Biot and Savart Integral using Vectors

[eliw00d]

How would I go about setting up a Biot and Savart Integral using Vectors?

Here is an exercise we had in class:

An L-shaped current carrying conductor has its elbow on the origin with one leg of length L points in the positive x-direction and one leg of length L points in the positive y-direction. The current I is flowing toward the negative x-direction and then toward the positive y-direction. The point P is located at a distance of L/2 above the horizontal segment and L/2 to the right of the vertical segment.

Use the Biot-Savart Law to find a general formula for the magnetic field at point P in terms of μ₀, I and L.

I tried to set it up using Vectors, and figured dl to be <-L,L> and r to be <½L, -½L>. After the cross product, I am not sure how to handle the integration, since r can vary. Would it be a double integral, or am I overthinking it?

The instructor gave me the answer, and I have been trying to figure out a way to use Vectors to solve it, since I am fairly comfortable with them. Any help would be appreciated!

 

[Dr. Courtney]

Not a double integral. The sum of two line integrals.

 

[eliw00d]

Alright, so I split it up into two parts, r₁ and r₂.

Both have a magnitude of L / sqrt(2).
r₁ has a direction vector of <√2 / 2, √2 / 2, 0>,
r₂ has a direction vector of <√2 / 2, -√2 / 2, 0>.

Then, dl₁ is <-L, 0, 0> and dl₂ is <0, L, 0>.

Both cross products are L * (√2 / 2) in (-k) direction.

Do I integrate from L to 0 for the horizontal part, and 0 to L for the vertical part?

The answer he supplied is (μ0 / 4π) * (I * (√2 / L)) in (-k) direction, but I am not getting the same answer.