Modifying Ampere's Law with Displacement Current to Understand Electromagnetic Waves, Study notes of Physics

Download Modifying Ampere's Law with Displacement Current to Understand Electromagnetic Waves and more Study notes Physics in PDF only on Docsity! Prelecture 21: Slide 2 Today we will introduce a new quantity, the displacement current, in order to make a necessary modification to Ampere’s law, one of the fundamental laws of electricity and magnetism. With this modified version of Ampere’s law, we will begin to develop an understanding of one of the most pervasive and important features of our modern world, electromagnetic waves. We’ll begin by identifying a lack of symmetry between Faraday’s law and Ampere’s law that will manifest itself in a real problem when applying Ampere’s law to a circuit that is used to charge a capacitor. We will solve this problem by introducing Maxwell’s displacement current and modifying Ampere’s law to include its ability to create magnetic fields. After a very brief review of travelling waves, we will demonstrate that Faraday’s law and the new Ampere’s law can be used to extract the wave equation for both electric and magnetic fields. We’ll close by examining some properties of these electromagnetic waves. In particular we will evaluate the velocity of these waves and the relationship between the phase and amplitudes of these electric and magnetic waves. Prelecture 21: Slide 3 We begin today with a review of the four fundamental equations of electricity and magnetism that we have developed so far. First, we have Gauss’ law which states that the integral of the electric flux through a closed surface is proportional to the charge enclosed by that surface. The corresponding equation for magnetic fields states that the integral of the magnetic flux through a closed surface is equal to zero. Since the integral of the magnetic flux through a closed surface should be equal to the magnetic charge, we say that magnetic charge does not exist. Next, we have Ampere’s law that states that the integral of B dot dl around a closed loop is proportional to the current that passes through that loop. Finally, we have Faraday’s law that states that the line integral of E dot dl around a closed loop is proportional to the time rate of change of the magnetic flux through that loop. These four equations are called Maxwell’s equations and do form the basis of all of electricity and magnetism. You may wonder why they are called Maxwell’s equations when we have not yet mentioned his name in connection with any of these laws. The reason we assign Maxwell’s name to these equations is that he was the first to point out that the equations as we have written them are inconsistent with the conservation of charge and offered a modification to one of these equations to rectify that situation. It should not be obvious to you that these equations are inconsistent with the conservation of charge, but you may notice a lack of symmetry in the way E and B fields are represented in these equations. In particular, Faraday’s law states that an electric field can be induced from a magnetic flux that changes in time, but Ampere’s law as written does not allow for the analogous situation: that magnetic fields might be induced from an electric flux that changes in time. In fact, Maxwell did propose an addition to Ampere’s law that explicitly predicts exactly how a changing electric flux can induce a magnetic field. We will develop this argument further on the next slide. Prelecture 21: Slide 6 We have now changed Ampere’s law so that a changing electric field can now create a magnetic field. When coupled with Faraday’s law in which a changing magnetic field can create an electric field, we now have the capability of self-sustaining electric and magnetic fields in empty space. In other words, we now have everything we need to explain the existence of electromagnetic waves, one of the most pervasive and important features of our current civilization. Before we discuss electromagnetic waves, though, it makes sense to do a brief review of the relevant wave formalism that was developed in the mechanics course. We start with the one-dimensional wave equation: d2h/dx2 = (1/v2) times d2h/dt2. Here h is the variable that describes the disturbance, for example, in a wave on a string, h(x,t) represents the height of the rope at a given position x at a time t. The general solution to his equation has the form: h(x,t) = h1(x-vt) + h2(x+vt), where h1 describes the waveform travelling in the positive x-direction and h2 describes the shape of a wave travelling in the negative x- direction. h1 and h2 can take any form, but the most common example we will use in this course is the harmonic plane wave. For example, the solution for a harmonic plane wave moving in the positive x-direction is given by: h(x,t) = Acos(kx – ωt). A represents the amplitude of the oscillations, the maximum value for disturbance h. The spatial form at any time t is specified by the wave number k = 2π/λ, where λ is the wavelength of the wave, the distance in x that it takes the form to repeat itself. The time dependence of h at any position x is specified by the angular frequency ω = 2π/T, where T is the period of the wave, the time it takes for the waveform to repeat itself. We also sometimes refer to the frequency of the wave f = 1/T = ω/2π. The spatial and time representations are linked by the velocity of the wave, v = λf = ω / k. The velocity of the wave is determined by the medium in which the wave is a disturbance. Armed with this knowledge of waves, we will demonstrate on the next slide that Maxwell’s equations (specifically the combination of Ampere’s law and Faraday’s law) have plane wave solutions. Prelecture 21: Slide 7 We will now demonstrate that Maxwell’s equations have plane wave solutions. We’ll start with Ampere’s law and Faraday’s law in empty space (i.e. ρ and I are zero). Here we see that in both cases we have a line integral equal to the time rate of change of flux. If we assume the plane wave is traveling in the z-direction (therefore E and B can only depend on z and t), we can obtain relations between Ex and By, for example, by choosing a Faraday loop in the x-z plane and an Amperian loop in the y-z plane. We start with a Faraday loop in the x-z plane as shown. The magnetic flux through the loop is given by By∆x∆z. The only contributions to the line integral will occur along the sides at different z, yielding the integral of E dot dl to be Ex(z2) – Ex(z1) times the length ∆x. Now the difference ∆Ex = Ex(z2) – Ex(z1) can be obtained from the partial derivative of Ex with respect to z: ∆Ex = ∑ Ex /∑ z times ∆z. Putting these equations together, we see that Faraday’s law becomes: ∑ Ex /∑ z = - ∑ By /∑ t . If we now apply Ampere’s law to a loop in the y-z plane we get a very similar expression relating ∑ By /∑ z to ∑ Ez /∑ t . Namely, that ∑ By /∑ z = - µ0ε0∑ Ex /∑ t . The only subtlety here was the extra minus sign which occurs because an increasing Ex leads to a path integral in the counterclockwise sense in the y-z plane when viewed from above. Consequently a positive dot product for a positive By occurs at z1 rather than at z2. We can eliminate reference to By in these two equations by differentiating both sides of the Ampere’s law equation with respect to time and differentiating both sides of the Faraday’s law equation with respect z. We then end up with ∑ 2Ex/∑ z 2 = µ0ε0∑ 2Ex /∑ t2. This final equation then has the form of the wave equation. We will discuss some features of these electromagnetic waves as revealed in this equation on the next slide. Prelecture 21: Slide 8 On the last slide, we combined Faraday’s law and Ampere’s law in empty space to arrive at the one- dimensional wave equation for Ex: namely that ∑ 2Ex/∑ z 2 = µ0ε0∑ 2Ex /∑ t2. We now want to use this equation to discuss some interesting properties of these electromagnetic waves. The first order of business is to calculate the velocity of these waves. We use the standard one-dimensional wave equation to identify the velocity of electromagnetic waves to be equal to 1/sqrt(µ0ε0). Now, µ0 and ε0 are constants that have been determined from electric and magnetic measurements. When we use the measured values for these constants, we determine that the velocity of electromagnetic waves is equal to 3.00 X 108 m/s. This is an amazing number; it happens to be identical to the well- measured value for the speed of light. Consequently, we identify light as an electromagnetic wave !! The velocity of the wave in the wave equation refers to its velocity with respect to the medium. What is the medium for electromagnetic waves? Maxwell named it the aether. The things that are waving in electromagnetic waves are the fields (E and B). Consequently, Maxwell viewed E and B fields as disturbances in the aether. A major experimental program was launched to determine the speed of the rest frame of the aether with respect to the Earth. All such experiments failed in the sense that they all measured the aether frame to be at rest with respect to the Earth, even though the Earth reverses its velocity every six months. This puzzle was not resolved until 1905 when Einstein proposed the special theory of relativity. In this theory, the speed of light is a constant, the same in all inertial reference frames. This property of light requires space and time to be quite different from what we have assumed in classical physics. Unfortunately, we don not have time here to develop this theory; we will simply accept that light is an electromagnetic wave that travels, in vacuum, at a constant speed, c = 3 X 108 m/s, with respect to all inertial observers. On the next slide we will discuss the relationships between the electric and magnetic fields in electromagnetic waves.