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Simple Derivation Maxwell Equations

  1. Feb 14, 2014 #1
    1. The problem statement, all variables and given/known data
    Derive the 2 divergence equations from the 2 curl equations and the equation of continuity.


    2. Relevant equations
    ∇°D=ρ
    ∇°B = 0
    ∇xE = -∂B/∂t
    ∇xH = J + ∂D/∂t
    ∇°J = -∂ρ/∂t (equation of continuity)


    3. The attempt at a solution
    1)∇xE = -∂B/∂t
    ∇°(∇xE) = ∇°(-∂B/∂t)
    0 =∇°(-∂B/∂t) (divergence of curl of vector field is 0)
    I'm stuck now I don't know what to do


    2)∇xH = J + ∂D/∂t
    ∇°(∇xH) = ∇°(J + ∂D/∂t)
    0 =-∂ρ/∂t + ∇°∂D/∂t
    Once again I am stuck here too


    Please guide me guys
     
  2. jcsd
  3. Feb 14, 2014 #2

    TSny

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    See this and apply it to your situation where one of the variables is time.
     
  4. Feb 14, 2014 #3
    You're on the right track, just remember that spacial derivitives (like the divergence) commute with the time derivitives.
     
  5. Feb 16, 2014 #4
    So
    0 = -∂/∂t(∇°B)
    Does the -∂/∂t and the divergence cancel? Or do they make a second order derivative?
     
    Last edited: Feb 16, 2014
  6. Feb 16, 2014 #5

    TSny

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    If the partial derivative with respect to time of a function of space and time is zero at all points of space, then what can you conclude about the function?

    Think of ##\small \nabla \cdot \bf{B}## as some function of space and time.
     
  7. Feb 16, 2014 #6
    That means that the function is constant regardless of time at all points of space
     
  8. Feb 16, 2014 #7

    TSny

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    Right, so ##\small \nabla \cdot \bf{B}## does not depend on time. It can be a function of space only.

    Strictly speaking, I think this is as far as you can go mathematically. But can you add a reasonable physical argument that will allow you to go further?
     
    Last edited: Feb 16, 2014
  9. Feb 16, 2014 #8
    oh okay
    thank you so much guys!
     
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