UNIVERSITY OF BRITISH COLUMBIA

Science 1 Physics

Sessional Examination

11 April 2000

Time: 2${1\over2}$ hours

Instructors: Jess H. Brewer & Domingo Louis-Martinez

1.
``QUICKIES''   [40 marks - 5 each]

(a)
To which of the laws represented by Maxwell's Equations did Maxwell actually make a direct contribution? Explain.

(b)
What is the difference between phase velocity and group velocity? Explain.

(c)
How do tropical fish and birds achieve brilliant colours without using pigments? Explain.

(d)
A certain whimsical physicist is fond of saying, ``Particles hate to be confined!'' What is he talking about? Explain.

(e)
In a hydrogen atom, the electrostatic force between the proton and electron is $2.3 \times 10^{39}$ times greater than the gravitational force. If we can adjust the distance between the two particles, at what separation will the electrostatic and gravitational forces between them be equal? Explain.

(f)
A dielectric is inserted between the plates of a capacitor, which is then charged. The dielectric is later removed. What qualitative effect (if any) does the removal of the dielectric have on the electrostatic energy stored in the capacitor? Explain.

(g)
Encircle any of the following circuits in which the current might exhibit simple exponential decay without any oscillations.
\begin{figure}\epsfysize 1.25in \epsfbox{PS/more_circuits.ps} \end{figure}

(h)
A diffraction grating is uniformly illuminated, producing the interference pattern shown below on a distant screen.
 
i) How many slits are illuminated?
\begin{figure}\epsfxsize 3.0in \epsfbox{PS/3slit_grating.ps} \end{figure}
ii) What is the ratio of the width of each slit to the spacing between adjacent slits?

2.
DHM   [10 marks] A 1 kg mass attached to an ideal massless spring experiences a viscous drag force Fd = - b v where b = 10 N-s/m and v is the velocity of the mass.
(a)
[3 marks] Write down the differential equation governing the motion of the mass, in terms of the mass m, the spring constant k and the drag coefficient b.

(b)
[4 marks] Solve this differential equation to find a general form for the position x of the mass as a function of time t, again in terms of the above constants. (If you don't remember how to solve this kind of DE, try a good guess.)

(c)
[3 marks] For what range of values of the spring constant k will the mass oscillate when the spring is stretched and released?

3.
Charged Ring   [10 marks]
\begin{figure}\epsfysize 1.0in \epsfbox{PS/ringofcharge.ps} \end{figure}
A net charge Q is uniformly distributed around the perimeter of a circular ring of radius R. What is the difference in electrostatic potential between a point at the geometrical centre of the ring and a point on its axis a distance z from the centre?

4.
Do $\vec{B}$   [10 marks]
\begin{figure}\epsfxsize 2.5in \epsfbox{PS/cyl_wire.ps} \end{figure}
A wire (i.e. a long, straight, solid cylindrical conductor) of radius R carries a steady current $I_\circ$ that is uniformly distributed over the cross-sectional area of the wire.
 
Calculate the magnetic field $\vec{B}$ at a distance r from the centre of the wire in both regions: $r \ge R$ and r < R.

5.
Rail Gun in Reverse   [10 marks]
\begin{figure}\epsfxsize 1.667in \epsfbox{PS/revrailgun.ps} \end{figure}
A vertical bar of length $\ell = 50$ cm is pulled to the right at a constant speed v = 4.20 m/s through a constant, uniform magnetic field B = 0.675 T normal to the plane of the circuit. Assuming that the resistance of the bar and rails is negligible compared to that of the resistor ( $R = 12 \; \Omega$), find the power dissipated in the resistor.

6.
Double Slit Plus   [10 marks] A double-slit interference experiment is performed in vacuum. Then a thin glass plate with index of refraction n = 1.56 is placed in front of one slit. The fifth interference maximum now appears at the angle where the second interference maximum previously appeared. If the incident light has a wavelength of 480 nm, how thick is the plate?

7.
YOUR CHOICE   [10 marks]   Answer ONE of the following two questions:

(a)
Hot Charge: In the middle of a large space habitat (where gravity can be neglected), a large flat sheet of insulating material is given a uniform positive charge of 10-12 C/m2. A small glass bead containing a single excess electron (which we will assume is ``stuck'' in the bead) is placed next to the charged sheet and allowed to reach thermal equilibrium with the ambient air at T=300 K. Assume that the air is perfectly still (no air currents).
i.
[4 marks] If P(h) is the probability of finding the bead a distance h away from the plate, how does this probability vary with h?
ii.
[6 marks] At what h is P(h) = P(0)/e? (Here e is the base of the natural logarithm.)

(b)
Star Dreck: Captain Picard places the starship Enterprise in a circular orbit about a 1-Solar-mass neutron star (radius approximately 9 km) at what seems a safe distance of 1000 km from its centre.
i.
[3 marks] What is the centripetal acceleration g of the centre of gravity of the Enterprise?
ii.
[3 marks] Show that the rate of change of the centripetal acceleration with the distance r from the centre of the star is given by   ${\displaystyle {dg \over dr} \; = \; - 2 \; {g \over r} }$.
iii.
[4 marks] If Picard's head is 1.8 m further from the star than his feet, what is the ``tidal'' difference between the gravitational acceleration of his head and that of his feet? Comment.


Jess H. Brewer
2000-04-12