8 03 at ESG Supplemental Notes Electromagnetic Fields in Fields in Matter You have likely seen the expressions for electric and magnetic fields in matter in some guise or another The purpose of these notes is to reconcile B B s notation with more common usages Start by combining the expressions in Eq 6 27 with the expressions for the fields D and H as given on the next page 414 Maxwell s equations then become D B 0 E B 0 t D J H t Note that the homogeneous equations those without charges densities or currents are unchanged Note also that taking the divergence of the equation involving H and recognizing that the divergence of a curl vanishes the continuity equation J t is recovered This is mentioned partly as a reminder that the and J that appear in the above equations are the free charge density and the conduction current So these are fine and admirable For now note that the divergence of H need not vanish and that it is possible for B to have a non zero curl even if J is absent and D is constant in time To proceed let s make two wild assumptions In fact everything else in this chapter will presume that E D and B H We have discussed briefly the situations where this is not the case Including the more general situation can be done but at the expense of introducing relative permittivity as a tensor and relative permeability as an invertible tensor Such considerations are necessary when dealing with materials such as birefringent crystals or LCDs So we forge ahead let B B M H 0 D 0 E P E We can solve these for P 0 E M 1 1 1 0 B Now introduce the notation on page 420 Eq 6 34b This leads to P 0 E E M M B 0 1 M To get Eq 6 34a 0 is eliminated in favor of and this is what accounts for the seeming assymetry in the two expressions that comprise Eq 6 34a It may seem to be a typo but it s not A few further points We still have as of February 24 1997 B 0 but M and H need not be zero Indeed this is the situation for a permanent magnet If or varies spatially not unusual we need to account for or 1 quantities that are not pleasant but shouldn t be frightening If you re gonna be a physics major or EE or something that will involve radiation from antennae you ll be using the fields D and H If you re interested I can recommend the text by Lorrain Corson a venerable copy of which I have and keep in my bookshelf and which you are welcome to borrow One of the neat things about the book is that it s well illustrated mit esg pers watko 8 03 notes mats tex 2