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Homework Statement
Imagine there are two parallel current-carrying coils, radius R, perpendicular to and centered on the x-axis, with centers at 0.5D and -0.5D. Both carry a current I in the same direction (clockwise). We would like to adjust D so that the magnetic field created by the coils is as constant as possible along the x-axis near x=0.
a) use equation 9.14 (below) to show the first derivative of Bx, with respect to x, is zero for all values of D just because of the arrangement of coils being symmetric about the origin.
b) If we place the coils a distance D apart, the second derivative of Bx will also be zero at X=0. Find this distance in terms of R.
Homework Equations
The equation is B=(2*pi*K*I*R^2) / ((c^2 (x^2 + R^2)^1.5))
The Attempt at a Solution
I had to derive that from the Biot-Savart law in a different problem. Anyway, if I recall, partial derivatives in respect to say X would be just like treating everything else a constant and X the only variable. In that case, I'd get constant 2piKIR^2/C^2...and then that multiplied by (X^2 + R^2)^-1.5. That derivative in terms of X would be
-3X / ((X^2 + R^2)^2.5).
B'=-6piKIR^2/(c^2*(x^2 + R^2)^2.5)).
A hint is given it might help to argue x should be replaced by X plus or minus 0.5D in this problem...but I don't see where that would help, especially in terms of computing the derivative.