Relativistic Formulation of MAXWELL EQUATIONS

To understand how electromagnetism arise from relativity we need to know the following :

1. Lorentz transformation equation.2. Charge is an invariant quantity; charge of a particle is a fixed number independent of how fast it is moving.3. Transformation rules are same no matter how the fields were produced; electric fields generated by changing magnetic fields transform the same way as those set up by stationary charges.

1. TOOLS OF RELATIVISTIC FORMULATION

1.1 Special Theory of RelativityIt is a theory, formulated essentially by Albert Einstein, which says that all motion must be defined relative to a frame of reference and that space and time are relative, rather than absolute concepts.

It is based on two postulates:1. The principle of relativity: The laws of Physics apply in all inertial reference systems.2. The universal speed of light: The speed of light is the same for all inertial observers, regardless of the motion of the source. From these two postulates Lorentz transformation equations can be derived which give us a way to transform space-time coordinates from one inertial reference frame to another.

1.2 LORENTZ TRANSFOMATION EQUATIONS

Here The inverse Lorentz transformations are:

I.

Simplification by introduction of FOUR VECTOR CONCEPT

The concept of four-vectors is used to simplify the expressions as will be shown below.We define the position-time four vector, as follows: = () or where = , In terms of , the Lorentz transformations take on a more symmetrical form.

Where

The above equations can also be written in short way as :

The coefficients may be regarded as the elements of matrix :

⋀ = Using Einstein’s summation convention (which says that repeated indices one as subscript, one as superscript) are to be summed from 0 to 3.

1.3 INVARIANTA quantity which has the same value in any inertial system is called an invariant. Here when we go from S to S’, there is a particular combination of them that remains the same: 1.3.1 This invariant can be written in the form

Where the components of is displayed as the matrix Now let us define a covariant four-vector

From this we get

Similarly we also can find out invariance of Proper Velocity and Momentum.

1.3.2 Invariant of Proper Velocity

Proper velocity is defined by the distance travelled (measured in lab frame or S frame) divided by the proper time.

Here Therefore In fact proper velocity is a part of four-vector:

Thus we get

Since speed of light (c) is same in all inertial frames of reference i.e. speed of light is invariant therefore “ is also an invariant quantity”.

1.3.3 Invariant of Momentum

In relativity momentum is defined as mass times proper velocity.

Or Here and relativistic energy is defined as . Thus we get

and thus energy-momentum four vector or four momentum is

And which is an invariant quantity.

2. RELATIVISTIC FORMULATION OF MAXWELL EQUATION

2.1 Transformation of Fields

The Lorentz transformation equation when applied to electric field gives shows magnetism as a relativistic phenomenon. It shows that “one observer’s electric field is another’s magnetic field.” Now, let us see the general transformation rules for electromagnetic fields which can be derived by Lorentz transformation equations

Let us take two capacitors and place it in the way as shown:

It is a system named where charges are at rest and there is no magnetic field.

Taking those plates to system , which moves to the right with speed we know from , the plates appear to move towards left with same speed.

This frame has both electric and magnetic field.

We again take a system which travels to the right with speed relative to and relative to .

Let be the electric and magnetic fields as observed from and let be the electric and magnetic fields as observed from . The set of transformation rules are :

Combination through FIELD TENSOR

Electric and magnetic fields are combined into a single entity through the Field Tensor. It is written as

where is a second rank anti symmetric tensor. A second rank tensor is an object with two indices, which transform with two factors of (one for each ⋀index),. Thus the Field Tensor transforms according to

and we get the required transformed fields.

DUAL TENSOR

By the substitution we obtain another tensor to relate the transformation equations. This tensor is called dual tensor

Field Sources in Relativistic FormulationTo know about relativistic formulation of Maxwell equations, knowing about the transformation of the sources of the fields, and J, is must. Charge density and current density go together to make a four-vector

whose components are .Now the Continuity Equation is

In terms of ,

and applying these two equations, we get ,

This is the continuity equation in relativistic formulation

Maxwell’s Equations in Relativistic Formulation

All the four Maxwell equations can be written in compact form in the following two equations :

Gauss’s Law

which is Gauss’s law.•

Ampere’s Law with Maxwell’s correction.

Combining this with the corresponding results for gives which is Ampere’s law with Maxwell’s correction.

Using and putting and expanding as above, we get

Faraday’s Law

Using and putting and expanding, we get .

References

• Concept of Modern Physics by Arthur Beiser• Concepts of Physics by H.C Verma• Introduction to Particle Physics by D.J Griffiths• Electrodynamics by D.J Griffiths

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