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Charge distribution in a conductor (using maxwell's equations)

  1. Mar 8, 2009 #1
    1. The problem statement, all variables and given/known data
    Show that any charge distribution in a conductor of conductivity σ and relative
    permittivity κ vanishes in time as ρ = ρ0exp(−t/ζ) where ζ = κǫ0
    σ


    2. Relevant equations
    Maxwell's equation
    ∇ · D = ρfree

    equation of continuity for a free charge density
    ∇ · Jfree = −∂(ρfree)/∂t


    ohms law
    J = σE



    3. The attempt at a solution

    I can see that ρ will get smaller and smaller as time 't' increases according to
    ρ = ρ0exp(−t/ζ) and clearly some sort of substitution is required of the equations but I can't see how substitution will result in a exponential appearing. Basically I don't know where to start so any help would be appreciated or a push in the right direction.

    Thanks
     
  2. jcsd
  3. Mar 8, 2009 #2

    gabbagabbahey

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    Try expressing J in terms of D then combine your two maxwell's equation into a single DE for [itex]\rho[/itex]
     
  4. Mar 8, 2009 #3
    I could sub the first eq into the second to get the divergence of current density equals the negative differential of the divergence of the electric displacement field, or

    ∇ · Jfree = −∂(∇ · D)/∂t

    Can Jfree and total J be considered to be the same thing in this example? Also I will lose ρ if i do this.
     
  5. Mar 8, 2009 #4

    gabbagabbahey

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    Is there ever any bound current in a conductor? If not, then Jfree and J are the same right?

    And you don't want to combine the equations in that manner....start with expressing D in terms of E.... there should be an equation for that
     
  6. Mar 8, 2009 #5
    Okay so Jfree=J. also D=εE so E=D/ε
    and therefore J=σD/ε

    edit:
    so

    ∇ · σD/ε = −∂(ρfree)/∂t
    or
    (∇σ/ε)· D + (σ/ε)(ρfree) = −∂(ρfree)/∂t

    Doesn't feel like im getting anyhere
     
    Last edited: Mar 8, 2009
  7. Mar 8, 2009 #6
    Not sure if your allowed to bump threads on this forum...o:)
     
  8. Mar 8, 2009 #7

    gabbagabbahey

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    σ/ε is a constant, so (∇σ/ε)=0 and therefor (σ/ε)(ρfree) = −∂(ρfree)/∂t

    Also, is ρfree a function of any other variables besides time, inside a conductor?....if not, then−∂(ρfree)/∂t=−d(ρfree)/dt and you have a seperable ordinary differential equation for ρfree.
     
  9. Mar 8, 2009 #8
    aha. I get it, Thanks alot thats brilliant. Don't think I would have ever got that on my own:smile:
     
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