Magnetic field due to parallelnwires

[vs5813]

Magnetic field due to parallel wires

Homework Statement

Two very long thin wires carrying equal and opposite currents +/-I are p laced parallel to the x-axis at y=0 and z=+/-a. Find an expression for B field at a point P in the xy-plane (z=0) and show that its maximum gradient occurs for y = a/sqrt(3).

Homework Equations

Field due to long wire at distance d away:
B=(mu_0*I)/(2*PI*d)

The Attempt at a Solution

By symmetry, i expect B field components to cancel except in y direction..So i end up getting total field:
B=B_y=(mu_0*I)/(PI*d) * cos(theta)
where theta is the angle that d makes to the y axis. Because of geometry of situation, this is equivalent to saying:
B=B_y=(mu_0*I*y)/(PI*d^2)
Then for the gradient to be maximized, I need maximum of:
dB/dtheta=-(mu_0*I)/(PI*d) * sin(theta)
...which should occur when sin(theta) is a max...but i don't understand how i could show that the maximum is for y = a/sqrt(3) any thoughts..? thankyou!

 

[aiy1tm]

You are correct about symmetry. You can write down the magnetic field as a function of y only in the x-y plane. Then, I suspect, you can take the derivative with respect to y (TWICE) to find the maximum in the gradient. The maximum field strength on the x-y plane is trivial by symmetry (the origin).

Closer to your proposed approach, this means you need to acknowledge that in the function you wrote down, "d" and "theta" are related. (You can probably write them as a function of one another, and y.) (for example, you know the angle between d must approach zero and d gets longer and longer)...