Hi I need help with higher Physics This is the question The

Hi I need help with higher Physics. This is the question:

The homogeneous Maxwell equations are contained in the Lorentz covariant equation delta^etaF^muv + delta^vF^etamu + delta^muF^veta = 0. (Notice that the indices vary cyclically between terms.) Show that the choice of indices (eta, mu, v) = (1, 2, 3) gives nabla middot B = 0, and that the three separate choices (eta, mu, v) = (0,1, 2); (eta, mu, v) = (0,1,3); (eta, mu, v) = (0,2, 3) give the three components of Faraday\'s law. What do the other combinations such as (eta, mu, v) = (0,1,1) and (eta, mu, v) = (1,3,3) yield?

Solution

14down vote

We vary the action

Ldt=(A,A)d3xdt=0Ldt=(A,A)d3xdt=0

So,

(AA+(A)(A))d3xdt=0(AA+(A)(A))d3xdt=0

(A(A))Ad3xdt=0A(A)=0(A(A))Ad3xdt=0A(A)=0

=JA+140FF=JA+140FF

A=JA=J

(A)=140((A)FF)=140((A)((AA)(AA)))=140((A)(AAAAAA+AA))(A)=140((A)FF)=140((A)((AA)(AA)))=140((A)(AAAAAA+AA))

(A)=120((A)(AAAA)).(A)=120((A)(AAAA)).

(A)(AA)=A(A)(A)+A(A)(A)=A+ggA(A)(A)=2A.(A)(AA)=A(A)(A)+A(A)(A)=A+ggA(A)(A)=2A.

We have:

(A)(AA)=2A.(A)(AA)=2A.

So,

((A))=10(AA)=10F.((A))=10(AA)=10F.

F=0J.

14down vote

We vary the action Ldt=(A,A)d3xdt=0Ldt=(A,A)d3xdt=0 (A,A)(A,A) is the density of lagrangian of the system. So, (AA+(A)(A))d3xdt=0(AA+(A)(A))d3xdt=0 By integrating by parts we obtain: (A(A))Ad3xdt=0A(A)=0(A(A))Ad3xdt=0A(A)=0 We have to determine the density of the lagrangian. One terms deals with the interaction of the charges with the electromagnetic field, JAJA. The other term is the density of energy of the electromagnetic field: this term is the difference of the magnetic field and the electric field. So we have: =JA+140FF=JA+140FF We have: A=JA=J so: (A)=140((A)FF)=140((A)((AA)(AA)))=140((A)(AAAAAA+AA))(A)=140((A)FF)=140((A)((AA)(AA)))=140((A)(AAAAAA+AA)) The third and the fourth are the same of first and the second terms. You can do kk: (A)=120((A)(AAAA)).(A)=120((A)(AAAA)). But (A)(AA)=A(A)(A)+A(A)(A)=A+ggA(A)(A)=2A.(A)(AA)=A(A)(A)+A(A)(A)=A+ggA(A)(A)=2A. We have: (A)(AA)=2A.(A)(AA)=2A. So, ((A))=10(AA)=10F.((A))=10(AA)=10F. The lagrangian equations provide the non homogeneus maxwell equations: F=0J.