Mathematical Physics Vol 1

Chapter 4. Field theory

96

it follows that the movement of the vector potential vector, along any closed contour, is equal to the magnetic flux through any surface bordered by that contour, which is valid in the general case. Gauge or gradient invariance of electromagnetic field Note that the functions of scalar and vector potential for a given electromagnetic field are not unambiguous. This is a consequence of the fact that they appear only in the form of their derivatives, and are thus determined only with the accuracy to the terms that are shortened in operations within the specified formulas. For practice, show that Maxwell’s equations do not change (they are invariant) if A and Φ are changed as follows Φ o = Φ − ∂ f ∂ t ; A o = A + grad f , where f = f ( x , y , z , t ) is a function of the variables x , y , z , t . Since Maxwell’s equations determine the values of E i B , it follows that an entire family of vector and scalar potentials that satisfy equations (4.66) and (4.68). can be defined for an electromagnetic field. The simplest physical explanation (simplified for the stationary case) is the example of the electrostatic field where the gradient invariance allows us to choose the reference level (the level at which the potential energy is equal to zero), in relation to which the potential energy and the potential are calculated. This means that in the definition of potential it is not necessary to assume that the test charge goes from infinity, but rather from some point in space, which thus becomes the reference (zero) level. Regardless of how the reference level is defined, the strength of the electrostatic field remains unchanged. φ is here practically the potential of the chosen reference level with respect to infinity. Electromagnetic potential equations Let us observe what is obtained when the scalar and vector potentials of the electromagnetic field are inserted in Maxwell’s equations for vacuum, where the field sources (current density j and charge density ρ ) appear. E = grad ( Φ + φ )= grad Φ φ = const .

∂ E ∂ t

rot B = µ o j + ε o µ o

,

∂ ∂ t −

∂ t

∂ A

rotrot A = µ o j +

grad Φ −

ε o µ o ,

− ∆ A + graddiv A = µ o j − grad

∂ 2 A ∂ t 2

∂ Φ ∂ t

ε o µ o , ∂ t ∂ Φ

+

= − µ o j + grad div A + ε o µ o

∂ 2 A ∂ t 2

∆ A − ε o µ o

.

Given the gradient invariance of the potential, A ( x , y , z , t ) and Φ ( x , y , z , t ) can be chosen so that they satisfy the expression div A + ε o µ o ∂ Φ ∂ t = 0 , and it follows that ∆ A − ε o µ o ∂ 2 A ∂ t 2 = − µ o j .