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Math Help - A Physical System :particularly hard lol

  1. #1
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    A Physical System :particularly hard lol

    a really hard one here. would appreciate if you could start me off:


    a physical system is governed by the following:

    curl E = -\frac{\partial B}{\partial t},
    div B = 0,
    curl B = J + \frac{\partial E}{\partial t},
    div E = \rho
    where t = time, and time derivatives commute with \nabla

    ............................................
    how could you show that \frac{\partial p}{\partial t} + div J = 0
    ............................................
    when \rho = 0 and J = 0 everywhere how can you show that:
    \nabla^2E - \frac{\partial^2E}{\partial t^2} = 0
    and
    \nabla^2B - \frac{\partial^2B}{\partial t^2} = 0
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  2. #2
    Super Member Rebesques's Avatar
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    Playing around with Maxwell's equations?

    Name the given relations (1)-(4). For the first question, we get from (4) that
    <br />
\frac{\partial p}{\partial t}=\frac{\partial (div {\bf E})}{\partial t}=div\frac{\partial {\bf E}}{\partial t}<br />
and substitute J and curlB from (3) to get <br />
\frac{\partial p}{\partial t}=div\frac{\partial {\bf E}}{\partial t}=div(curl {\bf B}-{\bf J})=-div{\bf J},<br />
as divcurl=0.

    For the second one, the identities div{\bf B}=0, div{\bf E}=0 mean the vector fields are solenoidal, so there exist scalar functions B,E such that {\bf B}=curlB, {\bf E}=curlE. Use now (1) and (3), remembering that  curl(curl E)=\nabla(divE)-\nabla^2 E.
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