# Question on Biot-Savart Law for Finite Length Filamentary Conductor

• Bizkit
In summary, the Biot-Savart Law is used to calculate the magnetic field due to a current traveling along a straight conductor of finite length. The equation is: \vec{H} = \frac{I}{4\pi\rho}(cos(\alpha_2) - cos(\alpha_1))\hat{a}_\phi.
Bizkit
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.

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?

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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.

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

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.

## 1. What is the Biot-Savart Law?

The Biot-Savart Law is a fundamental law in electromagnetism that describes the magnetic field produced by a steady current in a conductor. It is named after physicists Jean-Baptiste Biot and Félix Savart, who first described the law in the early 19th century.

## 2. What is a finite length filamentary conductor?

A finite length filamentary conductor is a conductor that has a finite length and carries a steady current. It can be thought of as a wire or any other conductor with a length that is not infinitely long.

## 3. How is the Biot-Savart Law applied to a finite length filamentary conductor?

The Biot-Savart Law is applied to a finite length filamentary conductor by calculating the magnetic field at a point in space using the formula B = μ0I/4π∫dl×r/r^3, where μ0 is the permeability of free space, I is the current in the conductor, dl is a small element of the conductor, r is the distance from the element to the point, and × represents the cross product.

## 4. What is the significance of the Biot-Savart Law for finite length filamentary conductors?

The Biot-Savart Law for finite length filamentary conductors is significant because it allows us to accurately calculate the magnetic field produced by a current-carrying conductor with a finite length. This is important for understanding and predicting the behavior of magnetic fields in a variety of practical applications, such as motors, generators, and electromagnets.

## 5. Are there any limitations to the Biot-Savart Law for finite length filamentary conductors?

One limitation of the Biot-Savart Law for finite length filamentary conductors is that it assumes the current is constant and the conductor is straight. In reality, currents may vary and conductors may have curved or irregular shapes, which can affect the accuracy of the calculations. Additionally, the law only applies to steady currents and does not account for the effects of changing magnetic fields.

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