## Anti-Helmholtz coil configuration

Hi

Say I have to coils in the anti-Helmholtz configuration as in the attached picture. It is pretty easy to find the field along the z-direction, as this is introduced in introductory EM. However say that I would like to know the field along the x-direction. This I don't know how to find.

What I *do* know is that the Maxwell Equations (div B = 0) tell me that
$$\frac{dB}{dx} = \frac{dB}{dy} = -\frac{1}{2}\frac{dB}{dz}$$
But does this imply that the field along x, B(x), is simply -B(z), the negated B-field along the z-direction?

Best,
Niles.
Attached Images
 coils.bmp (248.1 KB, 8 views)
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 Mentor div B = 0 is equal to $\frac{\partial B_x}{\partial x} + \frac{\partial B_y}{\partial y} = - \frac{\partial B_z}{\partial z}$ Using symmetry, x and y must be the same, therefore $\frac{\partial B_x}{\partial x}= - \frac{1}{2}\frac{\partial B_z}{\partial z}$ This does not give you any magnetic field! It is just the derivative of the field at some specific point - probably along the central axis. Looking at the (x,y)-plane right in the middle of the coils, you have a field going radially inwards/outwards (depending on the orientation).
 Thanks, that was kind of you.

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