M. Fabbrichesi , SISSA

Lagrangian for the electromagnetic field

In this last lecture of this course a general expression of the Lagrangian of the electromagnetic field is discussed for the case of specified external sources of charge and current [Jackson, sec. 12.7]. Equations of motion of the electromagnetic fields are then found by writing the Euler-Lagrange equations that correspond to the covariant form of the (inhomogeneous) Maxwell's equations. Next follows an explanation of the meaning of the time-time and time-space components of the canonical stress tensor [Jackson, eq. 12.103].The solution of the Wave equation in covariant form ∂^{α}∂_{α}A^{β}=4π/cJ^{β }is obtained through a k-space Green function approach that naturally leads to retarded and advanced solutions [Jackson, sec. 12.11]. As an application, Liénard-Wiechert potentials are then derived from the solution [Jackson, sec. 14.6]. Finally a covariant generalization of the Larmor's formula for β∼1 is given and Liénard result is discussed by showing the angular distribution of radiation emitted by an accelerated charge (synchrotron radiation) [Jackson Sec 14.2 and 14.3].