Magnitude of magnetic field at the center of a semi-circle

[nuagerose]

Homework Statement

A very long wire carrying a current I = 5.0 A is bent into a semi-circle with a diameter of 10 cm as shown in the figure. What is the magnitude of the magnetic field at the point P, halfway between the ends of the straight sections of wire and at the center of curvature of the semi-circle?

http://ezto.mhecloud.mcgraw-hill.com/13252699451002736673.tp4?REQUEST=SHOWmedia&media=image015.png

Homework Equations

For the magnetic field due to a wire:
B=(μ_0 * I) / (2 * pi * r)

For the magnetic field due to the semi-circle:
B= (μ_0 * I) / (4 * pi)

The Attempt at a Solution

I said that the magnetic fields from both wires and from the semi-circle all point in the same direction, into the page.

So I added up the magnetic fields from each component:

(μ_0 * I) / (2 * pi * r) + (μ_0 * I) / (2 * pi * r) + (μ_0 * I) / (4 * pi)

I get an answer that is not listed in the available choices.
Where am I going wrong?

 

[mfb]

Your formula for the wire assumes a wire that extends to infinity on both sides, but your straight sections have an end in this problem.

 

[nuagerose]

I am not given the length of the wire, so how do I account for that?

 

[Enigman]

You won't need the length just the angles the ends make with the point if you find the expression correctly.

 

[mfb]

Towards the left, the wire extends to infinity. On the other side, it ends where the semi-circle begins. This is also the closest point an infinite straight line would have, so you can use symmetry.

 

[nuagerose]

Sorry, I'm still confused. So you're saying one end of the wire is infinite while the other is not?
Then would I use this equation instead?
B = (μ_0*I) * (cos θ_1 + cos θ_2) / (4*pi*a), which is the equation for a finite wire?
Then how would I find the two angles?

 

[mfb]

Sorry, I'm still confused. So you're saying one end of the wire is infinite while the other is not?

One end of the straight sections.
See attachment, the red dot is the end of the red line.

Then would I use this equation instead?
B = (μ_0*I) * (cos θ_1 + cos θ_2) / (4*pi*a), which is the equation for a finite wire?

I would not do that, but it is possible.

Then how would I find the two angles?

Simple geometry.

 

[nuagerose]

The angle from the red dot to point P is 0, but I don't know how to find the angle between P and the infinite stretch of wire, since, well, it's infinite. There must be some flaw in the way I am looking at this problem. Help?

 

[mfb]

The angle from the red dot to point P is 0, but I don't know how to find the angle between P and the infinite stretch of wire, since, well, it's infinite.

You can consider the limit, if that is easier to visualize.

 

[nuagerose]

Would both angles be zero then?

 

[mfb]

No. Did you draw a sketch? What is the angle between the two directions? Certainly not zero.

 

[Septim]

If I were in your shoes, I would possibly use the Biot-Savart law you can find the relevant equations inside your textbook just look at the index and go to the page it directs -Griffiths has a nice discussion about it as far as I remember-. You are going to write the magnetic field as the sum of the infinitesimal contributions from wire elements and integrate over the wire. You may need to separate the integral into three parts, but I am not sure. The formula for the magnetic field is:

$\vec{B} = \ \frac{\mu_0}{4 \pi} \ \int \frac{ I \ \vec{dl}\times \vec{r}}{r^3}$​

For clarification: $\vec{r}$ points from the line element on the wire to the field point (the point at which the field is being measured).