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HW9
Due: 11:00pm on Tuesday, April 27, 2010
Note:
To understand how points are awarded, read your instructor's
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.
Dipole Motion in a Uniform Field
Description:
A dipole is released from rest and allowed to rotate in an electric field. Find the angular velocity when the dipole is aligned with the field (using conservation of energy). Then find the period of small
oscillations about the minimum of potential energy.
Consider an electric dipole located in a region with an electric field of magnitude
pointing in the positive
y
direction. The positive and negative ends of the
dipole have charges
and
, respectively, and the two charges are a distance
apart. The dipole has moment of inertia
about its center of mass.
The dipole is released from angle
, and it is allowed to rotate freely.
Part A
What is
, the magnitude of the dipole's angular velocity when it is pointing along the
y
axis?
Hint A.1
How to approach the problem
Because there is no dissipation (friction, air resistance, etc.), you can solve this problem using conservation of energy. When the dipole is released from rest, it has potential energy but no kinetic energy. When the
dipole is aligned with the
y
axis, it is rotating, and therefore has both kinetic and potential energy. The sum of potential and kinetic energy will remain constant.
Hint A.2
Find the potential energy
Find the dipole's potential energy
due to its interaction with the electric field as a function of the angle
that the dipole's positive end makes with the positive
y
axis. Define the potential energy to be zero
when the dipole is oriented perpendicular to the field:
.
Hint A.2.1
The formula for the potential energy of a dipole
The general formula for the potential energy of an electric dipole with dipole moment
in the presence of a uniform electric field
is
.
Hint A.2.2
The dipole moment
The dipole moment of the electric dipole
, when it makes an angle
with the positive
y
axis can be written as
.
Express your answer in terms of
, ,
, and .
A%SWER:
=
Hint A.3
Find the total energy at the moment of release
Find
, the total energy (kinetic plus potential) at the moment the dipole is released from rest at angle
with respect to the
y
axis. Use the convention that the potential energy is zero when the dipole is oriented
perpendicular to the field:
.
Express your answer in terms of some or all of the variables
,
,
, and
.
A%SWER:
=
Hint A.4
Find the total energy when
Find an expression for
, the total energy (kinetic plus potential) at the moment when the dipole is aligned with the
y
axis. Use the convention that the potential energy is zero when the dipole is oriented
perpendicular to the field:
.
Hint A.4.1
What is kinetic energy as a function of angular velocity?
What is the kinetic energy
of a body rotating with angular velocity
around an axis about which the moment of inertia is ?

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