Two coils, proving some equations

[nothing19]

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.

 

[nothing19]

sorry to pester, but any ideas? thanks.

 

[nlightner]

I would like to clarify that the provided equations and arrangement of coils are not sufficient to accurately determine the magnetic field along the x-axis near x=0. More information is needed, such as the distance between the coils (D) and the current (I) flowing through them. Additionally, the provided equation 9.14 is not complete and may be missing crucial terms.

However, based on the given information, I can provide a response to the content as follows:

a) From the given arrangement of coils, it can be observed that the magnetic field created by each coil at a point on the x-axis will have equal magnitude and opposite direction, resulting in a cancellation of the magnetic field at that point. This is due to the coils being symmetrically placed about the origin and carrying current in the same direction. Therefore, the first derivative of Bx with respect to x will be zero for all values of D.

b) To find the distance D in terms of R where the second derivative of Bx is also zero at x=0, we can differentiate the first derivative of Bx with respect to x again. This will give us the expression -18πKIR^2/(c^2(x^2+R^2)^3.5). Setting this equal to zero and solving for D, we get D=R/√3. Therefore, placing the coils at a distance of D=R/√3 will result in a constant and zero second derivative of Bx at x=0.