Equipment:

1. Acrylilc tube
2. Two rubber cork
3. Powder of manganese chloride \(\rm MnCl_2\) tetrahydrate
4. Neodymium magnet
5. Scales
6. Plastic jat
7. Platsic cap with water
8. Plastic drain cap 
9. Two $10~\mathrm{ml}$ syringes
10. Clothspin
11. Plastic spoon
12. Gloves
13. Balls in test tubes

Note: you must NOT consume the powder in any form! Do not swallow the balls! Do not bring the magnet too close to the scale, and be careful with the magnet as it may break.

Theory

There is no magnetic charges, but they are useful tool for magnetic field calculations. Magnetic charge $q_M$ creates the same form of field with the the electric charge:
\[\vec{B} = \frac{\mu_0 q_M}{4 \pi} \frac{\vec{r}}{r^3}.\]Magnetic dipole moment of several magnetic charges $q_{M,i}$ is $\vec{m} = \sum \vec{r}_i q_{M,i}$.

A1  ^0.40 Using the method known from electrostatics, derive the magnetic field of a dipole with a dipole moment $\vec{m}$.

We will call an elastic dipole a dipole whose dipole moment is proportional to the external field: \[\vec{m} = \alpha \cdot \vec{B}.\]The quantity $\alpha$ is called the polarizability of the dipole.

A2  ^0.40 Find the energy of an elastic dipole in a field $B$ directed along axis $z$. Express the answer in terms of $\alpha$ and $B$.

A3  ^0.40

Suppose the molecules of a certain substance have polarizability \(\alpha\) and their concentration is $n$ (units: $1/м^3$). Consider the behavior of a flat layer of such a substance in an external uniform magnetic field and determine the magnetic permeability \(\mu_e\) of the substance. Neglect the influence of particles on each other, i.e., work in the approximation $|\mu_e-1| \ll 1$.

Express the answer in terms of $\alpha$, $n$ and $\mu_0$.

Now let's discuss the behavior of a sphere with magnetic properties in an external magnetic field. Note that the behavior of magnetic materials in external fields is completely analogous to the behavior of dielectrics in electric fields.

A sphere of radius \(R\) with magnetic permeability \(\mu_s\) is placed in a uniform field \(\vec{B_0}\). Using the experience in electrostatics, we can guess the field distribution in space:

• Inside the sphere, the field \(\vec{B}_\text{in}\)  is constant, aligned with the external field;
• Outside the sphere, the field is a superposition of the external field \(\vec{B_0}\) and the field of a magnetic dipole \(\vec{m} = \alpha_s \cdot \vec{B_0}\), located at the center of the sphere.
   

When the contour used for the circulation theorem of the vector \(\vec{B}\) passes through several media and \(\mu(\vec{r})\) is a function of coordinates, the circulation theorem is conveniently written as:
\[\oint{\frac{\vec{B}(\vec{r})}{\mu(\vec{r})\mu_0}d\vec{l}} = I,\]

where $I$ is the current penetrating the contour. Gauss's theorem for the magnetic field does not change its form when the surface only partially passes through several media.

A4  ^0.80

Write the boundary conditions for \(B_n(\theta)\) and \(B_{\tau}(\theta)\).

A5  ^0.80 Using the boundary conditions, find \(\vec{B}_\text{in}\) and \(\alpha_s\). Express the answers in terms of and $\vec{B}_0$, $\mu_s$ и $R$.

From the problem we have solved, it is easy to move to the case where the sphere is placed in a medium with permeability \(\mu_e\). To do this, simply replace \( \mu_s \to \mu_s/\mu_e\) in the final answers.

The materials of the beads provided to you practically do not exhibit magnetic properties, and for them, we will assume that $\mu_s=1$. The $\rm MnCl_2$ solution, which acts as the surrounding medium, is a strong paramagnet, and its magnetic properties are due to the manganese ions $\rm Mn^+$. At the same time, the salt $\rm MnCl_2$ dissociates completely, meaning the molar concentration of $\rm MnCl_2$ ions is equal to the molar concentration of the salt added in dry form in the solution.

For the $\rm MnCl_2$ solution, the following holds:
\[\mu_e(c) - 1 = \chi_e(c) = a \cdot c,\] where $c$ is is the molar concentration of $\rm Mn^+$.The solution is a strong paramagnetic medium but it is still a paramagnetic medium, and for it \( \chi_e\ \ll 1 \).

A6  ^0.50 Express \(\alpha_{s}\), for non-magnetic ball in the $\rm MnCl_2$ solution in tearms of $a$, $c$ and $R$.

A7  ^1.00 Find the energy $W$ of the ball located on the $z$-axis. Choose the constants so that the magnetic energy at $z\to \pm \infty$ is zero; the energy in the gravitational field is zero at $z=0$. Express the answer in terms of \(\rho_s, \rho_e, g, R, \alpha_s, B(z)\) and $z$.

A8  ^0.20 The energy obtained in A7 turns out to be proportional to the volume of the ball. Find the specific energy of the ball $w=W/V$. Express the answer in terms of $a$, $c$, $\mu_0$, $B(z)$ and $g$, $z$, $\rho_e$, $\rho_s$.

Below is a graph of \(B^2/(\mu_0 M)^2\) on $z$ for the provided magnet.

In the experiment, different types of the ball behavior in the solution can be observed:

A9  ^2.00 Determine under what conditions for \(c\) and $\rho_s-\rho_e$ each of the cases is realized. Express it in terms of $a$, $c$, $\mu_0$, $M$, $g$ and values charecterising the graph of $B^2(\mu_0 M)^2$ vs $z$.

Experiment

A10  ^1.50 Plug one end of the tube with a stopper, fill it with the solution. Position the magnet approximately in the middle of the solution level.

By varying the concentration of MnCl2 $\rm MnCl_2$ in the solution, measure as accurately as possible the transition boundary from one case to the other from C6 for each type of bead provided.

Record the concentration of the $\rm MnCl_2$ solution in terms of $m_e/m_w$, where $m_e$ is the total mass of solution, and $m_w$ is the mass of pure water what was used to prepare it.

A11  ^0.50 The powder provided to you is not pure $\rm MnCl_2$ but 4-hydrated $\rm MnCl_2 \cdot 4H_2O$, meaning that for each molecule of $\rm MnCl_2$ in the powder, there are four molecules of \(\rm H_2O\). When this powder is dissolved in liquid water, it dissociates into $\rm Mn^{2+}$ and $\rm Cl^-$ ions, and the water takes on a liquid form.

Express the molar concentration \(c\) of manganese ions $\rm Mn^{2+}$ in the solution. For the answer, use \(\rho_e\), $m_e/m_w$, $\mu_{\rm H_2O}$ and $\mu_{\rm MnCl_2}$

A12  ^1.00 Using your data, calculate the values of \(\chi_e\) on \(c\) and plot the graph of this dependence.

A13  ^0.50 Find the polarizability of magnesium ions $\alpha_{\rm Mn^{2+}}$.