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21 January 2004 Physics 218, Spring 2004 1

Today in Physics 218: gauge transformations

More updates as we move from

quasistatics to dynamics: #3: potentials

For better use of potentials:

gauge transformations

The Coulomb and Lorentz

gauges #4: force, energy, and momentum in electrodynamics

The spectre of the Brocken. Photo by

Galen Rowell.

21 January 2004 Physics 218, Spring 2004 2

Update #3: potentials

In electrodynamics the divergence of Bis still zero, so according to the Helmholtz theorem and its corollaries (#2, in this case), we can still define a magnetic vector potential as However, the curl of Eisn't zero; in fact it hasn't been since we started magnetoquasistatics. What does this imply for the electric potential? Note that Faraday's law can be put in a suggestive form: BA 1, or 1 0.ct ct EA A E

21 January 2004 Physics 218, Spring 2004 3Thus Corollary #1 to the Helmholtz theorem allows us to

define a scalar potential for that last bracketed term: so we can still use the scalar electric potential in electrodynamics, but now both the scalar and the vector potential must be used to determine E.

Potentials (continued)

11VVct ct

w w A AEE

21 January 2004 Physics 218, Spring 2004 4

"Reference points" for potentials Our usual reference point for the scalar potential in electrostatics is at For the vector potential in magnetostatics we imposed the condition These reference points arise from exploitation of the built- in ambiguities in the static potentials: one can add any gradient to Aand any constant to V, and still get the same fields. So we decided to add whatever was necessary to make the second-order differential equations in Aand Vlook like

Poisson's equation (i.e. easy to solve).

In electrodynamics these choices no longer produce that last result: 0V .r 0. A

21 January 2004 Physics 218, Spring 2004 5

"Reference points" for potentials (continued)

For instance, Gauss's law gives us

which with still leaves us with a Poisson equation, but Ampère's law gives 2 144
1 4,V ct V ct A E A 0A 2 22
2

41 1Vcctct

w A AJ AA 22
22

114or .V

ct c ct A A AJ (P.R. #11)

21 January 2004 Physics 218, Spring 2004 6

"Reference points" for potentials (continued) This latter equation does not of course reduce to a Poisson equation with any of the reference conditions we have imposed. Thus we must look harder to use the built-in ambiguity of the potentials to make the differential equations simpler. The general way to do this, which we will cover next time, is called a gauge transformation.

21 January 2004 Physics 218, Spring 2004 7

Gauge transformations

In electro- and magentostatics, we showed that we could always choose our conventional reference points, without placing any peculiar constraints on Eor B. Now we have two, more complicated equations to simplify, and a more general approach is more fruitful. Consider performing a transformation on Aand V: add a vector to Aand a scalar to V, giving newpotential functions:

0 as Vr

0 A VV AA

21 January 2004 Physics 218, Spring 2004 8

Gauge transformations (continued)

Now, we can't add just any old thing to the potentials; we need for the fieldsarising from the new potentials to be the same as those from the old: where is a scalar function of rand t. 11 11VV ct ct ct ct E E c c c wwc ww wwc wwEE AA EE AA

0 , or

BB AA O c

21 January 2004 Physics 218, Spring 2004 9

Gauge transformations (continued)

Combine those last two results:

and integrate the second one over volume, applying the fundamental theorem of calculus: 1 1 0ct ct w c w c wquotesdbs_dbs12.pdfusesText_18

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