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Homework Help: (Electric) Scalar and vector potential

  1. Aug 24, 2012 #1
    1. The problem statement, all variables and given/known data

    In the problem, the electric scalar and vector potentials are,
    [itex]\phi=0, \vec{A}=A_0 e^{i(k_1 x-2k_2y-wt)}\vec{u_y}[/itex]

    I have to find E, B and S.

    Then, I have to calculate [itex]\phi '[/itex] that satisfies [itex]div\vec{A}+\frac{\partial \phi '}{\partial t}=0[/itex] Then calculate E and B.

    Is it possible to find [itex]\vec{A}'[/itex] and [itex]\phi'[/itex] that satisfy the previous equation and produce the same E and B as [itex]\vec{A}[/itex] and [itex]\phi[/itex]?

    2. Relevant equations
    [itex]\vec{E}=-grad\phi-\frac{\partial \vec{A}}{c\partial t}=0[/itex]

    3. The attempt at a solution

    Using the equations I find:

    [itex]\vec{E}=A_0wi/c e^{i(k_1x-2k_2y-wt)}\vec{u_y}[/itex]

    [itex]\vec{B}=A_0ik_1 e^{i(k_1x-2k_2y-wt)}\vec{u_k}[/itex]

    [itex]\vec{S}=\frac{c}{4\pi} \vec{E}x\vec{B}=1/(4\pi) A_0^2wk_1sin^2(k_1x-2k_2y-wt)\vec{u_x}[/itex]

    For the next part I find,

    [itex]\phi ' = - 2K_2 c A_0/(w) e^{i(k_1x-2k_2y-wt)}+ constant(x,y)[/itex]

    Then, I calculate E and B as before.

    I don't know how to answer the last part. Any idea?

    Thank you.
    Last edited: Aug 25, 2012
  2. jcsd
  3. Aug 24, 2012 #2


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    Your expression for the vector potential isn't a vector.

  4. Aug 24, 2012 #3
    Presumably [itex]A_0[/itex] is itself a vector, which makes that definition perfectly valid.

    Imagine taking your current definitions for [itex]A[/itex] and [itex]\phi[/itex] (I'm renaming your [itex]\phi'[/itex] to [itex]\phi[/itex] for simplicity, and so that I can reuse the symbol [itex]\phi'[/itex] below), and adding new quantities to them. So something like:
    [tex]A \rightarrow A + A'\\
    \phi \rightarrow \phi + \phi'[/tex]

    Now plug those definitions into your previous equations, and see if you can use them to come up with some constraints on the forms that [itex]A'[/itex] and [itex]\phi'[/itex] would have to take, in order to cause [itex]E[/itex] and [itex]B[/itex] to still come out the same.
    Last edited: Aug 24, 2012
  5. Aug 24, 2012 #4


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    I don't think so. A0 appears in the expressions for the electric and magnetic fields along with unit vectors. It's pretty clear that the OP meant for A0 to denote a scalar.
    Last edited: Aug 24, 2012
  6. Aug 24, 2012 #5
    Hmm, you're right. Guess I didn't look close enough at those E/B solutions.

    lailola, can you clarify what the original problem is, and how you came up with your definitions for E and B?
  7. Aug 25, 2012 #6
    I forgot to write the vector in the expression of the vector potential. I've written it.
  8. Aug 25, 2012 #7
    Ok, thank you!
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