B.Thide - Electromagnetic Field Theory

We know that in vacuum the signal (field) from the charge q0 at x0 propagates to x with the speed of light c so that

Now we want to find the correspondence to the rest system solution, Equation (4.55) on the preceding page, in an arbitrary inertial system. We note from Equation (4.37) on page 59 that in the rest system

As all scalar products, u R is invariant, which means that we can evaluate it in any inertial system and it will have the same value in all other inertial systems. If we evaluate it in the rest system the result is:

We therefore see that the expression

subject to the condition R R = 0 has the proper transformation properties (proper tensor form) and reduces, in the rest system, to the solution Equation (4.55) on the preceding page. It is therefore the correct solution, valid in any inertial system.

According to Equation (4.37) on page 59 and Equation (4.60) above

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where in the last step the definition of the four-potential, Equation (4.48) on page 61, was used. Writing the solution in the ordinary 3D-way, we conclude that for a very localised charge volume, moving relative an observer with a velocity v, the scalar and vector potentials are given by the expressions

These potentials are called the Liénard-Wiechert potentials.

4.3.3 The electromagnetic field tensor

Consider a vectorial (cross) product c between two ordinary vectors a and b:

c =a b = i jkaib j xˆk =(a2b3 a3b2)xˆ1 +(a3b1 a1b3)xˆ2 +(a1b2 a2b1)xˆ3

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In other words, the pseudovector c = a b can be considered as an antisymmetric tensor of rank two.

The same is true for the curl operator r . For instance, the Maxwell equation

We know from Chapter 3 that the fields can be derived from the electromagnetic potentials in the following way:

From this, we notice the clear difference between the axial vector (pseudovector) B and the polar vector (“ordinary vector”) E.

Our goal is to express the electric and magnetic fields in a tensor form where the components are functions of the covariant form of the four-potential, Equation (4.48) on page 61:

Inspection of (4.78) and Equation (4.77) above makes it natural to define the four-tensor

This anti-symmetric (skew-symmetric), four-tensor of rank 2 is called the electromagnetic field tensor. In matrix representation, the contravariant field tensor

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The covariant field tensor is obtained from the contravariant field tensor in the usual manner by index contraction (index lowering):

It is perhaps interesting to note that the field tensor is a sort of four-dimensional curl of the four-potential vector A . The matrix representation for the covariant field tensor is

is immediately observed by explicitly setting = 0 in this covariant equation and using the matrix representation Formula (4.80) for the covariant component form of the electromagnetic field tensor F , to obtain

which is the Maxwell source equation for the electric field, Equation (1.45a) on page 15.

For = 1, Equation (4.84) yields

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and similarly for = 2;3. In summary, in three-vector form, we can write the result as

which is the Maxwell source equation for the magnetic field, Equation (1.45d) on page 15.

The two Maxwell field equations

Hence, Equation (4.83) on the facing page and Equation (4.91) above constitute Maxwell's equations in four-dimensional formalism.

Bibliography

[1]J. AHARONI, The Special Theory of Relativity, second, revised ed., Dover Publications, Inc., New York, 1985, ISBN 0-486-64870-2.

[2]A. O. BARUT, Electrodynamics and Classical Theory of Fields and Particles, Dover Publications, Inc., New York, NY, 1980, ISBN 0-486-64038-8.

[3]R. BECKER, Electromagnetic Fields and Interactions, Dover Publications, Inc., New York, NY, 1982, ISBN 0-486-64290-9.

[4]D. BOHM, The Special Theory of Relativity, Routledge, New York, NY, 1996, ISBN 0-415-14809-X.

[5]W. T. GRANDY, Introduction to Electrodynamics and Radiation, Academic Press, New York and London, 1970, ISBN 0-12-295250-2.

[6]L. D. LANDAU AND E. M. LIFSHITZ, The Classical Theory of Fields, fourth revised English ed., vol. 2 of Course of Theoretical Physics, Pergamon Press, Ltd., Oxford . . . , 1975, ISBN 0-08-025072-6.

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[7]F. E. LOW, Classical Field Theory, John Wiley & Sons, Inc., New York, NY . . . , 1997, ISBN 0-471-59551-9.

[8]C. MØLLER, The Theory of Relativity, second ed., Oxford University Press, Glasgow . . . , 1972.

[9]H. MUIRHEAD, The Special Theory of Relativity, The Macmillan Press Ltd., London, Beccles and Colchester, 1973, ISBN 333-12845-1.

[10]W. K. H. PANOFSKY AND M. PHILLIPS, Classical Electricity and Magnetism, second ed., Addison-Wesley Publishing Company, Inc., Reading, MA . . . , 1962, ISBN 0-201-05702-6.

[11]J. J. SAKURAI, Advanced Quantum Mechanics, Addison-Wesley Publishing Company, Inc., Reading, MA . . . , 1967, ISBN 0-201-06710-2.

[12]B. SPAIN, Tensor Calculus, third ed., Oliver and Boyd, Ltd., Edinburgh and London, 1965, ISBN 05-001331-9.

[13]A. N. WHITEHEAD, Concept of Nature, Cambridge University Press, Cambridge . . . , 1920, ISBN 0-521-09245-0.

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5

Electromagnetic Fields and Particles

In previous chapters, we calculated the electromagnetic fields and potentials from arbitrary, but prescribed distributions of charges and currents. In this chapter we study the general problem of interaction between electric and magnetic fields and electrically charged particles. The analysis is based on Lagrangian and Hamiltonian methods, is fully covariant, and yields results which are relativistically correct.

5.1Charged Particles in an Electromagnetic Field

We first establish a relativistically correct theory describing the motion of charged particles in prescribed electric and magnetic fields. From these equations we may then calculate the charged particle dynamics in the most general case.

5.1.1 Covariant equations of motion

We will show that for our problem we can derive the correct equations of motion by using in 4D L4 a function with similiar properties as a Lagrange function in 3D and then apply a variational principle. We will also show that we can find find a Hamiltonian-type function in 4D and solve the corresponding Hamilton-type equations to obtain the correct covariant formulation of classical electrodynamics.

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Lagrange formalism

Let us now introduce a function L(4) which fulfils the variational principle

Z 1

L(4)(x ;u ) d = 0 (5.1)

0

where d is the proper time defined via Equation (4.18) on page 55, and the endpoints are fixed. We shall show that L(4) acts as a kind of generalisation to the common 3D Lagrangian.

We require that L(4) fulfils the following conditions:

1.The Lagrange function must be invariant. This implies that L(4) must be a scalar.

2.The Lagrange function must yield linear equations of motion. This im-

plies that L(4) must not contain higher than the second power of the fourvelocity u .

According to Formula (M.97) on page 187 the ordinary 3D Lagrangian is the difference between the kinetic and potential energies. A free particle has only kinetic energy. If the particle mass is m0 then in 3D the kinetic energy is m0v2=2. This suggests that in 4D the Lagrangian for a free particle should be

For an interaction with the electromagnetic field we can introduce the interaction with the help of the four-potential given by Equation (4.78) on page 65 in the following way

We call this the four-Lagrangian and shall now show how this function, together with the variation principle, Formula (5.1) above, yields covariant results which are physically correct.

The variation principle (5.1) with the 4D Lagrangian (5.3) inserted, leads

to

= Z 1 m0u u +q A u +u @ A x d = 00

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According to Equation (4.37) on page 59, the four-velocity is

which means that we can write the variation of u as a total derivative with respect to :

Inserting this into the first two terms in the last integral in Equation (5.4) on the preceding page, we obtain

Z1

L(4)(x ;u ) d

Partial integration in the two first terms in the right hand member of (5.7) gives

Z1

L(4)(x ;u ) d

where the integrated parts do not contribute since the variations at the endpoints vanish. A change of irrelevant summation index from to in the first two terms of the right hand member of (5.8) yields, after moving the ensuing common factor x outside the partenthesis, the following expression:

Z1

L(4)(x ;u ) d

Applying well-known rules of differentiation and the expression (4.37) for the four-velocity, we can express dA =d as follows:

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By inserting this expression (5.10) into the second term in right-hand member of Equation (5.9), and noting the common factor qu of the resulting term and the last term, we obtain the final variational principle expression

Z1

L(4)(x ;u ) d

Since, according to the variational principle, this expression shall vanish and

x is arbitrary between the fixed end points 0 and 1, the expression inside

in the integrand in the right hand member of Equation (5.11) above must vanish. In other words, we have found an equation of motion for a charged particle in a prescribed electromagnetic field:

With the help of Equation (4.79) on page 65 we can express this equation in terms of the electromagnetic field tensor in the following way:

This is the sought-for covariant equation of motion for a particle in an electromagnetic field. It is often referred to as the Minkowski equation. As the reader can easily verify, the spatial part of this 4-vector equation is the covariant (relativistically correct) expression for the Newton-Lorentz force equation.

Hamiltonian formalism

The usual Hamilton equations for a 3D space are given by Equation (M.102) on page 188 in Appendix M. These six first-order partial differential equations are

where H(pi;qi;t) = piq˙i L(qi;q˙i;t) is the ordinary 3D Hamiltonian, qi is a generalised coordinate and pi is its canonically conjugate momentum.

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