Magnetic Field at a point due to two parallel wires

[smithisize]

Homework Statement

Two long parallel wires are a center-to-center distance of 4.90 cm apart and carry equal anti-parallel currents of 3.70 A. Find the magnetic field intensity at the point P which is equidistant from the wires. (R = 10.00 cm).

http://imageshack.us/a/img11/3812/twoparallelwires.jpg

Homework Equations

dB = μ_o*I*(dL\vec{}xdR\vec{})/(4*pi*r^3)

The Attempt at a Solution

Well first I tried multiplying the equation for magnetic field of an infinite line (μ_o*I/(2*pi*r) by two since there are two wires. then I realized that since the current is flowing in opposite directions, the y-component of the field, so to speak, would cancel out, and now I'm stuck.

Here is the solution, but I want to know how to arrive here (and more specifically I'd like to know why we're multiplying the infinite line equation by the y-component): d*I*μ_o/(2*pi*(R^2+ (d/2)^2)

 

[tiny-tim]

hi smithisize!

… then I realized that since the current is flowing in opposite directions, the y-component of the field, so to speak, would cancel out …

no, if you draw a vector diagram (with arrows), you'll find the x components cancel out

 

[smithisize]

hi smithisize! no, if you draw a vector diagram (with arrows), you'll find the x components cancel out

Well, the vector equations for dr are R\hat{i} -(d/2)\hat{j} and R\hat{i}+(d/2)\hat{j} therefore the vector sum states that the vertical component cancels out.

But, I ended up figuring it out. If you draw the b-field for the top wire, and manipulate theta (of the upper left corner of a triangle drawn on the top wire) a bit, you can end up with the proper configuration and see that where it all comes from.

 

[tiny-tim]

Well, the vector equations for dr are R\hat{i} -(d/2)\hat{j} and R\hat{i}+(d/2)\hat{j}

no, if you draw the arrows, you'll see that one goes to the left, and the other to the right,

so it's -R\hat{i} +(d/2)\hat{j} and R\hat{i}+(d/2)\hat{j}