a) Consider an electron of mass m, travelling in a circular orbit (radius r) around the atom's nucleus (at constant speed). The angular momentum of such an electron is: |Z| = m|ū|r. The current created by this circulating electron is its charge divided by its orbital period. | Show that the magnitude of the orbiting electron's magnetic moment is: 2m Quantum mechanics tells us that |Z| can only take on values that are integer multiples of a constant, called Planck's constant i = 1.054x10-34] · s. So, läl = 1|Ã| = ħn where nis some integer 2m b) Iron has a density of 7900 kg/m³. Estimate the number of atoms that are contained in our bar magnet, using the atomic mass of iron and the density. c) Most electrons in atoms are paired with other electrons, which orbit in the opposite direction, so that the magnetic fields created by the two electrons will cancel out. Assume that in each iron atom, there is only one unpaired electron, and assume that for each electron I=4 (the integer number that defines the atom's magnetic moment). Further, assume that magnetic moment of every single atom in the entire cylinder are perfectly aligned along its long axis. Using your answers from parts a) and b), along with the equation for the magnetic field of a magnetic dipole, calculate the magnitude of the total magnetic field generated by this magnet. For simplicity, you can assume that the point P is roughly the same distance from each atom and that P lies pretty close to the central axis of each atom. *Note, for all atoms, the atomic current loops will be very small compared to their separation from point P.

a) Consider an electron of mass m, travelling in a circular orbit (radius r) around the atom's nucleus (at constant speed). The angular momentum of such an electron is: |Z| = m|ū|r. The current created by this circulating electron is its charge divided by its orbital period. | Show that the magnitude of the orbiting electron's magnetic moment is: 2m Quantum mechanics tells us that |Z| can only take on values that are integer multiples of a constant, called Planck's constant i = 1.054x10-34] · s. So, läl = 1|Ã| = ħn where nis some integer 2m b) Iron has a density of 7900 kg/m³. Estimate the number of atoms that are contained in our bar magnet, using the atomic mass of iron and the density. c) Most electrons in atoms are paired with other electrons, which orbit in the opposite direction, so that the magnetic fields created by the two electrons will cancel out. Assume that in each iron atom, there is only one unpaired electron, and assume that for each electron I=4 (the integer number that defines the atom's magnetic moment). Further, assume that magnetic moment of every single atom in the entire cylinder are perfectly aligned along its long axis. Using your answers from parts a) and b), along with the equation for the magnetic field of a magnetic dipole, calculate the magnitude of the total magnetic field generated by this magnet. For simplicity, you can assume that the point P is roughly the same distance from each atom and that P lies pretty close to the central axis of each atom. *Note, for all atoms, the atomic current loops will be very small compared to their separation from point P.

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3) Estimate the maximum strength of a permanent, iron magnet. Assume the magnet is a cylinder that is 2.0 cm long and has a 1.0cm in diameter, and that we want to know the magnetic field at a point (P), say 10cm from its center along its long axis. We’ll work through the problem sequentially:

a) Consider an electron of mass m, travelling in a circular orbit (radius r) around the atom's
nucleus (at constant speed). The angular momentum of such an electron is: |Z| = m|d]r.
The current created by this circulating electron is its charge divided by its orbital period.
Show that the magnitude of the orbiting electron's magnetic moment is:
läl = 1|Ã] = 2m
e
%3D
Quantum mechanics tells us that L can only take on values that are integer multiples of a
constant, called Planck's constant h = 1.054x10-34] · s. So,
lāl = 1|Ã| =
ħnwhere nis some integer
2m
b) Iron has a density of 7900 kg/m³. Estimate the number of atoms that are contained in
our bar magnet, using the atomic mass of iron and the density.
c) Most electrons in atoms are paired with other electrons, which orbit in the opposite
direction, so that the magnetic fields created by the two electrons will cancel out. Assume
that in each iron atom, there is only one unpaired electron, and assume that for each
electron n=4 (the integer number that defines the atom's magnetic moment). Further,
assume that magnetic moment of every single atom in the entire cylinder are perfectly
aligned along its long axis. Using your answers from parts a) and b), along with the equation
for the magnetic field of a magnetic dipole, calculate the magnitude of the total magnetic
field generated by this magnet. For simplicity, you can assume that the point P is roughly the
same distance from each atom and that P lies pretty close to the central axis of each atom.
*Note, for all atoms, the atomic current loops will be very small compared to their
separation from point P.

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