Biot-Savart Law: Magnetic Fields on an equilateral triangle

[alovesong]

Homework Statement

Two long straight wires sit at the lower corners of an equilateral triangle and carry current I. Find the magnitude and direction of the B field at the top vertex of the triangle for the case where:

a) the current in both lower wires flows out of the page
b) the current in the left wire flows out of the page, the current of the right wire flows into the page.

Homework Equations

dB=(μ_o Idlsinθ)/(4πr^2 ) for the magnetic field at a point P in space

The Attempt at a Solution

First, I don't really know what effect being physically connected to the current-bearing wires has on P. Assuming it's negligible, though, then working with just one wire

r= length of triangle side from P to current-bearing wire
R= distance directly from P to the wire?

B= (u_o*I/4π) integral[dlsinθ/r^2]

dl= R(csc^2θdθ = r^2dθ/R

B= (u_o*I/4πR) [int from θ=0 to π] sinθdθ = -(u_o*I/4πR)cosθ evaluated from 0 to π

This is based mostly around an example for another problem, but I think most of it applies... However I am confused on the integral boundaries (if it's an equilateral triangle, shouldn't θ be fixed at 60?) and am unsure how how incorporate the second wire. Help, please!

 

[Nabeshin]

I'm not sure if you have a correct picture of what's going on here in your head, because P is not connected to the current-bearing wires. Imagine an equilateral triangle. Now, the two lower VERTICES have wires passing through them perpendicular to the plane of the page with directions given in a) and b). This should help to simplify the problem into a point experiencing a force due to two magnetic fields from two wires.

 

[alovesong]

Okay, yeah, you're right - for some reason I thought that the "equilateral triangle" was physical, when it's not... but I'm still confused about what to do with the two separate wires.

 

[Nabeshin]

Homework Equations

dB=(μ_o Idlsinθ)/(4πr^2 ) for the magnetic field at a point P in space

Well you're going to need to know how to calculate the magnetic field due to the wires, which it seems you need to do via biot-savart.
The biot-savart equation, in its differential form, is actually this:
$$dB=\frac{\mu_{o}IdLx\hat{r}}{4 \pi r^{2}}$$
Where $$\hat{r}$$ is the unit vector in the direction of the point. Now, think about what the theta from the cross product represents, and you should be able to develop the general form for magnetic field a distance r away from a charge carrying wire. Hint: What is constant and what is changing?