MIT OpenCourseWare http ocw mit edu Electromagnetic Field Theory A Problem Solving Approach For any use or distribution of this textbook please cite as follows Markus Zahn Electromagnetic Field Theory A Problem Solving Approach Massachusetts Institute of Technology MIT OpenCourseWare http ocw mit edu accessed MM DD YYYY License Creative Commons Attribution NonCommercial Share Alike For more information about citing these materials or our Terms of Use visit http ocw mit edu terms chapter 9 radiation 664 Radiation In low frequency electric circuits and along transmission lines power is guided from a source to a load along highly conducting wires with the fields predominantly confined to the region around the wires At very high frequencies these wires become antennas as this power can radiate away into space without the need of any guiding structure 9 1 9 1 1 THE RETARDED POTENTIALS Nonhomogeneous Wave Equations Maxwell s equations in complete generality are 0B at VxE 1 aD VxH J 2 at V B 0 3 V D pf 4 In our development we will use the following vector identities Vx V V O 5 6 V VxA 0 2 Vx Vx A V V A V A 7 where A and V can be any functions but in particular will be the magnetic vector potential and electric scalar potential respectively Because in 3 the magnetic field has no divergence the identity in 6 allows us to again define the vector potential A as we had for quasi statics in Section 5 4 8 B VxA so that Faraday s law in 1 can be rewritten as Vx E A at 0O Ot The Retarded Potentials 665 Then 5 tells us that any curl free vector can be written as the gradient of a scalar so that 9 becomes E aA VV 10 at where we introduce the negative sign on the right hand side so that V becomes the electric potential in a static situation when A is independent of time We solve 10 for the electric field and with 8 rewrite 2 for linear dielectric media D eE B H Vx VxA p Jt 1 vaV a aA1 at c 1 1 V 11 The vector identity of 7 allows us to reduce 11 to S 1 Vi 1 a2A V V2 cA 2jC2 at2 t 12 Thus far we have only specified the curl of A in 8 The Helmholtz theorem discussed in Section 5 4 1 told us that to uniquely specify the vector potential we must also specify the divergence of A This is called setting the gauge Examining 12 we see that if we set 1 av 1A c at V A 13 the middle term on the left hand side of 12 becomes zero so that the resulting relation between A and J is the nonhomogeneous vector wave equation V2A 2 A Jrf c 14 The condition of 13 is called the Lorentz gauge Note that for static conditions V A 0 which is the value also picked in Section 5 4 2 for the magneto quasi static field With 14 we can solve for A when the current distribution J1 is given and then use 13 to solve for V The scalar potential can also be found directly by using 10 in Gauss s law of 4 as VV a VA P E at 15 The second term can be put in terms of V by using the Lorentz gauge condition of 13 to yield the scalar wave equation 1 2 V aa C at p 16 666 Radiation Note again that for static situations this relation reduces to Poisson s equation the governing equation for the quasi static electric potential 9 1 2 Solutions to the Wave Equation We see that the three scalar equations of 14 one equation for each vector component and that of 16 are in the same form If we can thus find the general solution to any one of these equations we know the general solution to all of them As we had earlier proceeded for quasi static fields we will find the solution to 16 for a point charge source Then the solution for any charge distribution is obtained using superposition by integrating the solution for a point charge over all incremental charge elements In particular consider a stationary point charge at r 0 that is an arbitrary function of time Q t By symmetry the resulting potential can only be a function of r so that 16 becomes 1 a2V I9 1 r y 0 4r r O 17 where the right hand side is zero because the charge density is zero everywhere except at r O By multiplying 17 through by r and realizing that I a a r aV r V r 18 we rewrite 17 as a homogeneous wave equation in the variable rV a I a 0 ar rV P c at rV 19 which we know from Section 7 3 2 has solutions rV f t 1 f 20 We throw out the negatively traveling wave solution as there are no sources for r 0 so that all waves emanate radially outward from the point charge at r 0 The arbitrary function f is evaluated by realizing that as r 0 there can be no propagation delay effects so that the potential should approach the quasi static Coulomb potential of a point charge lim V Q Q Q f t r o 4irer 41s 2t1 21 Radiationfrom Point Dipoles 667 The potential due to a point charge is then obtained from 20 and 21 replacing time t with the retarded time t rlc V r t Q t r c 4rer 22 The potential at time t depends not on the present value of charge but on the charge value a propagation time r c earlier when the wave now received was launched The potential due to an arbitrary volume distribution of charge pf t is obtained by replacing Q t with the differential charge element p1 t dV and integrating over the volume of charge V r t chare pf t t rqplc rc dV 23 where rQp is the distance between the charge as a source at point Q and the field point at P The vector potential in 14 is in the same direction as the current density Jf The solution for A can be directly obtained from 23 realizing that each component of A obeys the same equation as 16 if we replace pIle by l J 1 A r t 9 2 9 2 1 V 41rQp faIl current 24 RADIATION FROM POINT DIPOLES The Electric Dipole The simplest building block for a transmitting antenna is that of a uniform current flowing along a conductor of incremental length dl as shown in Figure 9 1 We assume that this current varies sinusoidally with time as i t Re fe j 1 Because the current is discontinuous at the ends charge must be deposited there being of opposite sign at each end q t Re Q e i t I dt ijoC zdq dl 2 2 This forms an electric dipole with moment p q dl …
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