Chapter 14 MAXWELL S EQUATONS ntroduction Electrodynamics before Maxwell Maxwell s displacement current Maxwell s equations: General Maxwell s equations in vacuum The mathematics of waves Summary NTRODUCTON The contributions of James Clerk Maxwell to science are remarkable, they are of comparable importance to those of Newton. Maxwell summarized the laws of electricity and magnetism, developed by Coulomb, Faraday, and Ampère, and expressed them in the following concise mathematical form in terms of flux and circulation, and most importantly added a missing term that is of crucial importance. The four Maxwell Equations summarize the fundamental laws that underlie all of electricity and magnetism. ELECTRODYNAMCS BEFORE MAXWELL Flux: Gauss s Law: Coulomb s Law implies that the electric flux out of a closed surface is: Φ = 1 E S = 0 The Biot Savart Law implies that the magnetic flux out of a closed surface is: Φ = B S =0 Although these two flux equations were derived only for statics, they also are obeyed for electrodynamics. Circulation: Faraday s Law implies that the circulation of the electric field is coupled to the rate of change of magnetic flux. E l = B S Ampère s law gives the circulation of the magnetic field: B dl = 0 j S Charge conservation is an important law of nature that must be obeyed by the laws of electrodynamics. Charge conservation implies that the net current flowing out of a closed surface must equal the rate of loss of charge from the enclosed volume. j S + =0 Maxwell realized that the above four equations for electrodynamics violate charge conservation. He showed that an additional term, called the displacement current,thatdependsontherateofchangeofelectric field, missing from Ampère s Law; is needed to satisfy charge conservation. This lecture will first discuss the need for the introduction of a new displacement current term to satisfy charge conservation. This will be followed by a summary of the fundamental Maxwell Equations. MAXWELL S DSPLACEMENT CURRENT Maxwell showed that Ampère s law could be generalized and made to satisfy charge conservation, by including an additional term to the current density called the displacement current density, j, where: j 0 E Addition of this displacement current density to the real current density j gives the Maxwell-Ampère law: B l = 0 j S+0 0 E S The second integral equals 0 R j S that is the Maxwell-Ampère includes the sum of the the real current density and the displacement current. 107
Figure 1 Evaluation of Ampère s Law when the closed loop C 0 A formal proof for the need of the displacement current term is as follows. Consider that the closed loop C, in figure 1, shrinks to zero, that is 0 then the surface integral becomes a closed surface. That is: =0 B l =0=0 j S + 0 0 E S Gauss law gives: 1 E S = 0 therefore, the time derivative of Gauss s law; E S = 1 0 nserting this into the above line integral for a closed loop C gives j S + =0 that is, the charge conservation relation is obtained. nclusion of the displacement current term leads to the second integral which is essential to this proof. Charging capacitor An understanding of Maxwell s displacement current density can be obtained by considering the case of a capacitor charging with current as illustrated in figure 2. The current in the wire produces a measurable magnetic field circling around the wire carrying the current. For surface S 1, bounded by the closed loop C, there is a real current passing inwards through this surface which is related to the non-zero circulation of B around the closed loop C given by Ampère s law. However, as shown in figure 2, surface S 2 which passes between the capacitor plates, has no real current flowing through its surface, and thus Ampère s law implies that there is 108 Figure 2 Two surfaces S 1 and S 2 bounded by the same curve C. The current passes through the surface S 1 but not S 2. However, there is a changing electric flux through S 2 no magnetic field around C. However, there remains a measurable magnetic field around the wire carrying the current. Clearly something is wrong with Ampère s Law. Maxwell s displacement current corrects this flaw in Ampère s Law, and satisfies charge conservation. nclusion of the displacement current does not change the integral of current though surface S 1 since the displacement current is zero for this surface because 0 R 1 E S =0 whereas the real current is. However, for 2, the real current density is zero but the displacement current is non-zero since the electric field between the capacitor plates in increasing. t is shown below that the net current through both 1 and 2 are the same if both the real and displacement currents are summed. ntegrating the displacement current over the surface S 2 gives: j E S = 0 2 2 S Consider Gauss law for the closed surface defined by 1 + 2 1+ 2 E S = 1 0 where 1 is used to reflect the fact that the righthand rule connecting C and 1 was taken such that 1 is taken pointing into the volume. The time derivative of this gives; E 1+ 2 S = 1 0 The left-hand integral has a zero contribution for surface 1 Thus the net displacement current is given by: 0 j S+ S = 0 1 2 2 E S =
nserting this in the charge conservation relation j S + =0 and using the fact that the real current density is nonzero only for 1 and displacement current is non-zero only for 2 gives j S + j S =0 1 2 or j S = j S = 1 2 where the direction of the first integral has been reversed. That is, the total displacement current flowing through 2 equals the total real current flowing through 1 Thus the Maxwell-Ampère s law can be written: E B l = 0 j S + 0 0 1 2 S where the first integral equals 0 for surface S 1 and zero for surface S 2, while the second integral equals 0 for surface S 2 and is zero for surface S 1 That is, the circulation of B around the loop C is the same whether surface 1 or surface 2 are used. Thus Maxwell s corrected version of Ampères Law satisfies charge conservation, that is, the circulation of magnetic field is independent on whether a current is flowing or the electric flux is changing, that is whether surface S 1 or surface S 2 are used for the closed loop C. Addition of the displacement current implies that the circulation of the B field around a closed loop depends on both the current and the rate of change of electric flux through the loop. This is analogous to Faraday s Law which relates the circulation of the E field around a closed loop to the rate of change of the magnetic flux through the loop. The enormous significance of the displacement current will become more apparent after we summarize Maxwell s Equations. Maxwell s displacement current states that a changing electric flux through a closed curve induces a circulation of the magnetic field around that circuit. This is a direct analogy to Faraday s Law where a changing magnetic flux though a circuit induces a circulation of the electric field around the circuit. This analogy can be illustrated by considering the induced magnetic field between the plates of a parallel-plate capacitor with circular plates, see figure 3. Consider a concentric circular path of radius in the plane of the radius circular capacitor plates. By symmetry using Maxwell-Ampère s law, plus the fact that the current density j between the plates is zero, gives B l = 0 j S + 0 0 E S Figure 3 Parallel-plate capacitor having circular plates of radius Current of charges the plates. Calculate the magnetic field around the concentric circular path of radius. 2 = 0 0+ 0 0 2 Using Gauss s Law at the surface of the capacitor plates gives that Thesegivethat = = 0 2 = 0 2 2 = 0 2 2 This result is the same as for the magnetic field inside a cylindrical conductor of radius having a uniform current density However, this result was obtained by 2 calculating the induced magnetic field due to a changing flux of electric field. MAXWELL S EQUATONS: GENERAL Aftermuchhardworkwefinally have arrived at the basic laws of electromagnetism as first developed by Maxwell. These are derived from the experimental laws of Coulomb, Faraday and Ampère plus superposition. Maxwell expressed these laws in terms of the flux and circulation of the electric and magnetic vector fields, as well as adding the missing term, the displacement current. The most general form of the Maxwell s Equationsisgivenbelow. Maxwell Equations, General form. Flux Gauss s Law for electric field: Φ = 1 E S = 0 109
= 1 0 (Enclosed charge) Gauss s law for magnetism Φ = Circulation: Faraday s Law B S =0 F = ( E + v B) Maxwell used his four equations to predict that electromagnetic waves can occur and these will travel at a velocity in vacuum of = 1 0 He was astonished to discover that this equalled the measured ve- 0 locity of light convincing him that light was an electromagnetic wave. t was not until after Maxwell s death that Hertz demonstrated that electromagnetic waves are created via oscillating electric fields. E l = Ampère-Maxwell law: B dl = 0 B S ( E j + 0 ) S = 0 (Net real and displacement currents flowing through the loop) Remember that the total flux of the electric field out of a closed surface is independent of the size or shape of the Gaussian surface because the electric field for a point charge has a 1 dependence. Gauss s law 2 for the electric field gives the strength of the enclosed charge. The total flux for the magnetic field also is independent of the size and shape of the surface because it also has a 1 dependence. However, the net magnetic flux is always zero because the magnetic field 2 for an current element is tangential not radial. The fact that the net flux is zero is equivalent to the statement that magnetic monopoles do not exist. That is, north and south poles only occur in pairs in contrast to charge which comes in two flavors, positive or negative. The circulation of the electric field is a statement of Faraday s law of induction. Also it includes the fact that for statics, the circulation is zero implying that the electric field from a point charge is radial, not tangential. A consequence for statics is that the line integral R E l is path independent allowing the use of the concept of electric potential. The circulation of the magnetic field is a statement of the Ampère-Maxwell law. Remember to use a righthanded definition relating the line integral and the direction of the flux through the loop when using either of the circulation relations. The complete description of electromagnetism requires the four Maxwell equations plus the Lorentz force relation that defines the electric and magnetic fields. That is: MAXWELL S EQUATONS N VACUUM t helps to consider the special case of Maxwell s Equations in vacuum, that is, where current and charge densities are zero, j = =0 Then Maxwell s equations in vacuum simplify to the following. Maxwell Equations for Vacuum Flux Circulation E l = E S =0 B S =0 B dl = 0 0 B S E S Note the symmetry between these pairs of equations both for the flux and circulation. The right-hand side of the last equation comes from Maxwell s displacement current. The only non-symmetric aspects are the negative sign in Faraday s law, and the product of the constants, 0 0 This product equals the square of the velocity of light in vacuum, as will be shown next lecture. That is: 0 0 = 1 2 110
where the phase is fixed with respect to the waveform shape. tcancorrespondtomotionineitherdirection. General wave motion can be described by solutions of a wave equation. The wave equation can be written in terms of the spatial and temporal derivatives ofthewavefunctionψ() Consider the first partial derivatives of Ψ() =( ) =() Figure 4 Travelling wave, moving at velocity v in the direction, at times t=0 and t= 0 Thus the system of units used for the velocity of light is the only parameter in Maxwell s equations. Remember, that the velocity of light in vacuum is taken to be a fundamental constant. That is, the equations are independent of the system of units used in defining 0 and 0. The symmetry of the Maxwell equations and the direct inclusion of the velocity of light is related to the direct connection of the electric and magnetic fields predicted by Einstein s Theory of Relativity. Next lecture will use Maxwell s equations in vacuum to demonstrate that they lead to the prediction of electromagnetic radiation. The discussion of electromagnetic radiation requires a knowledge of wave motion which will be reviewed next. MATHEMATCS OF WAVES Consider a travelling wave in one dimension. f the wave is moving, then the wave function Ψ ( ) describing the shape of the wave, is a function of both and. The instantaneous amplitude of the wave Ψ ( ) could correspond to the transverse displacement of a wave on a string, the longitudinal amplitude of a wave on a spring, the pressure of a longitudinal soundwave,theelectricormagneticfields in an electromagnetic wave, a matter wave, etc. f the wave train maintains its shape as it moves, then one can describe thewavetrainbythefunction () where the coordinate is measured relative to the shape of the wave, that is, it is like a phase. Consider that ( =0) corresponds to the peak of the travelling pulse shown in figure 4. f the wave travels at velocity v in the direction, then the peak is at =0for =0 and is at = at time t. That is: the fixed point on the wave profile () moves in the following way = = + moving in +x direction moving in -x direction That is, any arbitrary shaped wave form travelling in either direction can be written as: Ψ() =( ) =() Ψ = Ψ = Ψ and Ψ = Ψ Factoring out Ψ = Ψ for the first derivatives gives Ψ = Ψ The sign in this equation is independent of the shape of the waveform but depends on the sign of the wave velocity making it not a generally useful formula. Consider the second derivatives and 2 Ψ 2 2 Ψ 2 Factoring out 2 Ψ 2 = 2 Ψ 2 = 2 Ψ 2 = 2 Ψ 2 =+2 2 Ψ 2 gives 2 Ψ 2 = 1 2 2 Ψ 2 This wave equation in one dimension is independent of both the shape of the waveform and the sign of the velocity. There are an infinite number of possible shapes of waves travelling in one dimension, all of these must satisfy this one-dimensional wave equation. The converse is that any function that satisfies this one dimensional wave equation must be a wave in this one dimension. One example of a solution of this one-dimensional wave equations is the sinusoidal wave Ψ() = sin([ ]) = sin( ) where = Thewavenumber = 2 where is the wavelength of the wave, and angular frequency = 2. Note that this satisfies the above wave equation where the wave velocity equals = = The Wave Equation in three dimensions is 2 Ψ 2 Ψ 2 + 2 Ψ 2 + 2 Ψ 2 = 1 2 Ψ 2 2 There are an infinite number of possible solutions Ψ to this wave equation, any one of which corresponds to a wave motion with velocity v. 111
The Wave Equation is applicable to all forms of wave motion, both transverse and longitudinal. That is, it applies to waves on a string, siesmic waves, water waves, sound waves, electromagnetic waves, matter waves, etc. n the subsequent discussion of electromagnetic waves, the Maxwell Equations will be used to deduce a wave equation. The existence of a wave equation is equivalent to proving the existence of electromagnetic waves of any wave form, frequency or wavelength travelling with the velocity given by the wave equation. SUMMARY The Maxwell equations, which are the fundamental laws of electromagnetism, have been obtained. They play a crucial role in most branches of science and engineering. The Maxwell Equations apply to all manifestions of electromagnetism and they contain Einstein s Theory of Special Relativity. For example Maxwell s equations predict that the velocity of light is independent of motion of the frame of reference. Maxwell did not realize this important facet of his equations: it took Einstein to discover this crucial fact which he presented in his seminal 1905 paper entitled On the electrodynamics of moving bodies. Reading assignment: Giancoli, Chapter 32, Review the physics of waves, Giancoli, Chapter 15. 112