EM: Magnetic induction at a point due to a line of current, or square coil.

[Seiun]

Hi everyone,

(nb: I posted this in the introductory physics section, but maybe it should be here? I'm not exactly sure how to divide physics into introductory and advanced. I hope this isn't against the rules - it's only my second post!)

I am trying to understand my EM course again, and I have tried to solve this example for a couple of hours, but I cannot get the integration right. The example is in my notes, and is a worked problem... but the diagram is not very helpful, and I cannot see why particular changes of variable were made. Can anyone enlighten me on this problem?

I will try to explain the example, but a lot of my problem in understanding the example is the lack of a clear diagram and clear definition of the variables (in my opinion).

Let me know if I haven't said enough, keeping in mind that the example is not clear enough in my notes.

Homework Statement

The problem is to find the magnetic induction, B, at a point ("field point") due to a line of current. I guess you can assume the current continues to infinity, but the problem only considers from a point "a" to a point "b" on the line.

"dl" is a line element running from "b" to "a". The current "I" runs from "b" to "a". "r" is a vector running from the line element to the "field point". I assume this is the direction, an arrow has not been drawn but this is the standard definition, I believe. "d" is the shortest (perpendicular) distance form the line of current to the field point. $$\theta_{1}$$ is the angle (smallest) between "r" at point "b" and the current line. Similarly, $$\theta_{2}$$ is the angle (smallest) between "r" at point "a" and the current line.

I know this can easily be solved by Ampere's Law, however the example first uses the Biot-Savart law and produces a result that is used later on in a few occasions in the course.

Homework Equations

Biot-Savart Law:

$$B = \frac{\mu_0I}{4\pi}\int^{b}_{a} \frac{dl \times r}{r^{3}}$$

3. Relevant Result

This is the relevant result from the above problem, and what's used later in the course.

$$B = \frac{\mu_0I}{4\pi d} ( \cos \theta_1 + \cos \theta_2 )$$


Thanks!

 

[Mindscrape]

Well, it can't be an infinite line current because the magnetic field from an infinite line current is consant, right? If you are close enough to the wire it will look start to look like the infinite current line.

Maybe you can tell us where you start to get lost in the solution you have, or in your own work, and that will let us know where to start helping.