Magnetic field from two current carrying loops and variable extreme.

[boroscots]

Homework Statement

Two loops carry identical currents counter-clockwise, I and share the same radius, R. They are separated by the distance R with the z axis passing through the center of each loop. Find the magnetic field at a point x distance above the center of the two loops in terms of I, R, and x.
a) Use superposition from known results or the Biot-Savart law.
b) Find the value of x that extremizes the magnetic field by an appropriate calculus-based argument and the magnetic field at that point.
c) Prove this point is a maximum or minimum using an appropriate calculus-based argument.

Homework Equations

dB = [(mu knot)/4*pi]*(I ds x rhat)/r^2
and perhaps (1+x)^n approximately = 1+ nx + n(n-1)x^2/2! + ...

The Attempt at a Solution

From what I understood, using Biot-Savart's law, B = [(mu knot) * I ] / [R * (1 + ( R/2 + x )^2 / R^2 )^3/2] using z=R/2 +x and factoring out R^2 from the following equation...
B=[mu knot * IR / (R^2 + z^2)^3/2]

If this is correct, I thought to use the approximation to say that at x<<R, x=0 yields a magnetic field... 5 * mu knot * I / 8 * R. (solving part b)?

How do I go about determining if this is a maximum or minimum (assuming what I've done so far is correct)?

Thank you!

 

[anathema]

a) To find the magnetic field at a point x distance above the center of the two loops, we can use superposition by considering the magnetic field produced by each loop separately and then adding them together. This is based on the principle that the total magnetic field at a given point is the vector sum of the individual magnetic fields produced by each current element.

Using the Biot-Savart law, we can express the magnetic field produced by a single current loop as B = [(mu knot)/4*pi] * (I * ds x rhat)/r^2, where mu knot is the permeability of free space, I is the current, ds is the differential current element, rhat is the unit vector pointing from the current element to the point of interest, and r is the distance between the current element and the point of interest.

For two identical counter-clockwise loops with the same radius R and separated by a distance R along the z axis, the magnetic field at a point x above the center of the loops is given by the superposition of the magnetic fields produced by each loop. The magnetic field produced by the first loop is directed along the positive z axis, while the magnetic field produced by the second loop is directed along the negative z axis. Therefore, the total magnetic field at the point of interest is given by:

B = [(mu knot)/4*pi] * (I * ds x rhat)/r^2 + [(mu knot)/4*pi] * (I * ds x rhat)/r^2

= [(mu knot)/4*pi] * (2I * ds x rhat)/r^2

= [(mu knot)/2*pi] * (I * ds x rhat)/r^2

= [(mu knot)/2*pi] * (I * ds x rhat)/[R^2 + (R/2 + x)^2]^3/2

= [(mu knot)/2*pi] * (I * ds x rhat)/[R^2 + (R^2/4 + Rx + x^2)]^3/2

= [(mu knot)/2*pi] * (I * ds x rhat)/[(5/4)*R^2 + Rx + x^2]^3/2

= [(mu knot)/2*pi] * (I * ds x rhat)/[(5/4)*R^2 + (x + R/2)^2