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```P216 final exam
Apr. 11, 2000
Page 1 of 4
UNIVERSITY OF VICTORIA
EXAMINATIONS APRIL 2000
PHYSICS 216 S01
DURATION: 3 Hours
INSTRUCTOR: R. V. Kowalewski
STUDENTS MUST COUNT THE NUMBER OF PAGES IN THIS EXAMINATION
PAPER BEFORE BEGINNING TO WRITE, AND REPORT ANY DISCREPANCY TO
THE INVIGILATOR.
THE NUMBER OF PAGES IN THIS QUESTION PAPER IS INDICATED IN THE
HEADER AT THE TOP OF EACH PAGE.
The examination will be graded out of 100 points.
Students are allowed to bring one formula sheet (one side of an 8½ x 11 page) and a
calculator.
P216 final exam
Apr. 11, 2000
Page 2 of 4
(25 pts) Question 1 - short answers
(a) Lines of magnetic flux always form closed loops - true or false?
(b) How much energy (in eV) is required to bring two electrons from an infinite
separation to a separation of 0.1 nm?
(c) A hoop of area A is rotating around the z axis with angular velocity  = k in
the presence of a B field oriented along the +x axis. If the hoop is in the y-z
plane at time t = 0, graph the induced EMF as a function of time between t = 0
and t = 2/ (be sure to label the axes clearly).
(d) Does the revolution frequency of a charged particle in a uniform magnetic field
depend on the particle’s energy?
(e) Graph the electric field produced by a hollow conducting spherical shell of
radius R containing charge Q as a function of the distance r from the center of
the spherical shell. Label the axes.
(f) Explain how the Hall voltage can be used to measure the drift velocity of the
charge carriers in a sample. (Give me an equation and a drawing!)
(g) A coil of wire is rotated in a magnetic field. The ends of the wire from the coil
are connected by a resistor. Does the mechanical torque needed to rotate the
coil depend on the size (in Ohms) of the resistor?
(h) What limits the maximum magnetisation M that can be achieved in a particular
substance?
(15 pts) Question 2
A conducting bar is sliding along frictionless conducting rails. The rails are
connected by a resistance R which is large compared to the resistance of the rails
and of the sliding bar. A uniform B field is oriented perpendicular to the plane
containing the bar and the rails as shown.
(a) Find the magnitude of the induced EMF as a function of the speed v of the
sliding bar.
(b) In what direction (clockwise/counterclockwise) is the induced current?
(c) What force must be supplied to keep the bar moving at speed v?
(d) Show that the power supplied to keep the bar moving (i.e. work per unit time) is
equal to the power dissipated in the resistor.
y
x
v
Binto page
R
L
P216 final exam
Apr. 11, 2000
Page 3 of 4
(15 pts) Question 3
Consider a cylindrical capacitor consisting of an inner conductor of radius a, an
outer conductor or radius b, and a length L >> b. The capacitor is initially
connected to a battery that provides a potential difference V0.
(a) Find the electric field in the region between the conductors.
(b) How much charge is stored on the inner conductor if the material between the
conductors is air?
(c) A material with dielectric constant  is inserted into the gap between the
conductors. What happens to the energy stored in the capacitor? Calculate the
ratio U / Uair .
(d) The capacitor is now disconnected from the battery. The dielectric material is
then removed. Find the potential difference between the plates, written in terms
of the initial voltage V0.
(15 pts) Question 4
J. J. Thomson discovered the electron using a cathode ray tube (we now know that
the particles given off by such a tube are electrons), a velocity selector (crossed E
and B fields) and a fluorescent screen as shown. Assume the beam contains
monoenergetic particles. For simplicity assume w/L<<1 so you can use
y = L tan as the deflection, where  is the exit angle w.r.t. the initial beam when
the particle leaves the region of width w. Further assume the  is small (<0.1) so
that   sin  tan.
Find in terms of B, E, w and L:
(a) the velocity producing no deflection when E is switched on
(b) the deflection at the screen for electrons when E is switched off
(c) Produce a set of values for E, B, w and L that would produce no deflection at
the screen with E on and a 1 cm deflection with E off. Make experimentally
reasonable choices and be sure to satisfy the constraints assumed above on 
and w/L. Note that me = 9.11*10-31kg.
(d) If the particle beam contains a range of velocities, what might one expect to
see with E on and with E off?
particle beam
Eup
screen
Bout
w
L
P216 final exam
Apr. 11, 2000
Page 4 of 4
(15 pts) Question 5
(a) Use Ampère’s Law to determine the magnetic field inside a long solenoid of
circular cross-section. Explain what you are doing. The number of turns per
unit length is n, the current in the windings is I and the cross-sectional area is A.
(b) The solenoid from part (a) is now formed into a square cross-section without
changing the number of turns per unit length and with the perimeter of each
square winding equal to the circumference of the initial circular winding.
Explain why the B field inside the solenoid remains unchanged.
(c) Calculate the inductance per unit length for both cases. Remember that you
were already told that B is the same in both configurations. Express your
answer in terms of the quantities given above and fundamental constants.
(d) Calculate the change in stored energy per unit length in going from circular to
square cross-section.
(15 pts) Question 6
Consider a current loop of radius a carrying current I as shown. Show that the
magnetic field at point P can be written B = -0 / (2d3) where  is the magnetic
moment of the loop and d is the distance from P to the center of the loop. The point
P is in the plane containing the loop, and d>>a. Use the Biot-Savart Law (explain
clearly what you are doing!) and use d>>a to simplify the integrand before
evaluating the integral. It may help to recall the law of cosines for the triangle
shown, namely r2 = d2 + a2 - 2ad cos.
y
x
angle  (between sides of length a and d)
I
a
P
d
```
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