Maxwell's Equations and Conservation Lawsphysics.gmu.edu/~joe/PHYS685/Topic6.pdf · Maxwell's...

of 25 /25
 Maxwell's Equations and Conservation Laws Reading:  Jackson 6.1 through 6.4, 6.7 Ampère's Law, since identically. Although for magnetostatics, generally Maxwell suggested:  Use Gauss's Law to rewrite continuity eqn: is called the “displacement current”. Maxwell's Eqns: 1

Transcript of Maxwell's Equations and Conservation Lawsphysics.gmu.edu/~joe/PHYS685/Topic6.pdf · Maxwell's...

Page 1: Maxwell's Equations and Conservation Lawsphysics.gmu.edu/~joe/PHYS685/Topic6.pdf · Maxwell's Equations and Conservation Laws Reading: Jackson 6.1 through 6.4, 6.7 Ampère's Law,

 

Maxwell's Equations and Conservation Laws

Reading:  Jackson 6.1 through 6.4, 6.7

Ampère's Law, since identically.

Although for magnetostatics, generally

Maxwell suggested:  Use Gauss's Law to rewrite continuity eqn:

is called the “displacement current”.

Maxwell's Eqns:

1

Page 2: Maxwell's Equations and Conservation Lawsphysics.gmu.edu/~joe/PHYS685/Topic6.pdf · Maxwell's Equations and Conservation Laws Reading: Jackson 6.1 through 6.4, 6.7 Ampère's Law,

Lorentz force law:

This set of 5 eqns, along with Newton's 2nd Law, provides a completedescription of the classical dynamics of interacting charged particlesand electromagnetic fields.

Vector and Scalar Potentials; Gauges   (Jackson sec 6.2, 6.3)

is satisfied identically if we adopt the vector pot:

Faraday's Law:

This is satisfied identically if , or,

This reduces the number of equations from 4 to 2.

Recall that

2

Page 3: Maxwell's Equations and Conservation Lawsphysics.gmu.edu/~joe/PHYS685/Topic6.pdf · Maxwell's Equations and Conservation Laws Reading: Jackson 6.1 through 6.4, 6.7 Ampère's Law,

In vacuum, the 2 inhomogeneous Maxwell eqns are

(1)

(1b;  generalization of Poisson's eqn)

(2)

3

Identity (in Jackson cover):

Page 4: Maxwell's Equations and Conservation Lawsphysics.gmu.edu/~joe/PHYS685/Topic6.pdf · Maxwell's Equations and Conservation Laws Reading: Jackson 6.1 through 6.4, 6.7 Ampère's Law,

(2b)

For to remain unchanged as well, we require

is called a “gauge transformation”.

Invariance of the fields under such transformations is called “gauge invariance” or “gauge freedom”.

4

Page 5: Maxwell's Equations and Conservation Lawsphysics.gmu.edu/~joe/PHYS685/Topic6.pdf · Maxwell's Equations and Conservation Laws Reading: Jackson 6.1 through 6.4, 6.7 Ampère's Law,

We can use gauge freedom to specify useful conditions on

Lorenz gauge: (“Lorenz condition”)

Suppose do not satisfy the Lorenz condition.  We seek a scalarfield  such that gauge­transformed potentials do:

This is the wave eqn for .  Since the wave eqn has a solution for anysource, we can always find a  that will yield potentials satisfyingthe Lorenz condition.

Applying the Lorenz condition to (1b) and (2b):

(3)

5

Page 6: Maxwell's Equations and Conservation Lawsphysics.gmu.edu/~joe/PHYS685/Topic6.pdf · Maxwell's Equations and Conservation Laws Reading: Jackson 6.1 through 6.4, 6.7 Ampère's Law,

Thus, we have 2 uncoupled wave eqns for These are mucheasier to solve than the coupled eqns.  But, we must only employ solns that satisfy the Lorenz condition!

The Lorenz condition does not uniquely specify the potentials.  If

satisfy it, then so do

if (from eq 3)

Coulomb gauge: (also called “radiation” or “transverse” gauge)

In this case, (1b)  => , identical to Poisson's eqn of electrostatics

(“instantaneous Coulomb potential”)

6

Page 7: Maxwell's Equations and Conservation Lawsphysics.gmu.edu/~joe/PHYS685/Topic6.pdf · Maxwell's Equations and Conservation Laws Reading: Jackson 6.1 through 6.4, 6.7 Ampère's Law,

We will make use of Helmholtz's Theorem:  Any vector fieldwhose divergence and curl vanish at infinity can be expressed as the sum of a “longitudinal”, or “irrotational”, term with

and a “transverse”, or “solenoidal”, term with

with

(using the continuity eqn)

Then, (4) becomes

7

SP 6.1

Page 8: Maxwell's Equations and Conservation Lawsphysics.gmu.edu/~joe/PHYS685/Topic6.pdf · Maxwell's Equations and Conservation Laws Reading: Jackson 6.1 through 6.4, 6.7 Ampère's Law,

Note:  In Coulomb gauge,  propagates instantaneously.  That is, achange in  at results in a change in  at without any delay.But, propagates at the speed of light.   and together determine

in such a way that the fields are modified according to aspeed­of­light delay.

Green Functions for the Wave Equation        (Jackson sec 6.4)

In both the Lorenz and Coulomb gauges, we find wave eqns for the potentials.  The wave eqn has the form:

For a simple demonstration that the solns are waves, consider a scalar field  in 1­D and no source term (i.e., f = 0; homogeneous eqn).

8

Page 9: Maxwell's Equations and Conservation Lawsphysics.gmu.edu/~joe/PHYS685/Topic6.pdf · Maxwell's Equations and Conservation Laws Reading: Jackson 6.1 through 6.4, 6.7 Ampère's Law,

A wave with speed c has the form(­ for waves traveling in the +x direction, + for ­x direction)

=>  the wave is indeed a soln

9

Page 10: Maxwell's Equations and Conservation Lawsphysics.gmu.edu/~joe/PHYS685/Topic6.pdf · Maxwell's Equations and Conservation Laws Reading: Jackson 6.1 through 6.4, 6.7 Ampère's Law,

For the 3­D problem with non­zero source, we'll use a Green fcn method.  Assume that the region of interest is all space  =>  no bounding surfaces.

Recall the use of the Green function in electrostatics:

Introduce Green function such that

Soln for all space:

For all space, Green's Thm yields

10

Page 11: Maxwell's Equations and Conservation Lawsphysics.gmu.edu/~joe/PHYS685/Topic6.pdf · Maxwell's Equations and Conservation Laws Reading: Jackson 6.1 through 6.4, 6.7 Ampère's Law,

Check:

Now, we have

Introduce a Green fcn satisfying

(1)

Then, (2)

Check:

The physical interpretation of this Green fcn is odd:  delta­fcn source in both space and time  =>  source that flits into existence just for an instant, then ceases to exist

11

Page 12: Maxwell's Equations and Conservation Lawsphysics.gmu.edu/~joe/PHYS685/Topic6.pdf · Maxwell's Equations and Conservation Laws Reading: Jackson 6.1 through 6.4, 6.7 Ampère's Law,

First, use a Fourier transform (Jackson 2.44 and 2.45) to eliminate the explicit time dependence:

(3)

Note the common, but potentially confusing, practice of using the samevariable, with different arguments, for the fcn and Fourier coeffs.

Also, recall that (Jackson 2.47)

The wave eqn for G becomes:

This must be true for all t'  =>

(4)

12

Page 13: Maxwell's Equations and Conservation Lawsphysics.gmu.edu/~joe/PHYS685/Topic6.pdf · Maxwell's Equations and Conservation Laws Reading: Jackson 6.1 through 6.4, 6.7 Ampère's Law,

This reduces to the eqn for the electrostatic Green fcn when  = 0.

Since we are considering all space, we may place the origin anywhere

separately.  We also have complete freedom re. orientation of axes=>  G depends only on

(5)

Adopting spherical coords, eqn (4) for G(R) becomes

(4)

Everywhere but R = 0,

13

Page 14: Maxwell's Equations and Conservation Lawsphysics.gmu.edu/~joe/PHYS685/Topic6.pdf · Maxwell's Equations and Conservation Laws Reading: Jackson 6.1 through 6.4, 6.7 Ampère's Law,

In the limit   0, this must yield the electrostatic Green fcn for all space:

Thus,

with (6)

These are outgoing (+) and incoming (­) spherical waves.  The value of Ais determined by the time boundary condition.

14

Page 15: Maxwell's Equations and Conservation Lawsphysics.gmu.edu/~joe/PHYS685/Topic6.pdf · Maxwell's Equations and Conservation Laws Reading: Jackson 6.1 through 6.4, 6.7 Ampère's Law,

From eqns (3), (5), and (6):

G(+) is the “retarded Green function” and G(­) is the “advanced Green function”.  For G(+), a disturbance occurs at and the news reaches

at a later time t, with the delay t = R/c.  The news travels at speed c.

From eqn (2), solns of the wave eqn are:

To match boundary conditions, solns of the homogeneous wave eqn are added.

15

Page 16: Maxwell's Equations and Conservation Lawsphysics.gmu.edu/~joe/PHYS685/Topic6.pdf · Maxwell's Equations and Conservation Laws Reading: Jackson 6.1 through 6.4, 6.7 Ampère's Law,

A simple, common situation is that   0 as t  ­∞, in which case the retarded soln applies.

where [...]ret

 indicates that the retarded time.

Recall the wave eqn for the potentials in the Lorenz gauge:

16

Page 17: Maxwell's Equations and Conservation Lawsphysics.gmu.edu/~joe/PHYS685/Topic6.pdf · Maxwell's Equations and Conservation Laws Reading: Jackson 6.1 through 6.4, 6.7 Ampère's Law,

In a sample problem, we'll verify that these potentials satisfy the Lorenzcondition.

Suppose the source is a moving point charge q with trajectoryvelocity

17

Page 18: Maxwell's Equations and Conservation Lawsphysics.gmu.edu/~joe/PHYS685/Topic6.pdf · Maxwell's Equations and Conservation Laws Reading: Jackson 6.1 through 6.4, 6.7 Ampère's Law,

Substitute

Note: Also,

18

Page 19: Maxwell's Equations and Conservation Lawsphysics.gmu.edu/~joe/PHYS685/Topic6.pdf · Maxwell's Equations and Conservation Laws Reading: Jackson 6.1 through 6.4, 6.7 Ampère's Law,

where we evaluate at the retarded time t', which is defined implicitly by  t' = t – R/c,  or,  R = c (t – t').

19

Page 20: Maxwell's Equations and Conservation Lawsphysics.gmu.edu/~joe/PHYS685/Topic6.pdf · Maxwell's Equations and Conservation Laws Reading: Jackson 6.1 through 6.4, 6.7 Ampère's Law,

The calculation is nearly identical for

These are known as the “Liénard­Wiechert potentials”.

Conservation of Energy and Momentum        (Jackson sec 6.7)

Power done by external fields on a charge q is

20

SP 6.2, 6.3

Page 21: Maxwell's Equations and Conservation Lawsphysics.gmu.edu/~joe/PHYS685/Topic6.pdf · Maxwell's Equations and Conservation Laws Reading: Jackson 6.1 through 6.4, 6.7 Ampère's Law,

Vector identity:

Suppose medium is linear with negligible dispersion or losses

 and  are real and frequency­independent

Define

Also, define

21

Page 22: Maxwell's Equations and Conservation Lawsphysics.gmu.edu/~joe/PHYS685/Topic6.pdf · Maxwell's Equations and Conservation Laws Reading: Jackson 6.1 through 6.4, 6.7 Ampère's Law,

(1)

RHS = negative of rate per unit volume at which fields do work on particles = rate at which field energy increases per unit volume

=>    u   = field energy density

= field energy flux density vector  (“Poynting vector”)

(1) expresses conservation of (mechanical plus electromagnetic field) energy.

To develop conservation of momentum, start with the electromagnetic force on a charge q :

If the sum of the momenta of all the particles in volume V is then

22

Page 23: Maxwell's Equations and Conservation Lawsphysics.gmu.edu/~joe/PHYS685/Topic6.pdf · Maxwell's Equations and Conservation Laws Reading: Jackson 6.1 through 6.4, 6.7 Ampère's Law,

= 0, so we're adding zero here

(2)

23

Page 24: Maxwell's Equations and Conservation Lawsphysics.gmu.edu/~joe/PHYS685/Topic6.pdf · Maxwell's Equations and Conservation Laws Reading: Jackson 6.1 through 6.4, 6.7 Ampère's Law,

Consider the term on the RHS.

If the Cartesian coords are x ( = 1, 2, 3), then the   = 1  component is

=>  ­component of RHS of (2)

with

24

Page 25: Maxwell's Equations and Conservation Lawsphysics.gmu.edu/~joe/PHYS685/Topic6.pdf · Maxwell's Equations and Conservation Laws Reading: Jackson 6.1 through 6.4, 6.7 Ampère's Law,

The integrand is the divergence of T 

:

(n = ­component of unit normal

(2) becomes

=>  electromagnetic field momentum

and = the flux of ­component of field                    momentum into volume V

=>  the field momentum density

“Maxwell stress tensor” T

 = rate  at which ­component of               momentum crosses unit area along the ­ direction.

25

SP 6.4—6 .6