How is one side of a magnetic material attracted to opposing poles of a magnet?

Question

Magnetic materials are commonly diamagnetic, paramagnetic or ferromagnetic. The following pdf (http://faculty.washington.edu/mrdepies/122/Workbook_122/WB_Solution_Ch32.pdf; see exercise #2) claims that magnetic materials are attracted to either pole of a magnet.

I understand that magnetic materials become magnetized when their domains are induced to spin in the same direction. The consistent spin somehow gives rise to a magnetic attraction.

How is it, however, that one given side of a magnetic object can attract to either side of a magnet. I would expect the magnetic material to behave as a magnet would and have poles.

Answer 1

That is true for paramagnetic material only. On the contrary, diamagnetic ball will be repelled from both poles. To see why, consider the potential energy of a magnetic moment $\boldsymbol\mu$ in an external magnetic field $\boldsymbol B$: \begin{equation} U = -\boldsymbol \mu \boldsymbol B. \tag{1} \end{equation} Now, notice that in linear approximation $\boldsymbol \mu = \chi\boldsymbol B$, where $\chi$ is positive or negative for a paramagnet or a diamagnet respectively. Substituting this into (1): \begin{equation} U = -\chi B^2. \tag{2} \end{equation} The force acting on the moment: \begin{equation} \boldsymbol F = -\nabla U = \chi \nabla(B^2). \tag{3} \end{equation} In the vicinity of any pole, the gradient of $B^2$ points towards the pole (denser field lines). Hence $\boldsymbol F$ points towards a pole for a paramagnet, and to outside for a diamagnet.

Obviously, a ferromagnet will be different because $\boldsymbol \mu \neq \chi\boldsymbol B$ in this case.