# What makes the magnetic field inhomogeneous in the Stern–Gerlach experiment?

The magnetic field depicted in Stern–Gerlach experiment is described as inhomogeneous. Is that depiction the only depiction of inhomogeneous magnetic field used, or are there other examples of inhomogeneous magnetic fields used that do not symmetrical?

Why the depicted magnetic filed that looks symmetrical is inhomogeneous? Is it possible to have symmetrical magnetic field that is inhomogeneous? or inhomogeneous magnetic field that is symmetrical?

• Homogeneity only means that in some part of space the quantity you are considering (e.g. magnetic field) has the same properties. The Helmholtz coil has a nearly homogeneous magnetic field between the coils (the force on a small test-magnet inside the coil would be of the same size and the same direction), the field lines are also symmetric. Outside the coil, the magnetic field isn't homogeneous but it is still symmetric. Nov 17, 2015 at 15:05
• A simple N/S magnet is symmetric (rotationally about the N-S axis), but not homogeneous - the magnetic field is not constant throughout space. You can create reasonably homogeneous fields in a defined volume to contain an experiment, but even those fields are not homogeneous outside of that volume. Nov 17, 2015 at 17:52
• As a general rule you have to take care to create homogeneity by arranging sufficient levels of symmetry. There are more ways to be inhomogeneous than homogeneous. Nov 17, 2015 at 20:46

A vector field

$$\mathbf V(x,y,z) = V_x (x,y,z) \ \hat x + V_y (x,y,z) \ \hat y + V_z (x,y,z) \ \hat z$$

is homogeneous if its value does not depend on the point in space where you measure the field. Mathematically,

$$\mathbf V(x,y,z) = \mathbf V (x',y',z')$$

For example, the 2D field

$$\mathbf V= (x^2,y^2)$$

is clearly not homgeneous. However, we can say that it is symmetrical with respect to the axis $x=y$, since exchanging $x$ with $y$ leaves the magnitude of $\mathbf V$ unchanged (notice that in the figure warmer colors correspond to higher field intensity): The 2D field

$$\mathbf V = (1,1)$$

is trivially homogeneous: In the Stern-Gerlach experiment you need an inhomogeneous field to observe the effect, as explained on Wikipedia:

If the particle is treated as a classical spinning magnetic dipole, it will precess in a magnetic field because of the torque that the magnetic field exerts on the dipole (see torque-induced precession). If it moves through a homogeneous magnetic field, the forces exerted on opposite ends of the dipole cancel each other out and the trajectory of the particle is unaffected. However, if the magnetic field is inhomogeneous then the force on one end of the dipole will be slightly greater than the opposing force on the other end, so that there is a net force which deflects the particle's trajectory

To conclude, I just want to point out that there is no such thing as a perfectly homogeneous magnetic field. This is because of Gauss' law,

$$\nabla \cdot \mathbf B = 0$$

that tells us that a magnetic field has no sources nor sinks: its field lines can only be closed loops. Therefore, a perfectly homogeneous magnetic field is impossible.

If the magnetic field were homogeneous then the atoms (tiny magnets) would have experienced only a turning moment, and no deflecting force. As such we could not obtain the deflected components inspite of the orientation of the atoms relative to the field.

Think of a single coil of wire, with current $I$. The field looks symmetrical about the axis of the coil, but as you move away from it the magnitude $|\mathbf{B}|$ will decrease $1/r^3$. Hence it is not homogeneous.

Same think with any bar magnet, as you go away from them, the field goes down, so its strength varies in space and it is not homogeneous.