Plane Wave Equations

1. Plane Wave Equations Alan Murray

2. Alan Murray � University of Edinburgh Maxwell's Equations � Completed! Here, yet again, are all of JCM�s equations. We�ll now take an extra interest in 3 and 4 � Ampere and Faraday, as they suggest that electric fields can create magnetic fields and vice versa.Here, yet again, are all of JCM�s equations. We�ll now take an extra interest in 3 and 4 � Ampere and Faraday, as they suggest that electric fields can create magnetic fields and vice versa.

3. Alan Murray � University of Edinburgh What does this mean? Physically, Ampere says that a current, or a changing flux (displacement current) causes a magnetic field. Faraday says that a changing magneic flux causes emf, current and an electric field. We know that electromagnetic waves exist in space (otherwise we�d all be in the dark!) � so these two effects obviously combine to produce self-sustaining fields that, once �launched� propagate more-or-less unchanged, for some distance. If we mash together Faraday�s law and Ampere�s, we may hope to get a desicription as to HOW such waves will actually behave in detail. All we are going to do is do something to both Faraday and Ampere that produces a common term that will allow us to treat them as simultaneuous equations and thus eliminate either the magnetic or electric field, thus solving for the electric or magnetic field respectively. OK, here goes � we�ll use the derivative form.Physically, Ampere says that a current, or a changing flux (displacement current) causes a magnetic field. Faraday says that a changing magneic flux causes emf, current and an electric field. We know that electromagnetic waves exist in space (otherwise we�d all be in the dark!) � so these two effects obviously combine to produce self-sustaining fields that, once �launched� propagate more-or-less unchanged, for some distance. If we mash together Faraday�s law and Ampere�s, we may hope to get a desicription as to HOW such waves will actually behave in detail. All we are going to do is do something to both Faraday and Ampere that produces a common term that will allow us to treat them as simultaneuous equations and thus eliminate either the magnetic or electric field, thus solving for the electric or magnetic field respectively. OK, here goes � we�ll use the derivative form.

4. Alan Murray � University of Edinburgh Cross-breed Ampere and Faraday! This is really just a page of maths that looks worse than it is. We have taken Ampere and Faraday and used B=mH and D=eE to write them in terms of E and H only. We have then taken the d/dt of Ampere and the curl of Faraday and found that this gives us a term of the form �xdH/dt in both cases, so we can eliminate that term to get an equation that only has time- and space- derivatives of E, and a few odds and ends of constants s, m and e, the conductivity, magnetic permeability and dielectric constant respectively. This is the plane wave equation. Note that if we had taken the curl of Ampere and then mashed him into Faraday, we would have got an identical-looking equation for H. This is really just a page of maths that looks worse than it is. We have taken Ampere and Faraday and used B=mH and D=eE to write them in terms of E and H only. We have then taken the d/dt of Ampere and the curl of Faraday and found that this gives us a term of the form �xdH/dt in both cases, so we can eliminate that term to get an equation that only has time- and space- derivatives of E, and a few odds and ends of constants s, m and e, the conductivity, magnetic permeability and dielectric constant respectively. This is the plane wave equation. Note that if we had taken the curl of Ampere and then mashed him into Faraday, we would have got an identical-looking equation for H.

5. Alan Murray � University of Edinburgh Cross-breed Ampere and Faraday! Just for completeness � here is the same calculation for H. We will refer back to the result � that H is governed by an identical differential plane wave equation to E and is therefore likely to behave in a VERY similar manner.Just for completeness � here is the same calculation for H. We will refer back to the result � that H is governed by an identical differential plane wave equation to E and is therefore likely to behave in a VERY similar manner.

6. Alan Murray � University of Edinburgh Now some simplifications � Let�s make life simple now. We know that the total electric field has a direction � it may be oscillating and therefore a double-ended arrow, but it has a direction. Let�s choose our xyz axes such that E = (0, EY, 0). That will make the maths easier immediately. We will also look for a solution that is a TRAVELLING WAVE, propagating in the x-direction, with only a y-component of electric field.Let�s make life simple now. We know that the total electric field has a direction � it may be oscillating and therefore a double-ended arrow, but it has a direction. Let�s choose our xyz axes such that E = (0, EY, 0). That will make the maths easier immediately. We will also look for a solution that is a TRAVELLING WAVE, propagating in the x-direction, with only a y-component of electric field.

7. Alan Murray � University of Edinburgh Travelling Waves These animations show that when the electric field E changes in time with a frequency w and in space (here direction x) with a spatial wavelength l=2p/b, we get a travelling wave. It looks like it is moving and is, in fact, carrying electromagnetic energy in the +x direction(or at least it will do when it acquires a matching magnetic field � see later). Note � all that is happening here is that the electric field is varying in time and space as described. The field is not �moving� from left to right � watch the blue rectangle in the animation to see this clearly.These animations show that when the electric field E changes in time with a frequency w and in space (here direction x) with a spatial wavelength l=2p/b, we get a travelling wave. It looks like it is moving and is, in fact, carrying electromagnetic energy in the +x direction(or at least it will do when it acquires a matching magnetic field � see later). Note � all that is happening here is that the electric field is varying in time and space as described. The field is not �moving� from left to right � watch the blue rectangle in the animation to see this clearly.

8. Alan Murray � University of Edinburgh Plane Wave As in many differential-equation solving exercises, the approach is to �guess� the form of the solution � and here we �guess� at a sinusoidal variation in space and time (think of waves at the seaside!), which we will, being good electrical engineers, express as exponentials and j�s! A PLANE WAVE is one where the amplitude of (in this case, so far) the electric field is everywhere the same in a plane perpendicular to the direction of travel (the yz plane here). So � E only varies with x (direction of travel) and time t.As in many differential-equation solving exercises, the approach is to �guess� the form of the solution � and here we �guess� at a sinusoidal variation in space and time (think of waves at the seaside!), which we will, being good electrical engineers, express as exponentials and j�s! A PLANE WAVE is one where the amplitude of (in this case, so far) the electric field is everywhere the same in a plane perpendicular to the direction of travel (the yz plane here). So � E only varies with x (direction of travel) and time t.

9. Alan Murray � University of Edinburgh Cross-breed Ampere and Faraday! Before that, we should explode the multiple curls and simplify with E=(0,EY,0) as decided a couiple of slides ago � as above. The whole lot comes down a simple (!) second derivative of Ey with respect to x.Before that, we should explode the multiple curls and simplify with E=(0,EY,0) as decided a couiple of slides ago � as above. The whole lot comes down a simple (!) second derivative of Ey with respect to x.

10. Alan Murray � University of Edinburgh Cross-breed Ampere and Faraday! Plane wave equation for E describes the variation in time and space of an electric plane wave With a y-component only (we have aligned the y-axis with E) propagating in the x-direction. There is an exactly equivalent equation for H Eliminate E, not H, from the combination of Ampere and Faraday. rather a waste of our time : in notes, but not lectured. We can, however, infer that whatever behaviour we get for Ey will apply to H, although we do not yet know the direction of H. Watch this space � This gets the whole nasty 3-dimensional mess down to a 1D equation in space and time (much nicer!). Let�s have a close look at it before we try to solve it.This gets the whole nasty 3-dimensional mess down to a 1D equation in space and time (much nicer!). Let�s have a close look at it before we try to solve it.

11. Alan Murray � University of Edinburgh What have we here? 3 terms � The d2E/dx2 is a little boring, although it does tell us that the electric field is going to depend upon x � the direction of travel. That�s a relief. A wave that is constant for all values of its direction of travel isn�t going anywhere � think about it! The terms on the right tell us that the E-field varies in time. The first term tells us that in a good conductor (where s>0), there�s a first-derivative dE/dt term � this term is absent in an insulator. The second term on the RHS is always there, as even a vacuum has non-zero permeability and dielectric constant (permittivity). Let�s take it in easy stages and look at insulators first, where we can drop the dE/dt term .. Then perhaps the solution for a conductor will be a variant of that for an insulator (if we�re lucky � are you feeling lucky?). 3 terms � The d2E/dx2 is a little boring, although it does tell us that the electric field is going to depend upon x � the direction of travel. That�s a relief. A wave that is constant for all values of its direction of travel isn�t going anywhere � think about it! The terms on the right tell us that the E-field varies in time. The first term tells us that in a good conductor (where s>0), there�s a first-derivative dE/dt term � this term is absent in an insulator. The second term on the RHS is always there, as even a vacuum has non-zero permeability and dielectric constant (permittivity). Let�s take it in easy stages and look at insulators first, where we can drop the dE/dt term .. Then perhaps the solution for a conductor will be a variant of that for an insulator (if we�re lucky � are you feeling lucky?).

12. Alan Murray � University of Edinburgh Start with an insulatorto make life easy (s=0) Let s=0 �.. an insulator. Now it�s nice and easy (!). Write the Ey=Ey0sin(?t-�x) as Ey=Ey0ej(?t-�x) and bash through the derivatives to find a relationship between (angular) frequency ? and �wave number� �=2p/? � in other words a relationship between frequency f and wavelength ?. So � the wavelike solution works and we can begin to infer some things about it � if we can avoid being spooked by the Greek symbols and maths, that is! Let s=0 �.. an insulator. Now it�s nice and easy (!). Write the Ey=Ey0sin(?t-�x) as Ey=Ey0ej(?t-�x) and bash through the derivatives to find a relationship between (angular) frequency ? and �wave number� �=2p/? � in other words a relationship between frequency f and wavelength ?. So � the wavelike solution works and we can begin to infer some things about it � if we can avoid being spooked by the Greek symbols and maths, that is!

13. Alan Murray � University of Edinburgh Still don�t know what it means � Travelling wave of the form OK � lets use v=f?, an expression we�ve known from the cradle, to calculate how fast this thing moves. Well, well � it moves with the speed of light. Are we surprised? We should not be. If it hits glass, for example, the dielectric constant e jumps from 8.85x10-12 to 6x8.85x10-12 and the wave slows down to a sedate 1.43x108 m/s. That�s a relief .. as we can now be sure that Snell�s Law (*) works and thus all lenses (including the ones in your eyeballs) work. If v wasn�t equal to v(1/ e�) we would all be seeing a blur. (*) nothing to do with our own Dr. Tony Snell, sadlyOK � lets use v=f?, an expression we�ve known from the cradle, to calculate how fast this thing moves. Well, well � it moves with the speed of light. Are we surprised? We should not be. If it hits glass, for example, the dielectric constant e jumps from 8.85x10-12 to 6x8.85x10-12 and the wave slows down to a sedate 1.43x108 m/s. That�s a relief .. as we can now be sure that Snell�s Law (*) works and thus all lenses (including the ones in your eyeballs) work. If v wasn�t equal to v(1/ e�) we would all be seeing a blur. (*) nothing to do with our own Dr. Tony Snell, sadly

14. Alan Murray � University of Edinburgh This is why lenses work � A nice picture of a lens and the effect of a wave slowing up through it.A nice picture of a lens and the effect of a wave slowing up through it.

15. Alan Murray � University of Edinburgh What is H up to? However, we�ve forgotten all about H � and it is certainly behaving in a similar manner. We developed an identical equation for H � so it�s also of the form H=H0ej(?t-�x) although we have to figure out the direction of H0 and its phase realtonship with E. Both are easy. If we push our E-field (0,Ey,0) through Faraday�s Law, we find that H only has a z-component, H=(0,0,Hz). So E and H (and thus D and B) are both in the yz plane and we do, indeed, have a PLANE wave. By actually equating the two sides of Faraday we find a constant that relates Ey0 to Hz0 and it is real in an insulator, so E and H are in phase. We call the constant the �characetistic impedance� of the medium in which the wave is travelling, Zi, such that E=ZiH, by analogy with Ohm�s Law, V=ZI.However, we�ve forgotten all about H � and it is certainly behaving in a similar manner. We developed an identical equation for H � so it�s also of the form H=H0ej(?t-�x) although we have to figure out the direction of H0 and its phase realtonship with E. Both are easy. If we push our E-field (0,Ey,0) through Faraday�s Law, we find that H only has a z-component, H=(0,0,Hz). So E and H (and thus D and B) are both in the yz plane and we do, indeed, have a PLANE wave. By actually equating the two sides of Faraday we find a constant that relates Ey0 to Hz0 and it is real in an insulator, so E and H are in phase. We call the constant the �characetistic impedance� of the medium in which the wave is travelling, Zi, such that E=ZiH, by analogy with Ohm�s Law, V=ZI.

16. Alan Murray � University of Edinburgh Summary so far : Insulator H and E both obey ej(wt-bx) H and E are in time-phase |E|=Zi|H|, Zi is the characteristic impedance Zi is real in an insulator Zi = 377O in free space (air!) Zi � 150O in glass Wave travels at a velocity v=vme 3x108 m/s in free space Here�s a summary of what we have found in insulators, with a few example numbers plugged in.Here�s a summary of what we have found in insulators, with a few example numbers plugged in.

17. Alan Murray � University of Edinburgh Now a conductor � Fields lead to currents Currents cause �Joule heating� (I2R) Leads to loss of energy Fields still oscillate, but they decay Multiply the solution we have already by a term e-ax? OK � now let�s put s back in and modify the solution Ey=Ey0ej(?t-�x) accordingly. What does intuition say? If s>0, currents will flow in the con ductor and heat I2R will be generated. We would expect this to cause the wave to lose energy, so let�s try including a term x e-ax that causes it to decay as it travels along the x-axis, Ey=Ey0ej(?t-�x) x e-ax .OK � now let�s put s back in and modify the solution Ey=Ey0ej(?t-�x) accordingly. What does intuition say? If s>0, currents will flow in the con ductor and heat I2R will be generated. We would expect this to cause the wave to lose energy, so let�s try including a term x e-ax that causes it to decay as it travels along the x-axis, Ey=Ey0ej(?t-�x) x e-ax .

18. Alan Murray � University of Edinburgh Now a conductor � s>0 Just plug in the new form of solution and do all the derivatives again � and we now find a relationship between ?, �, a, the constants that determine how the wave looks and moves, and e, � and s, the physical constants of the (conducting) material. It�s normal to set a+j� = ?, just to tidy things up, as a+j� appears all over the place. We now have two major parameters that describe the wave � 1) ?, which describes the temporal frequency of oscillation. 2) ?=a+j�, which describes the spatial wavelength (via �) and the spatial decay (via a). ?=a+j� is called the propagation constant.Just plug in the new form of solution and do all the derivatives again � and we now find a relationship between ?, �, a, the constants that determine how the wave looks and moves, and e, � and s, the physical constants of the (conducting) material. It�s normal to set a+j� = ?, just to tidy things up, as a+j� appears all over the place. We now have two major parameters that describe the wave � 1) ?, which describes the temporal frequency of oscillation. 2) ?=a+j�, which describes the spatial wavelength (via �) and the spatial decay (via a). ?=a+j� is called the propagation constant.

19. Alan Murray � University of Edinburgh Example : Good Conductor Here are some numbers plugged in for a typical �good conductor� � perhaps copper or aluminium. We find that the propagation constant is a complex number at 45�, so E and H are 45� out of phase .. We�ll have a closer look at this in a slide or two. The wave has slowed down dramatically and has what looks like a rapid decay with x.Here are some numbers plugged in for a typical �good conductor� � perhaps copper or aluminium. We find that the propagation constant is a complex number at 45�, so E and H are 45� out of phase .. We�ll have a closer look at this in a slide or two. The wave has slowed down dramatically and has what looks like a rapid decay with x.

20. Alan Murray � University of Edinburgh Example : Good Conductor Pretty rapid, I think you�ll agree! The animation shows what it would look like, if we could see E and H!Pretty rapid, I think you�ll agree! The animation shows what it would look like, if we could see E and H!

21. Alan Murray � University of Edinburgh Example : Good Conductor,E=ZiH �. Intrinsic Impedance As before (i.e. with the insulator) , let�s have a close look at what H is up to via Faraday. We now have an intrinsic impedance that is complex, so E =(0,Ey,0) and H =(0,0,Hz) aren�t in phase any more. In fact �As before (i.e. with the insulator) , let�s have a close look at what H is up to via Faraday. We now have an intrinsic impedance that is complex, so E =(0,Ey,0) and H =(0,0,Hz) aren�t in phase any more. In fact �

22. Alan Murray � University of Edinburgh Example : Good Conductor,E=Zi H�. Intrinsic Impedance H lags E by 45�.H lags E by 45�.

23. Alan Murray � University of Edinburgh Poynting Vector (introduction only) P=ExH is called the Poynting Vector direction of travel power Actually power/area For future notice � if we multiply (vector product) the Electric and Magnetic fields E and H, we get something called the Poynting Vector, P. It expresses the direction that a wave is travelling and also the energy density in the wavefront. This takes the form of the energy/unit area of wavefront. This will become important later in figuring out (for example) how much wave energy �hits�, for example, a satellite dish. It�s the Poynting vector, integrated over the surface of the dish. More anon �For future notice � if we multiply (vector product) the Electric and Magnetic fields E and H, we get something called the Poynting Vector, P. It expresses the direction that a wave is travelling and also the energy density in the wavefront. This takes the form of the energy/unit area of wavefront. This will become important later in figuring out (for example) how much wave energy �hits�, for example, a satellite dish. It�s the Poynting vector, integrated over the surface of the dish. More anon �

24. Alan Murray � University of Edinburgh Reflection at a Boundary We can now figure out what happens when a wave hits an �interface� between two different media � say air and glass. Some of it is transmitted and some reflected. Physics tells us that the total electric field just left of the boundary and just right of the boundary must be the same, as must the magnetic fields � so �We can now figure out what happens when a wave hits an �interface� between two different media � say air and glass. Some of it is transmitted and some reflected. Physics tells us that the total electric field just left of the boundary and just right of the boundary must be the same, as must the magnetic fields � so �

25. Alan Murray � University of Edinburgh Reflection at a Boundary We can equate them as above, to find that there is a clear relationship between the incident, reflected and transmitted waves that depends 100% on the impedances of the media. If they are impedance-matched, everything is transmitted. The bigger the difference between the impedance, the more is reflected. That�s why impedance matching matters � of which more anon from Drs Ewen and Flynn. So it�s goodbye from me � until VLSI in 4th year.We can equate them as above, to find that there is a clear relationship between the incident, reflected and transmitted waves that depends 100% on the impedances of the media. If they are impedance-matched, everything is transmitted. The bigger the difference between the impedance, the more is reflected. That�s why impedance matching matters � of which more anon from Drs Ewen and Flynn. So it�s goodbye from me � until VLSI in 4th year.