Diamagnets have the property that they "dampen" the effects of an external magnetic field by creating an opposing magnet

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Diamagnets have the property that they "dampen" the effects of
an external magnetic field by creating an opposing magnetic
field. The diamagnet thus has an induced dipole moment that
is anti-aligned, such that the induced north pole is closer to the
north pole creating the external field. An application of
this is that diamagnets can be levitated (Links to an
external site.).
Now, the mathematics of generally describing a force by a
non-uniform field on a dipole is a little beyond the scope of this
course, but we can still work through an approximation based on
energy. Essentially, whenever the theoretical loss of
gravitational potential energy from "falling" no longer can "pay
the cost" of increasing the magnetic potential energy, the object
no longer wants to fall.
Suppose a diamagnetic object floats above the levitator where
the magnitude of the magnetic field is 18 T, which is inducing* a
magnetic dipole moment of 3.2 μA⋅m2 in the object.
The magnetic field 2.0 mm below the object is stronger with a
magnitude of 33 T. What is the approximate mass of the
floating object?
Give your answer in units of g
(i.e., x10-3 kg), and use g
= 9.81 m/s2. You may assume the object's
size is negligible.
*Note: The induced dipole moment is proportional to the field
that induces it.
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If you shake a magnetic compass (Links to an external
site.) and then set it down, you can watch the needle bounce
back and forth around its equilibrium position. If this
motion is unimpeded by friction, it is an example of simple
harmonic motion (Links to an external site.). You may
learn more about simple harmonic motion in Phys 1230, but we'll be
seeing another example soon in Unit 6.
In this case, energy is being transferred continually between
magnetic potential energy and kinetic energy, and the energy is
conserved if there is no friction. When there is no kinetic
energy, the needle is the maximally deflected and its magnetic
potential energy is maximum; when the magnetic potential energy is
minimum, the needle is moving the fastest. Conservation of
energy shows that Etotal=KE(θ)+U(θ)=Umax−Umin=KEmax.
Note that only the 2nd and 3rd terms are functions of angle, but
everything else is parametrized by the initial condition.
For your compass, the measured angle between maximum deflection
and equilibrium is 55o. What percent of the
maximum kinetic energy does the needle have when it is only
deflected 33o from equilibrium?
Give your answer as a %.