Chapter 12. Electrodynamics and Relativity

Does the principle of relativity apply to the laws of electrodynamics?

12.3.1 Magnetism as a Relativistic Phenomenon

In the reference frame where q is at rest,

Because v- > v+, the Lorentz contraction of the spacing between negative charges is more severe;

the wire carries a net negative charge!

Conclusion: As a result of unequal Lorentz contraction of the positive and negative lines,a current-carrying wire that is electrically neutral in one inertial system will be charged in another.

net current:

12.3 Relativistic Electrodynamics

A point charge q traveling to the right at speed u < v

by the Einstein velocity addition rule, the velocities of the positive and negative lines are

Magnetism as a Relativistic Phenomenon

The line charge sets up an electric field:

In S system, the wire is attracted toward the charge by a purely electrical force.

The force can be transformed into in S (wire at rest) by (Eq. 12.68)

But, in the wire frame (S) the total charge is neutral ! what does the force F imply? Electrostatics and relativity imply the existence of another force in view point of S frame. magnetic force.

In fact,

In the reference frame where q is at rest,

, magnetic field, B =

One observer’s electric field is another’s magnetic field! Therefore, the relativistic force F is the Lorentz force in system S, not Minkowski!

12.3.2 How the Fields Transform

Let’s find the general transformation rules for electromagnetic fields: Given the fields in a frame ( ), what are the fields in another frame ( )?

consider the simplest possible electric field in a large parallel-plate capacitor in S0 frame.

S0 frame

S frame

In the system S, moving to the right at speed v0, the plates are moving to the left with the different surface charge :

The total charge on each plate is invariant, and the width ( w) is unchanged, but the length (I) is Lorentz-contracted by a factor

This rule pertains to components of E that are perpendicular to the direction of motion of S.

How the Fields Transform

Let’s find the general transformation rules for electromagnetic fields: Given the fields in a frame ( ), what are the fields in another frame ( )?

This case is not the most general case:we began with a system So in which the charges were at rest and where, consequently, there was no magnetic field.

To derive the general rule we must start out in a system with both electric and magnetic fields.

For parallel components, consider the capacitor lined up with the y z plane.

the plate separation (d) that is Lorentz-contracted, whereas I and w (and hence also ) are the same in both frames.

How the Fields Transform

S frame

To derive the general rule we must start out in a system with both electric and magnetic fields.

Consider the S system, there is also a magnetic field due to the surface currents:

: relative to S S

0: relative to S S

0 0: relative toS S

0SS S

By the right-hand rule, this field points in the negative z direction;

: velocity of relative to S S

0: velocity of relative to S S

0 0: velocity of relative to S S

What we need to derive the general rule is an introduction of another frame ,

then, derivation of the transformation of fields in S system into fields in system. ,E B ,E B

also, since

How the Fields Transform

Similarly, simply align the same capacitor parallel to xy plane instead of xz plane

How the Fields Transform

the field component s parallel to the motion is unchanged.

: relative to S S

How the Fields Transform

: relative to S S

Two special cases:

(1) 0 , ( 0);If B in S frame E

(2) 0 , ( 0);If E in S frame B

If either E or B is zero (at a particular point) in one system, then in any other system the fields (at that point) are very simply related.

12.3.3 The Field Tensor

The components of E and B are stirred together when you go from one inertial system to another. What sort of an object is this, which has six components and transforms according to the above relations? It's an antisymmetric, second-rank tensor.

Remember that a 4-vector transforms by the rule

Lorentz transformation matrix

A (second-rank) tensor is an object with two indices, which transform with two factors of (one for each index):

A tensor (in 4 dimensions) has 4 x 4 = 16 components, which we can display in a 4 x 4 array:

However, the 16 elements need not all be different.

The Field Tensor

10 distinct elements

6 distinct elements, and four are zero

Thus, the general antisymmetric tensor has the form

Let's see how the transformation rule works, for the six distinct components of an antisymmetric tensor.

The Field Tensor

Lorentz transformation of an antisymmetric tensor:

The complete set of transformation rules is

Now we can construct the field tensor F by direct comparison:

The Field Tensor

Dual tensor

PropertiesAntisymmetry:

Six independent components: In Cartesian coordinates, the three spatial components of (Ex, Ey, Ez) and (Bx, By, Bz).

Inner product: If one forms an inner product of the field strength tensor a Lorentz invariant is formed

meaning this number does not change from one frame of reference to another.

Pseudoscalar invariant: The product of the tensor with its dual tensor gives the Lorentz invariant:

Determinant:

F F

12.3.4 Electrodynamics in Tensor Notation

To begin with, we must determine how the sources of the fields, and J, transform.

Imagine a cloud of charge drifting by, we concentrate on an infinitesimal volume V, which contains charge Q moving at velocity u.

charge density current density

The charge density in the rest system of the charge:

Because one dimension (the one along the direction of motion) is Lorentz-contracted,

current density 4-vector.

Continuity equation in Tensor Notation

current density 4-vector.

The continuity equation in terms of

Transformation of the charge density and current density

The current density 4-vector is divergenceless.

Continuity equation

Current density 4-vector (charge and current densities)

4 Maxwell’s Equations

Maxwell’s Equations in Tensor Notation:

If = 0, Gauss's law:

If = 1, 2, and 3, Ampere's law with Maxwell’s correction:

Combine this with the corresponding results for = 2 and 3.

4 Maxwell’s Equations

If = 0,

If = 1, 2, and 3, Faraday's law:

Combine this with the corresponding results for = 2 and 3.

Maxwell’s Equations in Tensor Notation:

: Minkowski force (Lorentz force in relativistic notation)

Minkowski force in Tensor Notation

If = 1,

With a similar formula for = 2, and 3,

Lorentz force law in relativistic notation

12.3.5 Relativistic Potentials

: 4-vector potential

For = 0, = 1 (2,3):

For = 1, = 2 ( = 1, = 2) ( = 2, = 3):

The most elegant (and the simplest) formulation of Maxwell's equations(d' Alembertian)

Relativistic Potentials

: 4-vector potentialMaxwell’s Equations

The Lorentz gauge condition in relativistic notation,

In the Lorentz gauge, Maxwell’s Equations reduces to,

Introduction to Electrodynamics, David J. Griffiths

1. Vector analysis2. Electrostatics3. Special techniques4. Electric fields in mater5. Magnetostatics

6. Magnetic fields in matter7. Electrodynamics8. Conservation laws9. Electromagnetic waves

10. Potentials and fields11. Radiation12. Electrodynamics and relativity

4-vector potential

4-vector density