Why are magnets in form of hollow sphere not a magnetic monopole? [closed]

Question

If I place many micro-bar-magnets on the surface of hollow sphere such that the pole of every magnet points towards the same side then, you could imagine that a monopole would be produced. So, I just created a monopole. But we all know that monopoles don't exist. Then what is wrong in this procedure? Assuming that I got some kind of power that I can put the magnets together in one place without getting budged by the repulsive forces!

Answer 1

A good starting point to see that gluing small magnets to a sphere doesn't work is seeing what the field of just two magnets opposed gives you. A single dipole magnet has a field that falls off as $\frac{1}{r^3}$, while the two opposed has a field that dies faster, falling off as $\frac{1}{r^4}$. As you add more and more magnets (pointing in different directions), you cancel out successively more and more of the field (i.e. higher order moments), and eventually, in the limit of infinite tiny magnets all pointed outward, the sum of all the magnetic fields cancel, and you are left with, oddly enough, no magnetic field at all, inside or outside the sphere (except for inside the magnets themselves).

Edit: A good way to think about this is considering a magnetic dipole as two magnetic monopoles very close to one another. So this sphere of magnets can be thought of as two spheres of uniform magnetic charge and opposite sign. By the shell theorem, there is no field inside a uniformly charged shell, and outside the uniformly charged shell the field is (in some system of natural units) $\frac{Q}{r^2}-\frac{Q}{r^2}$. The only place the field is non-zero is between the shells. If we take $r_{out}-r_{in} \rightarrow 0$ while keeping $\frac{Q}{r_{out}-r_{in}}$ constant (which keeps the dipole moment of each dipole constant), this field vanishes as well.

Yet another edit: I was asked for more clarification, so I'll go about this another way:

First, it's clear that whatever field we find must be rotationally symmetric, as the system we have set up has spherical symmetry. This forbids the existence of any fields (using spherical coordinates) in the $\hat\phi$ or $\hat\theta$ directions (azimuthal or polar directions). So the only field we can have is radial, and cannot depend on angle: $$\vec B=f(r)\hat r$$

Now, we can apply Gauss' law for magnetic fields (with magnetic monopoles):

$$\int \vec B \cdot d\vec A = \mu_0Q_m$$

where the left hand side is a surface integral over an enclosed volume, and $Q_m$ is the amount of magnetic charge contained within. If we use a spherical volume, then the left hand side simplifies: (as $\vec B \mathbin{\|} \vec dA$ and $\left|\vec B\right|$ is constant)

$$\int \vec B \cdot d\vec A = \int\left|\vec B\right| \left|d \vec A\right| =\left|\vec B\right|\int\left|d\vec A\right| = \left|\vec B\right|4\pi r^2$$

And thus the total magnetic field is:

$$\vec B=\frac{\mu_0Q_m}{4\pi r^2}\hat r$$

In the absence of any magnetic monopoles, $Q_m=0$, and so this is zero. If you make this and notice a monopole moment, it means that there is a magnetic monopole that just happened to stop inside your sphere! Don't lose it- it's probably worth a Nobel prize.