Consider a plane polarized electromagnetic wave in the vacuum described by the vector and scalar potentials,A(r, t) Ae(kr-wt)D(r, t) = Poe(kr-wt)where the orientation of the vector Ao is arbitrary.a) Using Maxwell's equations, find the relationship that must hold between Ao and pob) Using the principal of gauge invariance, show that one can transform to a new but physically equivalent vector potential which istransversely polarized, i.e. Ao k-0. Explicitly find the gauge transformation function x that does this.c) What is the scalar potential p(r, t) in the gauge of part (b)?

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Asked Sep 14, 2019
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Consider a plane polarized electromagnetic wave in the vacuum described by the vector and scalar potentials,
A(r, t) Ae(kr-wt)
D(r, t) = Poe(kr-wt)
where the orientation of the vector Ao is arbitrary.
a) Using Maxwell's equations, find the relationship that must hold between Ao and po
b) Using the principal of gauge invariance, show that one can transform to a new but physically equivalent vector potential which is
transversely polarized, i.e. Ao k-0. Explicitly find the gauge transformation function x that does this.
c) What is the scalar potential p(r, t) in the gauge of part (b)?
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Consider a plane polarized electromagnetic wave in the vacuum described by the vector and scalar potentials, A(r, t) Ae(kr-wt) D(r, t) = Poe(kr-wt) where the orientation of the vector Ao is arbitrary. a) Using Maxwell's equations, find the relationship that must hold between Ao and po b) Using the principal of gauge invariance, show that one can transform to a new but physically equivalent vector potential which is transversely polarized, i.e. Ao k-0. Explicitly find the gauge transformation function x that does this. c) What is the scalar potential p(r, t) in the gauge of part (b)?

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Expert Answer

Step 1

a) Relation Between Ao and øo:

First Calculate electric and magnetic field by the scalar and vector potential.

For calculating electric field:

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E =-Vo- Where, and A is scalar potential andvector potential respectivey and E is electric field A(r.t)= Ae r,t)=ae

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Step 2

Now,

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uesa-a) ((-5+<at} a t+ik¢e{t+a7+z*at}77+ ikáe(=»5-zi>a* Ek Vo=ikae Vo=ikae--a -a)ft(j+pa)_{r{2»9e£, +e or Vo 3ikaea) and аа а Lett at Aekr-t}

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Step 3

Put the value in Electri...

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аA E V E-3ikpe)(-wsekr-a2) E -3ikaea) + @Ae(ks-ax) Maxwell's first relati on: V.E 0 v.(-3ikaea)+ WA.eks-ax)) - 0 -6i2k2ek-a 3ikwA.elk-ax) = 0 3ikwAe = 6i?k2he-ax) DA. 3ik 6ik er-a)

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