Physics Laboratory II – X3.1B – Spring 2016 Pre-Lab 3

Question # 00590503 Posted By: Prof.Longines Updated on: 09/18/2017 09:32 AM Due on: 09/18/2017
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In Lab 3, you will explore the Lorentz force law and Faraday’s law.
Torque on a Magnetic Dipole
The magnetic dipole moment vector
r
µ is a vector whose magnitude tells you how strong a magnet is and whose
direction tells you which end of the magnet is ‘north’. In a bar magnet, for example, the magnetic dipole moment
vector would point from the magnet’s south pole to the north pole. We saw many examples of magnetic dipoles in
Lab 2, including dipoles created by current-carrying wires rather than just bar magnets. Part of the lab included
sketching the field of a dipole. The dipole moment vector for a loop of wire can be constructed as follows:
• The magnitude is µ = NIA, where N is the number of coils in the loop, A is the area of the loop,
and I is the current flowing through the loop. We typically use the symbol µ for the dipole
moment, but it is very much a different thing than µ0, the magnetic constant! Be careful…
• The direction is found using the right-hand rule: curl your fingers in the direction of the current in
the loop and stick out your thumb. Your thumb points in the direction of the dipole moment
vector. (It also points in the direction of the magnetic field in the center of the loop.) The tip of
your thumb is the ‘north pole’ of the current loop.
The torque on a magnetic dipole of magnitude µ in a uniform magnetic field of strength B has magnitude µB sin?,
where ? is the angle between the direction that the dipole moment vector points and the direction of the magnetic
field. The torque acts to try to align the dipole moment vector with the magnetic field vector (just like in a
compass).
Consider a single square loop of wire of area A carrying a current I in a uniform magnetic field of strength B. The
field is pointing directly up the page in the plane of the page. The loop is oriented so that the plane of the loop is
perpendicular to the plane of the page (this means that the normal vector for the loop is always in the plane of the
page!). In the illustrations below the magnetic field is shown in red and the current through the current loop is
shown in blue. The loop starts out in orientation (i) and rotates clockwise, through orientations (ii) through (viii)
before returning to (i).
a) [1.5 points] For each of the eight configurations, draw in the dipole moment vector. [Hint: Right-hand rule!]
b) [1.5 points] For each of the eight configurations, determine whether the torque on the current loop is
clockwise, counterclockwise, or zero. In which two of the above eight orientations will the loop experience the
maximum magnitude of torque? What is the magnitude of torque in these two cases?
Let ? be the angle between the normal vector to the loop (that is, the direction of the magnetic dipole moment
vector) and the magnetic field, so configuration (i) has ? = 0. Let a positive angle be an angle clockwise of ? = 0 (so,
for example, if ? = 0 were pointing to the top of the page then ? = ?/4 (45°) would be to the top-right corner.
c) [1 point] Sketch a graph of the torque on the wire as a function of the angle ? that the normal vector (let
torque be defined as positive when the action of the torque wants to rotate the loop in a clockwise fashion).
The potential energy of a magnetic dipole of magnitude µ in a uniform magnetic field of strength B is given by U =
–µB cos ?. [Note: You should not be using the formula I?2
/2 in any of the following parts! Also, it is okay for the
potential energy to be negative! All this means is that the potential energy is smaller than the potential energy of the
“reference configuration.”]
d) [2 points] For each of the eight configurations, determine the potential energy of the loop. Be sure to explicitly
indicate if the potential energy is positive, negative, or zero.
e) [2 points] Suppose you release the loop of wire from rest at configuration (iii). At the instant of release, what
is the rotational kinetic energy of the loop? What is the total energy of the loop? [Hint: I again call to your
attention that the loop is starting from rest…]
The torque at configuration (iii) is CCW and so the loop will start to rotate, picking up angular speed past
configuration (ii) until it reaches configuration (i).
f) [2 points] What is the total energy of the loop in configuration (i)? What is the rotational kinetic energy of the
loop? [Hint: Conservation of energy is your friend! You also found the potential energy of this configuration in
part d.]
In configuration (i), even though the torque is zero the kinetic energy keeps the loop rotating (just like a stretched
spring released from rest will compress farther than the equilibrium length of the spring, since there is kinetic energy
and momentum when the force is zero!).
g) [2 points] Describe what happens to the loop in terms of velocities and torques as the loop enters configuration
(i) and beyond. (“At configuration (i) the angular velocity of the loop is CW/CCW/0 and the torque is
CW/CCW/0. The loop stays at (i)/rotates to (viii)/rotates to (ii), at which point… <repeat>”. In terms of total
energy, potential energy, and rotational kinetic energy, what is happening to the loop (“The
total/potential/kinetic energy is increasing/decreasing/staying the same until it reaches configuration (-) at which
point…”).
Now suppose that we turn the current off when the loop is in configurations (v) through (viii).
h) [2 points] Repeat part (f)!
Faraday’s Law
Faraday’s law is a relation between a changing magnetic flux and an induced voltage. Recall that the flux through a
surface is roughly a measure of how much field goes through the surface. If the magnetic field has uniform strength
B, the surface is a piece of plane of area A and the magnetic field vector makes an angle ? with the normal vector to
the surface, then the magnetic flux through the surface is ?B = BA cos?.
Consider again the loop of wire in the magnetic field as illustrated in the previous part. However, we will not have
any external current going through any of the loops in this part (any current in the loops is from the induced
voltage). Suppose the loop is rotating clockwise with a frequency f, so that at time t = 0 the loop is in orientation (i);
at time t = 1/(4f) the normal vector is in orientation (iii); at time t = 1/(2f) the loop is in orientation (v); at time t =
3/(4f) the loop is in orientation (vii); at time t = 1/f the loop is back in orientation (i). More precisely, the angle that
the normal vector makes with the magnetic field vector as a function of time is ? = 2?ft.
i) [2 points] Sketch a graph of the flux as a function of time. Which orientations will have the most positive
flux? The most negative flux? Zero flux?
j) [2 points] Sketch a graph of the rate of change of flux as a function of time. What orientations will have the
greatest rate of change of flux? What orientations will have (momentarily) no change in flux?
k) [2 points] What orientations will have the greatest magnitude of induced voltage? For each orientation state
whether the induced voltage is positive, negative, or zero? Sketch a graph of the induced voltage as a function
of time.
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  1. Tutorial # 00588684 Posted By: Prof.Longines Posted on: 09/18/2017 09:33 AM
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