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

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.

1. 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.

2. 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!