# Thread: A Physical System :particularly hard lol

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

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how could you show that $\displaystyle \frac{\partial p}{\partial t}$ + div J = 0
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when $\displaystyle \rho = 0$ and J = 0 everywhere how can you show that:
$\displaystyle \nabla^2E - \frac{\partial^2E}{\partial t^2}$ = 0
and
$\displaystyle \nabla^2B - \frac{\partial^2B}{\partial t^2}$ = 0

2. Playing around with Maxwell's equations?

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

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