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

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.

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.