Vetor Spaes in Phsis 8/6/15 Chapter 11. Mawell's Equations in Speial Relativit. 1 In Chapter 6a we saw that the eletromagneti fields E and B an be onsidered as omponents of a spae-time four-tensor. This tensor, the Mawell field tensor F, transforms under relativisti "boosts" with the same oordinate-transformation matri used to arr out the Lorent transformation on the spae-time vetor. Sine then we have introdued vetor differential alulus, entered around the gradient operator. This operator, operating on the fields E and B and the potentials and A, an be used to epress the four Mawell equations, giving a omplete theor of the eletromagneti field. In this hapter we will start to put these equations into "ovariant form," epressed in terms equall valid in an Lorent frame. In Chapter 6a we introdued the position four-vetor, t the eletromagneti-field four-tensor, 1 1 1 E E E 1 E B B F, 1 E B B 1 E B B and the Lorent transformation matri, 1 1 whih is used in the "usual tensor wa" to transform vetor and tensor indies from one rest frame to another. Contration of indies must alwas be between a upper and a lower inde, with the metri tensor One ma note fators of 1/ whih were not present in the form of the field-strength tensor introdued in Chapter 6a. This is a pesk issue of unit sstems. The form given here gives the orret onstants in the SI sstem of units. 11-1
Vetor Spaes in Phsis 8/6/15 1 1 g g 1 1 used to raise or lower indies. Upper indies are referred to as "ontravariant" indies, and lower indies, as "ovariant" indies, referring to details of tensor analsis whih we hope to avoid disussing here. More Four-Vetors Let's just see some other ombinations of a salar and a three-vetor whih form fourvetors. E p four-momentum p, p E p p p p p four-urrent, A four-potential: A, A A A A A A 1 t 1 four-gradient:, t Note the ool salar invariants formed b ontrating ertain of these vetors with themselves. 11 -
Vetor Spaes in Phsis 8/6/15 t p p E p m 4 1 t The proper-time interval, invariant under Lorent trans. A partile's invariant mass-squared. The wave-equation operator, or the d'alembertian. Here we are interested in seeing what important relations of eletromagnetism an be epressed simpl in ovariant language. Here is an interesting ontration to form a foursalar: Conservation of harge. t Remember? Positive divergene of requires a derease in the harge densit. Now let's show where Mawell's equations ome from. Sine the divergene of the eletri field equals the harge, probabl the divergene of the field-strength tensor equals the four-vetor ombination of harge and urrent. 1 1 1 E E E 1 E B B F t 1 E B B 1 E B B E E E t E B B This gives a stak of four equations, t E B B t E B B t 11-3
Vetor Spaes in Phsis 8/6/15 1 E E B B t E B B t E B B t Or, in old Earth-bound three-vetor notation, E E B t Here we have the two most ompliated of Mawell's equations, the soure equation. And ou might notie that the famous "displaement-urrent" term, invented b Mawell to make the wave-equation work, has appeared as b magi: E displaement t. Well, that is about as muh eitement as most people an bear. But if ou are good for more - - - nobod reall likes the url. Let's set the four-url of the field-strength tensor equal to ero. This will of ourse involve the four-dimensional version of the Levi-Civita totall anti-smmetri tensor, 1, an even permutation of 13 1, an odd permutation of 13 otherwise Then F The top line gives and the net three lines give B B B 1 3 3 E E B 3 1 E B E 1 B E E B 11-4
Vetor Spaes in Phsis 8/6/15 B E t These omplete Mawell's equations. Good enough for one da. 1 http://www.ph.duke.edu/~rgb/class/ph319/ph319/node135.html 11-5