Question on Biot-Savart Law for Finite Length Filamentary Conductor

[Bizkit]

When finding the angles for the finite length Biot-Savart formula of a filamentary conductor H = I*(cos(α₂) - cos(α₁))a_Φ/(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.

 

[arunma]

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:

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

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. dx[/tex], dy[/tex], 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?

 

[Bizkit]

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:

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

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 ( \vec{H} = \frac{I}{4\pi\rho}(cos(\alpha_2) - cos(\alpha_1))\hat{a}_\phi ) 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. I is the current, \rho is the perpendicular distance between the line of current and the point of interest, \alpha_1 and \alpha_2 are the angles between the line current and the lines which connect the ends of the conductor to the point of interest, and \hat{a}_\phi = \hat{a}_L \times \hat{a}_\rho, where \hat{a}_L is the unit vector along the line current and \hat{a}_\rho 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 \alpha_1 is the angle at the beginning of the current and \alpha_2 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.

 

[kcdodd]

Are you sure about the negative sign in between? Interchanging angles shouldn't change the direction of the field it should be symmetric.

 

[Bizkit]

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.