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!

Magnetic field due at a point due to a wire outlining an ellipse

  1. Apr 9, 2014 #1
    1. The problem statement, all variables and given/known data
    Find the magnetic field due to a curved wire segment.

    2. Relevant equations
    Biot-Savart Law (differential form)

    dB=[itex]\frac{\mu_{o}i}{4\pi}[/itex] [itex]\frac{d\vec{S}\times \hat{r}}{r^{2}}[/itex]

    3. The attempt at a solution

    In class we found the magnetic field at a point in space (point P) caused by the current running through a wire. Point P is equidistant from every point on the wire call this distance R. dS is a differential element that points along the wire and r hat points toward point P. The angle between dS and r hat is 90 degrees at all points.

    The point P is essentially the center of a circle and the wire outlines the edge of a circle.

    B=[itex]\frac{\mu_{o}i}{4\pi}[/itex]∫[itex]\frac{d\vec{S}\times \hat{r}}{r^{2}}[/itex]


    B=[itex]\frac{\mu_{o}i}{4\pi R^{2}}[/itex]∫[itex]dS(1)sin90^{o}[/itex]

    B=[itex]\frac{\mu_{o}i S}{4\pi R^{2}}[/itex] S=R[itex]θ_{1}[/itex]
    where [itex]θ_{1}[/itex] is the angle swept out between one end of the wire and point P and the other end the wire and point P.

    B=[itex]\frac{\mu_{o}i R θ_{1} }{4\pi R^{2}}[/itex]

    B=[itex]\frac{\mu_{o}i θ_{1} }{4\pi R}[/itex]

    OK punchline.

    I was thinking how could I find the magnetic field at point P due to a piece of wire that outlines a portion of an ellipse. This would mean that the angle between dS and r hat would be a different angle at every point on the wire and the distance between point P and the wire would be different at every point along the wire.

    So the integral would be

    B=[itex]\frac{\mu_{o}i}{4\pi}[/itex]∫[itex]\frac{dS sinθ}{r^{2}}[/itex] where S,θ, and r are all variables. How would I integrate this?

    dS=drdθ where dθ the angle between dS and r hat and dr is the infinitesimal change of the radius as the integral adds from one end of the wire to the other end.

    B=[itex]\frac{\mu_{o}i}{4\pi}[/itex]∫[itex]\frac{drdθ sinθ}{r^{2}}[/itex]

    This is as far as I can get. Any help would be appreciated. Thanks.

    In retrospect maybe I shouldn't say the wire outlines an ellipse. I just want a wire the satisfies the conditions that the angle between dS and r hat changes at every point along the wire and the distance between point P and the wire changes at every point along the wire. Thanks again.
    Last edited: Apr 9, 2014
  2. jcsd
  3. Apr 9, 2014 #2
    Are you thinking the wire is a portion of the ellipse or the whole thing? I got the impression you were talking about a piece of wire that was just curved, but if it is a whole ellipse then the field will be zero t the center just like a circle.

    Were I doing it I would use the equation for an ellipse in the integral and go from there.

    An ellipse is

    [itex]r(\theta) = \frac{ab}{\sqrt{(b^2\cos^2\theta)+(a^2\sin^2(\theta)}}[/itex]

    so plugging into B-S law:

    $$B = \frac{\mu_0I}{4\pi} \int \frac{r (b^2\cos^2\theta)+(a^2\sin^2\theta)d\theta}{(ab)^2}= \frac{\mu_0I}{4\pi (ab)}\int \sqrt{(b^2\cos^2\theta)+(a^2\sin^2 \theta)} d\theta$$

    which is a very ugly looking integral.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted