How Does the Biot-Savart Law Apply to a Right Angle Wire Configuration?

[ParoXsitiC]

Homework Statement

http://i.minus.com/1333003834/Q661ScjxBUxkrL2FfmVFPQ/iTEfM3UTAVtTa.png

Homework Equations

B = ∫ ([μ0 / 4pi] * I * ds-vector x r-hat) / r^2

The Attempt at a Solution

I know the horizontal line will not add anything to the magnetic field (B), so focusing on the vertical line.

I take a little bit of length (ds) which I will call dy.

dy x r-hat = dy sin θ
r = sqrt(x^2+y^2)
sin θ = x / r

do all your substitutions and get:

B = ([μ0 / 4pi] * I * x ) ∫ dy / (x^2+y^2)^(3/2)

At this point I am confused on my limits of integration, I know for an infinite long straight wire I use -∞ to ∞.

In my notes I have an example where it goes from -y1 to y2 and comes out with

B = ([μ0 / 4pi] * I ) / x * (cos θ1 - cos θ2)

where θ1 is the angle between -y1 and the point i am finding, and θ2 is 180 - θ1.
This whole θ thing is tripping me up, how did it get there ( I am assuming trig subsitutation). Further more how can I get a grasp on what θ1 would be?

I guess -y1 in my situation is just y, and y2 is 0.

so θ1 = inverse-tan (x/y) and thus θ2 = 180 - inverse-tan (x/y)?

 

[tiny-tim]

Hi ParoXsitiC!

(try using the X² and X₂ buttons just above the Reply box )

(and write arctan or tan^-1, not inverse-tan)

(You're rambling a bit , so I won't answer point-by-point)

θ isn't a substitutuion, it's the actual angle, between the current and the line from the point to P

You can either do ∫ dy, in which case your limits are the endvalues of y, in this case 0 and ∞

or you can do ∫ dθ, in which case your limits are the endvalues of θ, in this case 0 and π/2.

(btw, can't you just say it's half the value for a whole line?)

 

[ParoXsitiC]

That makes sense but how did they get to the two cosines mathematically?

I guess my understand is that the formula is a simplified equation that does not include an integral, but I don't understand how it was derived, nor do I understand which angle is taken for θ₂

 

[tiny-tim]

That makes sense but how did they get to the two cosines mathematically?

oh, cos = adj/hyp (adjacent/hypotenuse)

= y/√(x² + y²) ​

(and θ₂ is the angle PYO, where Y = (0,y), as y -> ∞)