1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I Maxwell’s equations with varying charge but constant current

  1. Sep 13, 2017 #21

    phyzguy

    User Avatar
    Science Advisor

    Well, let's look at it. If we look at the curl B Maxwell's equation in integral form (Ampere's law), we can write:
    [tex] \int B \cdot dL = \mu_0 [\int J \cdot dS + \epsilon_0 \frac{\partial}{\partial t} \int E \cdot dS][/tex]
    In a circuit, we can write [itex] J = \sigma E[/itex] or [itex]E = \rho J[/itex], where rho is the resistivity. Then we can write:
    [tex] \int B \cdot dL = \mu_0 [\int J \cdot dS + \epsilon_0 \rho \frac{\partial}{\partial t} \int J \cdot dS] = \mu_0 [I + \epsilon_0 \rho \frac{\partial I}{\partial t} ][/tex]
    or, for some frequency ω:
    [tex] \int B \cdot dL = \mu_0 [I + i \epsilon_0 \rho \omega I ][/tex]

    So the question becomes what is the magnitude of the combination ερω compared to 1. I think if you put in some numbers you will find that in any normal circuit where a reasonable current is flowing that the second term is negligible. It is only when the resistivity is so high that no significant current is flowing that the second term matters. It is hard to call this case a "circuit".
     
  2. Sep 13, 2017 #22
    Thank you but also may I ask you to look at my second question in post 20. Because I am aware that curl of time dependent magnetic field should depend on current density (conductivity * electric field)and change in electric field *free space permittivity but when I applied curl of jefimenko equation for time dependent magnetic field I can not obtain the desired result (current density (conductivity * electric field)and change in electric field *free space permittivity ) could you help me about that providing mathematical demonstration? ?????????
     
  3. Sep 13, 2017 #23

    phyzguy

    User Avatar
    Science Advisor

    I don't really understand your question. Maxwell's equations always hold, so we always have:

    [tex] \nabla \times B = \mu_0 J + \frac{1}{c^2} \frac{\partial E}{\partial t}[/tex]

    whether or not B is time dependent. Given any distribution of currents and charges, you can always calculate E and B by calculating the retarded potentials, as described here. However, taking the curl of B as given there and showing that it reduces back to the right-hand side of Maxwell's equations may be a difficult exercise.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Maxwell’s equations with varying charge but constant current
  1. Maxwell's Equations (Replies: 3)

Loading...