Bio-Savart Law, current through wire in semi circle

[arrowface]

Homework Statement

I solved this problem. I just have a general question at the very end.

A current flows in a wire that has straight sections on either side of a semicircular loop of radius b. Find the mag and direction of the magnetic field B at point P(center of loop).

......... periods = mag field pointing out
....._____...... x's = mag field pointing in
.../xxxxxxx\.... radius = "b" from point P to any part in the semi circle.
...../xxxxxxxxx\........y
-->--->--| xxxxPxxxx|--->---->--.....|
xxxxxxxxxxxxxxxxxxxxxxxxxxxxx...z(out)|___x

Homework Equations

Bio-Savart

The Attempt at a Solution

B = μ/4pi ∫(I(dl x r ))/r^2

I started out by pulling out the constants and everything I knew. Since dl is in the same direction as the current, the magnetic field from the straight pieces of the wire does not contribute to the magnetic field at point P due to the angle between dl and r:

B = μI/(4pi*b^2) ∫(dl x r )

As the current goes around the semi circle, a perfect 90 degrees is maintained between dl and r so r cancels out( sin(90)= 1 )

B = μI/(4pi*b^2) ∫ dl

The next step is finding dl which is the sum of which dl travels around the semi circle. Since it is half of a circle it would be pi multiplied by the radius which is b:

B = μI*pi*b/(4pi*b^2) >>>>>> B = -μI/4b in the negative z direction due to the magnetic field.--------------
My Question
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What if the semicircle was not a semicircle, but a square that was cut in half? How would I deal with finding dl then?

 

[antibrane]

It sounds like your confusion has to do with computing the cross product $d\mathbf{\vec{l}}\times \mathbf{\vec{r}}$. I would suggest breaking the wire up into the three separate sections, for example, the integral for the longest part of the half-square would have $d\mathbf{\vec{l}}= dx\mathbf{\hat{x}}$, since the current of that segment lies parallel to the $\hat{\mathbf{x}}$ axis. Then you just need to compute the cross product. Think about what $\mathbf{\vec{r}}$ could be; you are integrating along the current in the wire that lies at $y=b/2$ and $z=0$, from $x=\left[-b/2,b/2\right]$. You just need to compute the cross-product of the two quantities $d\mathbf{\vec{l}}\times \mathbf{\vec{r}}$ once you have this.

 

[rude man]

1. --------------
My Question
--------------

What if the semicircle was not a semicircle, but a square that was cut in half? How would I deal with finding dl then?

You would have to use the Biot-Savart law using vectors.

Set up an x-y coord. system with P at the origin. The square is bounded by (-a,0), (-a,2a), (a,2a) and (a,0). I use i and j for unit vectors.
Biot-Savart: dB = kI(dl x r)/r^3

For the left vertical part of your square (x = -a): dl = dy j and r = a i - y j. r = |r|.
Integrate from y = 0 to y = 2a.

For the horizontal stretch y = 2a, dl = dx i and r = -x i - 2a j. Integrate from x = -a to +a.
Etc. Get the idea?