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Question on Biot-Savart Law for Finite Length Filamentary Conductor

  1. Dec 9, 2009 #1
    When finding the angles for the finite length Biot-Savart formula of a filamentary conductor H = I*(cos(α2) - cos(α1))aΦ/(4πρ), is α1 supposed to be calculated at the start of the current, and α2 at the end? I'm just wondering because my book does it this way and vice-versa, so I'm not entirely sure which way is correct.
     
  2. jcsd
  3. Dec 9, 2009 #2
    Hi Bizkit. I'm slightly confused about your question. First, when you described the conductor in question, you wrote the magnetic field (that's what H is supposed to represent, right?). The Biot-Savart Law is an equation for finding the magnetic field due to a conductor. If you already have he magnetic field, then why do you want to use the Biot-Savart Law?

    Second, which "angles" are you referring to? The Biot-Savart Law, as it's usually presented in freshman physics books, is:

    [tex]\vec{B} = \int_C \dfrac{\mu_0}{4\pi}\dfrac{Id\vec{l}\times\hat{\vec{r}}}{r^2}[/tex]

    The Biot-Savart Law contains a line integral, and so it's going to have a number of integration variables equal to the number of dimensions in which the conducting wire exists (e.g. [itex]dx[/tex], [itex]dy[/tex], [itex]dz[/tex]). If you describe the conducting wire with angular variables, like in polar coordinates, then you'll have to integrate between two angles. Are these the angles you're talking about?
     
    Last edited: Dec 9, 2009
  4. Dec 9, 2009 #3
    Sorry for confusing you, I'll try to explain it better. The book I am using is a junior electromagnetics book. It states that the Biot-Savart Law is:

    [tex]\vec{H} = \int_{L}\frac{I d\vec{l} \times \hat{a}_R}{4\pi R^{2}}[/tex]

    This equation is for straight conductors. There are two other ones for surfaces and volumes which only differ by the current used (surface or volume current).

    The equation I put before ( [tex]\vec{H} = \frac{I}{4\pi\rho}(cos(\alpha_2) - cos(\alpha_1))\hat{a}_\phi[/tex] ) is based off of this equation. It is used to calculate the magnetic field made by a current traveling along a straight conductor of finite length. [tex]I[/tex] is the current, [tex]\rho[/tex] is the perpendicular distance between the line of current and the point of interest, [tex]\alpha_1[/tex] and [tex]\alpha_2[/tex] are the angles between the line current and the lines which connect the ends of the conductor to the point of interest, and [tex]\hat{a}_\phi = \hat{a}_L \times \hat{a}_\rho[/tex], where [tex]\hat{a}_L[/tex] is the unit vector along the line current and [tex]\hat{a}_\rho[/tex] is the unit vector along the perpendicular distance. Hopefully that all makes sense. This http://www.scribd.com/doc/4705015/Ch9Sources-of-Magnetic-Fields" will take you to a document where on the bottom of the third page you will find a picture similar to what I have (the variables are different, but the setup is the same). What I want to know is whether or not [tex]\alpha_1[/tex] is the angle at the beginning of the current and [tex]\alpha_2[/tex] is the angle at the end of the current. I'm wondering because my book does it like that, and vice versa, so I'm not entirely sure which way to do it. I need to know soon because my test is later this morning.
     
    Last edited by a moderator: Apr 24, 2017
  5. Dec 9, 2009 #4
    Are you sure about the negative sign in between? Interchanging angles shouldn't change the direction of the field it should be symmetric.
     
  6. Dec 9, 2009 #5
    That's what my book shows. Perhaps the author made a mistake. As for my earlier question; I'll just ask my teacher about it right before the test, which is coming up really soon here. Thanks anyways.
     
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