1. Not finding help here? Sign up for a free 30min 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!

Finding the constants in an Electric Field Equation

  1. Sep 27, 2013 #1
    1. The problem statement, all variables and given/known data
    The problem calls for finding the relationship between the constants A and B in the following equation of an instantaneous electric field:

    E=[A(x+y)[itex]\widehat{a}_{x}[/itex]+B(x-y)[itex]\widehat{a}_{y}[/itex]]cos(wt)

    The details of the medium in which the field exists are that it is homogenous, isotropic, linear, and is source-free.

    20130927_151144_zps7427c05e.jpg

    2. Relevant equations



    3. The attempt at a solution

    I really have no idea how to go about solving this problem. Initially I was thinking I was just playing around with the vector equation by seeing what it looks like when I plug in different values of x and y and when t=0 or 1. Here is what that looked like:

    20130927_154159_zps9ef8919b.jpg

    Can someone please help point me in the right direction?
     
  2. jcsd
  3. Sep 27, 2013 #2

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Have you studied the properties of the "divergence" of the electric field?
     
  4. Sep 28, 2013 #3

    vanhees71

    User Avatar
    Science Advisor
    2016 Award

    The only question is, what an "instantaneous" electric field might be. I think it's just an electric field, right?

    Then just use the Maxwell and constitutive equations for the given situation to constrain the functions [itex]A[/itex] and [itex]B[/itex] as much as you can.
     
  5. Sep 28, 2013 #4

    rude man

    User Avatar
    Homework Helper
    Gold Member

    And how about the curl, since the field is apparently time-invariant.
     
  6. Sep 28, 2013 #5
    Using Maxwell's equation

    [itex]\nabla \cdot E[/itex]=[itex]\frac{q_{ev}}{ε_{0}}[/itex]

    where [itex]q_{ev}[/itex] is the electric charge density and noting that [itex]q_{ev}[/itex]=0 for a source free medium, I get:

    [itex]\nabla \cdot E[/itex]={[itex]\frac{\partial}{\partial_{x}}[/itex][itex]\widehat{a}_{x}[/itex][itex]\cdot[/itex][[itex]\widehat{a}_{x}[/itex]A(x+y)]+[itex]\frac{\partial}{\partial_{y}}[/itex][itex]\widehat{a}_{y}[/itex][itex]\cdot[/itex][[itex]\widehat{a}_{y}[/itex]B(x-y)]}cos(ωt)=0

    ={[itex]\frac{\partial}{\partial_{x}}[/itex]Ax+[itex]\frac{\partial}{\partial_{x}}[/itex]Ay+[itex]\frac{\partial}{\partial_{y}}[/itex]Bx-[itex]\frac{\partial}{\partial_{y}}[/itex]By}cos(ωt)=0

    (A-B)cos(ωt)=0

    Which is true when ωt=[itex]\frac{\pi}{2}[/itex], or when A=B.

    I am not really sure how to interpret this result.
     
    Last edited: Sep 28, 2013
  7. Sep 28, 2013 #6
    Sorry I had some mistakes in there, I was making a few changes!
     
  8. Sep 28, 2013 #7

    TSny

    User Avatar
    Homework Helper
    Gold Member

    You might want to start with what you know about ##\vec{\nabla} \cdot \vec{D}## and then, as vanhees71 suggested, use the consitutive relations to relate ##\vec{D}## to ##\vec{E}##.

    Using all four of the assumptions: (1) source free (2) homogeneous (3) isotropic (4) linear medium, you can deduce what ##\vec{\nabla} \cdot \vec{E}## must equal.
     
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: Finding the constants in an Electric Field Equation
  1. Finding electric field (Replies: 2)

Loading...