Biot Savart, Intersecting Currents, and Calculating Magnetic Fields

[cwatki14]

Two infinitely long straight wires lie in the same plane and carry the currents depicted in Figure P.48.

(a) Find the magnetic field at the point P located 24 cm from the intersection of the wires along the bisector of the acute angle between them.
(b) Find the magnetic field at the point S, located 20 cm from the intersection of the wires along the bisector of the obtuse angle between the wires.

Here's my solution which is wrong...:
At point P a magnetic field is created by the wire with the 30 A current and the wire with the 10 A current. You should be able to arithmetically add these fields since they both lie in the same plane, the z plane.
The Bio-Savart Law states:
B(vector)=($$\mu$$/4$$\pi$$)(I)($$\int$$dlxr/r^2)
At point P: The magnetic field is equal to the sum of the field caused by the 30 A wire and the 10 A wire.
http://photos-g.ak.fbcdn.net/hphotos-ak-snc1/hs259.snc1/10621_1139688773814_1275240494_30745014_2380313_n.jpg
This image represents the triangle created by the point and the wires. I used this to calculate the l and r vectors.
The field of the 30 A on P is is:
($$\mu$$/4$$\pi$$)(30A)((.2078)/(.12)^2)
=4.3301e-5 T in the positive z direction
the cross product of l x r indicates that B has a positive value.
The field of 10 A on P is similarly calculated, but from the image of the wires, l x r should give a B field in the negative z direction. I am not going to plug all the numbers in because they are pretty much the same, just the values for I change and the direction of the B vector is now negative.
Thus the total B field on point A is the magnetic field due to 30A - magnetic field due to 10A.
I calculated for the point S in the same manner, except here I think the two B vectors are in the positive z direction so they add. So the total magnetic field at S=field caused by 10A + field caused by 30A.

I don't know if I am using the wrong equation, or my right hand rule is wrong. Please help, it's killing me!

 

[Gear300]

I'm not exactly sure how this system would work since one wire can exert a magnetic force on the other. But ignoring that, you could just take the net magnetic field at P by analyzing the magnetic field induced by each individual wire. For one wire, you'll have to integrate from -INF to +INF. You also have to take into account that the magnetic field is a vector (so you'll have to a take into account the vector sum when integrating).

 

[cwatki14]

I'm not exactly sure how this system would work since one wire can exert a magnetic force on the other. But ignoring that, you could just take the net magnetic field at P by analyzing the magnetic field induced by each individual wire. For one wire, you'll have to integrate from -INF to +INF. You also have to take into account that the magnetic field is a vector (so you'll have to a take into account the vector sum when integrating).

I thought dl was just the length of the wire that creates the the intersection of point p, which I drew out in the triangle picture. The opposite of the triangles represent the length r, and the adjacent components represent the magnitude of the l vector.

 

[Gear300]

The dl x r in the Biot-Savart Law is a cross product between the displacement vector of the current (dl) and the radial distance from a particular dl (r). Since the wires are infinitely long, the currents are traveling an infinite path (so you'll have to integrate dl from -INF to +INF for each wire). At the same time, you'll have to take into account that the angle between dl and r changes over this interval. You'll also have to take into account that the magnetic field is a vector (so when you're integrating from -INF to +INF some of the components of the magnetic field will cancel out). You should end up getting a general equation for the magnetic field at a point P from a wire -- you can apply that to both wires.

 

[ehild]

To get B at a distance R from a long, current carrying wire, you have to integrate Biot-Savart's Law for the whole length of the wire, and the integration is a bit tricky. But you certainly have learned about Ampere's law, use it instead.

ehild

 

[Carl_Weggel]

You are making the problem unnecessarily complicated. Use Ampere's Law:
B . dl = Mu0 N I, or
B . 2 Pi R = Mu0 N I, or
B = Mu0 N I / (2 Pi R)
Decompose the field from each of the two wires into its Cartesian components, add these components into the total field, and lastly compute the vector total field.

Carl_Weggel@Juno.com
978-474-0396