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!

Homework Help: Vector Identity problem

  1. Jun 5, 2013 #1
    1. The problem statement, all variables and given/known data
    I want to compute the electric field knowing the magnetic field using a vector identity

    2. Relevant equations

    E=i [itex]\frac{c}{k}[/itex] (∇[itex]\times[/itex]B)

    B(r,t)=(μ0ωk/4π) ([itex]\hat{r}[/itex]×[itex]\vec{p}[/itex])[1-[itex]\frac{1}{ikr}[/itex]](eikr/r)

    [itex]\vec{p}[/itex]=dipole moment,constant vector

    we have ti use the identity [itex]\nabla[/itex][itex]\times[/itex](A[itex]\times[/itex]B)=(B[itex]\cdot[/itex]∇)A-(A[itex]\cdot[/itex]∇)B +A(∇[itex]\cdot[/itex]B) +B(∇[itex]\cdot[/itex]A)

    the identy simplifies in this situtation because for some reason we take (A[itex]\cdot[/itex]∇)B=0 and A(∇[itex]\cdot[/itex]B)=0
    So applying this we have :

    E(r,t)=ic/k(μ0ωk/4π) [itex]\nabla[/itex][eikr/r2(1-[itex]\frac{1}{ikr}[/itex]]×(r×p)+ic/k(μ0ωk/4π)[eikr/r2(1-[itex]\frac{1}{ikr}[/itex]]∇×(r×p)
    E(r,t)=i(ω/4πε0c)[ik([itex]\frac{1}{r^2}[/itex]-[itex]\frac{1}{ikr^3}[/itex])]eikr r×(r×p) + i(ω/4πε0c)[(eikr/r^2)(1-[itex]\frac{1}{ikr}[/itex])][-∇[itex]\cdot[/itex]r)p+(p[itex]\cdot[/itex]∇)r] the this part says it's equal to -∇[itex]\cdot[/itex]r)p+(p[itex]\cdot[/itex]∇)r=-3p+p=-2p so

    E(r,t)=[itex]\frac{k^2}{4πε0}[/itex](r×p)×r (ei(kr-ωt)/r) + [itex]\frac{1}{4πε0}[/itex][3r(r[itex]\cdot[/itex]p)-p]([itex]\frac{1}{r^3}[/itex]-[itex]\frac{ik}{r^2}[/itex])ei(kr-ωt)

    My problem is i dont know how the vector identy is used here..with this tools we calculate magnetic and electric fields in the approximation zones( near,far-field) when vector potential is given. Can someone give a more simple example than this of what he did in this solution?
    Last edited: Jun 5, 2013
  2. jcsd
  3. Jun 5, 2013 #2

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper

    Lets make sure I follow you first:

    You want to find ##\vec{E}## given:
    $$\vec{E} = i\frac{c}{k}\vec{\nabla}\times\vec{B}\\
    \vec{B}(\vec{r},t)=\frac{\mu_0\omega k}{4\pi}\left (\vec{r}\times\vec{p} \right )
    \left [ 1 - \frac{1}{ik\vec{r}} \right ]\frac{1}{\vec{r}}e^{ikr}\\

    \vec{\nabla}\times(\vec{A}\times\vec{B})=(\vec B\cdot\vec{\nabla})\vec A-(\vec A\cdot\vec \nabla)\vec B + \vec A(\vec \nabla \cdot \vec B)+ \vec B(\vec \nabla \cdot \vec A)

    $$... skipping a bit for now:
    ... if I got the above right, it looks to me that when you do ##\vec \nabla \times \vec B## you will end up with a term involving ##\vec \nabla \times (\vec{r}\times\vec{p})## ... which is where the identity should have come in.

    BTW: the equation editor can be tricky to use.
    It is normally better just to type the LaTeX markup in directly.
  4. Jun 5, 2013 #3
    i figured out how the identity works. In this situation i don't know , but as it seems the problems i am into, don't require all of the above but simpler cases.

    Thanks a lot for your time
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted