therefore we find ( E) = ( B) LHS = ( E) 2 E = 2 E RHS = ( ) 1 E c 2 = 1c 2 E 2 2 ( ) 1 2 c E = 0 This vector equation means that each component of the vector E satisfies the same equation. ) Similarly, we can instead start from the 4th equation, eliminate E and derive the equation for B. This gives exactly the same form of the equation as above, namely ( ) 1 2 c B = 0 These equations are called the wave equations for the electromagnetic field. Note that the term expressing the displacement current plays the important role in deriving the wave equations. 13

14 Exercise 5.3 Derive the wave equation for B Plane wave solution of the wave equation Since the wave equation is a linear equation, a superposition of various solutions is again a solution. In other words, one should be able to express a general solution as a superpositions of suitable basic solutions. Very often the solutions called the plane wave solutions play the role of such basic solutions. Wave in one dimension : We will begin with the simplest situation, i.e. a wave propagating only in the x-direction. (This is the same as considering a wave in 3-dimensions which do not depend on y, z.) One dimensional one-component wave equation is given by ( ) 1 2 c 2 2 ζ(x, t) = 0 2 x 2 There are several different ways of solving this equation but we will describe the most elementary method which makes use of the factorization. Since / and / x commute, i.e. we can act them successively in different order, we can treat these differential operators as if they were ordinary numbers. Then we can use the simple factorization formula a 2 b 2 = (a + b)(a b) and write the equation as ( (ct) ) ( x (ct) + ) ζ(x, t) = 0. x To make it even simpler looking, we introduce a convenient new variables 14

15 (x +, x ) in place of the original variables (x, t) in the following way: x ± x ± ct, i.e. x = 1 2 (x+ + x ), ct = 1 2 (x+ x ), x = 1 ( + 2 x + ) (ct) x = 1 ( 2 x ) (ct) Then the wave equation can be written as x + x ζ(x+, x ) = 0 This can now be easily solved. First let us write ζ = ψ. Then the equation becomes + ψ = 0. This simply says that ψ does not depend on x + i.e. it can only be a function of x, which we call f(x ). Its functional form is arbitrary. Then the original equation becomes ζ(x +, x ) = f(x ). Let us integrate this equation with respect to x. Then we get ζ(x +, x ) = x f(t)dt + g(x + ), g(x + ) = arbitrary Note that we have added an arbitrary function of x + because such a term is killed by. Since the integral of an arbitrary function is again an arbitrary function, ζ(x +, x ) is a sum of an arbitrary function of x and an arbitrary funciton of x +. It will be convenient to rename these functions as ζ R (x ) and ζ L (x + ), where the meaning of the subscript R and L will become clear shortly. So the general solution of the one dimensional wave equation is ζ(x +, x ) = ζ R (x ) + ζ L (x + ) = ζ R (x ct) + ζ L (x + ct) (5.9) 15

16 Let us now understand the physical meaning of this solution. First, note that ζ R (x ct) takes a constant value for x ct = const.. This is equivalent to x = ct + const., meaning that such a point moves to the right with the velocity c. In other words ζ R (x ct) expresses a right-moving wave with an arbitrary fixed shape. This is dipicted as ζ R Similarly, ζ L (x + ct) expresses a left-moving wave. Basic solutions : As we have seen, the shape of the waves to be superposed is arbitrary. However, as one can imagine intuitively, the trigonometic functions are particularly useful. In fact from the theory of Fourier analysis, under most of the circumstances, they can be shown to be enough to express any waves by superposition. ζ R = a(k) sin(k(x ct)) or ζ R = a(k) cos(k(x ct)) They can be expressed together by the complex representation: ζ R = a R (k)e ik(x ct) Here k is called the wave number. Let us express various quantities which characterize a wave in terms of the wave number: 16

17 wave length λ kλ = 2π λ = 2π k period T kct = 2π T = 2π ck frequency ν ν = 1 T = ck 2π The left-moving wave can be written as ζ L = a L (k)e ik(x+ct) The general solution is a superposition of the left- and the righ-moving waves, namely ζ(x, t) = dk ( a R (k)e ik(x ct) + a L (k)e ik(x+ct)) Waves in more than two dimensions When the number of dimensions increases, the directions in which the wave propagates clearly increases. Therefore the wave number, the sign of which distinguishes the ±x-directions that it propagates in one dimension, must be generalized to a vector called the wave vector k expressing a general direction into which it propagates. The basic solution can then be taken as ζ ± k ( x, t) = a ± ( k)e i( k x k ct) Exercise ( 5.4 ) Verify that the wave above satisfies the wave equation ζ = 0. c 2 2 Physical meaning of the basic solution : We would like to see the shape of the plane wave. amplitude ζ + k takes a constant value is The place where the k ct k x = const. 17

18 To make it simpler, we can take the origin of the time coordinate so that the constant vanishes. Dividing by k we get k ˆk x = ct, ˆk k This describes a plane perpendicular to the unit vector ˆk, which moves in the ˆk direction at the speed c. So this solution is called the plane wave solution. ˆk ˆk x x t =const. plane Plane wave solution for the electromagnetic field In the case of the electromagnetic field, the complex representation of the plane wave solution is of the form E = ɛ E ( k)e i k x i k ct B = ɛ B ( k)e i k x i k ct Here, ɛ E, ɛ B are the vectors along which the amplitude varies and are called polarization vectors. In the complex form above, the polarization vectors are also complex in general. The actual electromagnetic field is obtained by taking the real part. Once one gets used to this way of writing, one appreciates the advantage over the use of sines and cosines. Characteristic properties of the plane electromagnetic wave : The plane wave solution was obtained from a particular combination of the Maxwell s equations. We must still check that the solution so obtained satisfy each of the original Maxwell s equations. This will give some conditions on the polarization vectors and the wave vector and such information dictates the properties of the wave. (1) Electromagnetic wave is a transverse wave That is, the direction of polarization is perpendicular to the direction of propagation. This is dictated by the following Maxwell s equations. E = 0 k ɛ E ( k) = 0 B = 0 k ɛ B ( k) = 0 18

19 (2) The vectors ɛ E, ɛ B, k form a right-handed orthogonal frame First consider the Faraday s equation E = B/ for the plane wave. We have LHS = E = ( ɛ E ( ) k)e i k x i k ct = i k ɛ E e i k x i k ct RHS = B = ( ɛ B ( ) k)e i k x i k ct = i k c ɛ B e i k x i k ct From this we get ɛ E k ɛe = k c ɛ B k Similarly, from B = (1/c)( E/) we obtain the relations ɛ B We have thus proved the assertion. k ɛb = 1 c k ɛ E So as we have seen, the Maxwell s equations predict that the electric and the magnetic fields propagate in the vacuum as a wave with the velocity c = 1/ µ 0 ɛ 0. Since this velocity was essentially equal to that of the light velocity, which was measured at that time with reasonable accuracy, the light was identified with the electromagnetic wave. The electromagnetic wave predicted by Maxwell was finally measured in 1888 by Herz, thereby established the correctness of the Maxwell s theory. 19