Atomic Structure of Magnets

Austin Bryan

Short Description of Topic

The Main Idea

The magnetic field produced by a magnet is the sum of the magnetic fields generated by each individual atom. These very small magnetic fields are generated much like those of circular current loops; however instead of being generated by electrons flowing through a wire, the field in each individual atom is produced in one three different ways:

1. An electron orbiting around the atomic nucleus.
2. An electron rotating around its axis.
3. The rotation of protons and neutrons within the nucleus of the atom.

All three of these situations produce a magnetic dipole proportional to the angular momentum. Together, the magnetic dipoles of all the atoms in the magnet sum to give the total magnetic dipole of the magnet. The magnetic field at an observation location can then be found from this dipole.

Although all atoms have electrons orbiting their nuclei, most materials are not magnetic. Each atom in these materials has a small magnetic dipole, however these dipoles are unaligned and disordered and therefore usually sum to zero. In magnetic materials, regions of magnetic dipoles line up. Although some of these regions cancel other regions out, enough regions align to produce a nonzero magnet field. This is allowed by interactions between atoms in certain elements (usually iron, nickel, cobalt, or alloys of these metals).

A Mathematical Model

The magnetic dipole in a magnet is analogous to the magnetic dipole in a current loop. In a current loop [math]\displaystyle{ {\mu = I \pi R^2} }[/math] when [math]\displaystyle{ R }[/math] is the radius of the loop. Since the units of [math]\displaystyle{ I }[/math] are [math]\displaystyle{ \frac{charge}{time} }[/math], the charge of an electron is [math]\displaystyle{ -e }[/math], and the period for one orbit around the nuclues is [math]\displaystyle{ t = \frac{2 \pi R}{v} }[/math] where [math]\displaystyle{ v }[/math] is the speed of the the electron, the magnetic dipole for one atom in a magnet simplifies to [math]\displaystyle{ \mu = \frac{e R v}{2} }[/math] where [math]\displaystyle{ R }[/math] is now the radius of the orbit.

The magnetic dipole is proportional to the angular momentum, [math]\displaystyle{ L }[/math] of the electron orbiting the nucleus. Assuming a circular orbit and assuming the speed of the electron is much less than the speed of light, [math]\displaystyle{ L = R p = R m v }[/math]. Multiplying the magnetic dipole by [math]\displaystyle{ \frac{m}{m} }[/math] reveals the proportionality of magnetic dipole and angular momentum [math]\displaystyle{ \mu = \frac{m}{m} \frac{e R v}{2} \Rightarrow \mu = \frac{e}{2 m} (R m v) = \frac{e}{2 m} L }[/math] From this equation the charge and mass of an electron and a proton can be plugged in to be compared. Because an electron weighs so much less than a proton, the magnetic dipole from the orbit of an electron is [math]\displaystyle{ 10^4 }[/math] times bigger than the magnetic dipole from the rotation of a proton or neutron in the nucleus, allowing the contributions from the protons and neutrons to be neglected.

For the purpose of calculating the magnetic dipole, it can be assumed that [math]\displaystyle{ L }[/math] is equal to Planck's Constant, [math]\displaystyle{ \hbar = 1.05 \times 10^{-34} \, J \cdot s }[/math]. Plugging in the charge and mass of an electron gives [math]\displaystyle{ \mu \approx 1 \times 10^{-23} \, A \cdot m^2 \, per \, atom }[/math].

Finally, if [math]\displaystyle{ \mu = N \mu_{atom} }[/math] where [math]\displaystyle{ N }[/math] is the number of atoms, this [math]\displaystyle{ \mu }[/math] is the magnetic dipole of the magnet.

Examples

Simple

A bar magnet made from iron has a mass of 72 g. What is the magnetic dipole of the bar magnet?

[math]\displaystyle{ N = \left (\frac{72 \, g \, iron}{56 \, \frac{g \, iron}{mol \, iron}} \right ) \left (6.022 \times 10^{23} \frac{atoms}{mol} \right ) = 7.74 \times 10^{23} \, atoms }[/math]

[math]\displaystyle{ \mu = (7.74 \times 10^{23} \, atoms) \left (1 \times 10^{-23} \, \frac{A \cdot m^2}{atom} \right ) = 7.74 \, A \cdot m^2 }[/math]

Middling

A bar magnet that is 15% mass iron and 85% mass nickel has a mass of 126 g. What is the magnetic dipole of the bar magnet?

[math]\displaystyle{ 126 \, g \cdot 0.15 = 18.9 \, g \, iron }[/math]

[math]\displaystyle{ 126 \, g \cdot 0.85 = 107.1 \, g \, nickel }[/math]

[math]\displaystyle{ N = \left (\frac{18.9 \, g \, iron}{56 \, \frac{g \, iron}{mol \, iron}} \right ) \left (6.022 \times 10^{23} \frac{atoms}{mol} \right ) = 2.03 \times 10^{23} \, atoms }[/math]

[math]\displaystyle{ N = \left (\frac{107.1 \, g \, nickel}{59 \, \frac{g \, nickel}{mol \, iron}} \right ) \left (6.022 \times 10^{23} \frac{atoms}{mol} \right ) = 1.09 \times 10^{24} \, atoms }[/math]

[math]\displaystyle{ 2.03 \times 10^{23} \, atoms + 1.09 \times 10^{24} \, atoms = 1.29 \times 10^{24} \, atoms }[/math]

[math]\displaystyle{ \mu = (1.29 \times 10^{24} \, atoms) \left (1 \times 10^{-23} \, \frac{A \cdot m^2}{atom} \right ) = 12.9 \, A \cdot m^2 }[/math]

Difficult

A compass originally points north. A bar magnet made of iron is placed [math]\displaystyle{ 20 \, cm }[/math] west of the compass on axis, with the north end of the magnet pointing towards the compass. The compass deflects [math]\displaystyle{ 55^\circ }[/math]. What is the mass of the magnet?

[math]\displaystyle{ \overrightarrow{B}_{net} = \overrightarrow{B}_{Earth} + \overrightarrow{B}_{magnet} }[/math]

[math]\displaystyle{ B_{magnet} = (2 \times 10^{-5} \,T) \tan{55^\circ} }[/math]

[math]\displaystyle{ B_{magnet} = 2.86 \times 10^{-5} \,T }[/math]

[math]\displaystyle{ B_{magnet} = \frac{\mu_0}{4 \pi} \frac{2 \mu}{r^3} }[/math]

[math]\displaystyle{ \mu = \frac{r^3 B_{magnet}}{2} \frac{4 \pi}{\mu_0} }[/math]

[math]\displaystyle{ \mu = \frac{(0.20 \, m)^3 (2.86 \times 10^{-5} \,T)}{2} (1 \times 10^7 \, T \cdot \frac{m}{A}) }[/math]

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