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. [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?

 

[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.