THE UNIVERSITY OF BRITISH COLUMBIA

Physics 108 Final Exam - 21 April 2005

SOLUTIONS

Jess H. Brewer

time: 3 hours

1. "QUICKIES"   [50 marks - 5 each]
   1. Given only the value of the speed of light,   c = 2.99792458×10⁸ m/s,  and that of the permeability of free space,   $$\mu_{{_0}}^{}$$ = 4$\pi$×10^-7 N/A²,  show how to calculate the value of the permittivity of free space,   $$\epsilon_{{_0}}^{}$$,  in units of C²/N-m². ¹

   2. The current  I  in any shorted coil will exhibit thermal fluctuations. If a 1 H coil is in thermal equilibrium at a temperature of 300 K, what is the root mean square current   I_rms = $\sqrt{{\langle I^2 \rangle}}$  flowing through the coil? ²
      

   3. To which of the laws represented by Maxwell's Equations did Maxwell himself actually make a direct contribution, and what was his contribution? ³

   4. A long straight solenoid of inductance L and resistance R is made by tightly wrapping one layer of copper conductor (square cross section, width w) around a circular spindle of radius r and length $\ell$. If all the dimensions are increased by a factor of 2 (i.e. the picture shown is unchanged but the scale is doubled), the time constant of the coil is . . . ⁴
      

      [encircle the best answer]

      . . . halved. . . . doubled. . . . quartered. . . . quadrupled. . . . unchanged.

      (Note: partial credit is possible only if you explain your answer.)

   5. Some tropical fish have bright red scales without any red pigments. Explain briefly how this is possible. ⁵

   6. A proton is initially at rest in a uniform magnetic field out of the page, as shown. A uniform electric field is then applied in the + y direction, perpendicular to the magnetic field. Describe the subsequent motion of the proton qualitatively, using a simple sketch and a few words.   (Remember, a uniform field is the same everywhere in space.) ⁶
      

   7. A semi-infinite flat sheet of conductor with thickness  d  is viewed "edge-on" at right. The sheet carries a uniform current density   $\vec{{\mbox{\boldmath $J$\unboldmath }}}\,$  into the page. Calculate the magnetic field   $\vec{{\mbox{\boldmath $B$\unboldmath }}}\,$  outside the sheet as a function of   $\vec{{\mbox{\boldmath $J$\unboldmath }}}\,$  and  d.   Explain each step of your reasoning. ⁷
      

   8. An uncharged spherical conducting ball is suspended by an insulating thread. A positive point charge is moved near the ball and held fixed.   The ball will . . .
      1. . . . be attracted to the point charge and swing toward it.
      2. . . . be repelled from the point charge and swing away from it.
      3. . . . not be affected by the point charge.
      [Indicate the best completion of the sentence.] ⁸
      (Note: partial credit is possible only if you explain your answer.)
      
      
      
      
      
      
      

   9. Shown below are four types of calculation problems and four "Laws" of Electricity and Magnetism. Match up (with connecting lines) each problem with the Law best suited to solving it. ⁹
      

   10. An AC power supply produces an oscillating voltage of 15 V amplitude at 60 Hz. It is driving a circuit consisting of a 30 $\Omega$ resistor in series with a 0.2345 F capacitor and a 30 $\mu$H inductance. Show that the circuit is being driven on resonance and then find calculate the maximum current flowing through the resistor. ¹⁰

2. Rationally Calibrated Thermometer   [10 marks] A thermometer using thermal expansion of a liquid in a bulb is calibrated in inverse electron volts (eV^-1). It is pictured at right with the bulb (not shown) in thermal equilibrium with a large heat reservoir.
   
   1. [3 marks] In what sense is this a "rational" thermometer calibration? ¹¹
   2. [3 marks] Which end is the bulb on?   [Explanation required!] ¹²
   3. [4 marks] What is the temperature of the reservoir in degrees Kelvin? ¹³

3. Charge in a Spherical Cavity in a Cylindrical Conductor   [10 marks] A positive charge Q is held fixed at the centre of a hollow spherical cavity in a long, solid cylindrical conductor of radius R and length $\ell$ $\gg$ R. The cavity is not centred on the axis of the cylinder, but it is far from the ends of the cylinder. The conductor itself has zero net charge.
   

   Describe the electric field as a function of position . . .

   1. [3 marks] . . . in the conductor; ¹⁴
   2. [3 marks] . . . inside the spherical cavity; ¹⁵
   3. [4 marks] . . . outside the conductor (r > R, where r is the perpendicular distance from the cylinder's axis), far fom either end of the cylinder. ¹⁶
   Be as complete as possible, with formulae and/or in words, and explain your reasoning.

4. Velocity Selector   [10 marks] A beam of protons with velocity v = 10⁶ m/s is directed along the + x axis between a pair of square metal plates of area A = 1 m² separated by a distance d = 1 cm. The plates have an initial charge of Q = 10^-6 C (the left plate has + Q and the right plate has - Q). A uniform magnetic field B = 0.01 T is applied in the + z direction. At t = 0 the switch S is closed, allowing the charge to flow off the plates through an R = 100 M$\Omega$ resistor. Assume that the magnetic field acts only in the space between the plates, and neglect "edge effects" on the electric field near the edges of the plates.
   
   1. [3 marks] For what value of the electric field E
      between the plates will the protons pass straight
      through without any deflection? ¹⁷
   2. [2 marks] How long will one proton spend inside the velocity selector at that E field? ¹⁸
   3. [5 marks] Calculate the time at which the protons will pass undeflected between the plates. ¹⁹

5. Spinerator   [10 marks] A small metal ball is whirling around in a circle on the end of an  R = 40 cm  long wire attached to a fixed, electrically insulated pivot. The angular frequency of rotation is   $\omega$ = 30 s^-1.  There is a uniform magnetic field  B = 0.15 T  into the page as shown, normal to the plane of rotation.  What is the voltage difference between the metal ball and the central pivot?  Be sure to indicate which end is positive and which negative. ²⁰
   

6. Understanding Gratings   [10 marks] A "diffraction grating" is produced by making  N  parallel scratches on a flat, transparent plate, so that the distance between any two adjacent scratches is  d  and the effective width of the "slit" formed by one scratch is  a.  Light shines on the grating normal to its surface and passes through it to make an interference pattern on a distant screen;  $\vartheta$  is the angle between the incident light and the direction to a given position on the screen.
   1. [2 marks] What is the meaning of the formula   d sin$\vartheta_{m}^{}$ = m$\lambda$ ? ²¹
   2. [2 marks] What is the meaning of the formula   a sin$\vartheta_{1}^{}$ = $\lambda$ ? ²²
   3. [2 marks] What is the meaning of the formula   d sin$\Delta$$\vartheta$ = $\lambda$/N ? ²³
   4. [2 marks] What is the advantage of large  N? ²⁴
   5. [2 marks] How many minima are there between any two adjacent principal maxima? ²⁵