THE UNIVERSITY OF BRITISH COLUMBIA

Physics 108 - SECOND MIDTERM - 11 March 2005

SOLUTIONS

Jess H. Brewer

time: 50 min

1. QUICKIES  [10 marks each - 50 total]
   1. Show with sketches how to combine
      1. two identical capacitors to give an equivalent capacitance half as big as either; ¹
      2. two identical resistors to give an equivalent resistance half as big as either; ²
      3. two identical coils to give an equivalent inductance half as big as either; ³
      4. two identical batteries to give an equivalent voltage half as big as either. ⁴
       
      
   2. Explain why a static magnetic field can't change the kinetic energy of a charged particle. ⁵ 
       
       
      Describe in quantitative detail what happens after the switch is closed at $t=0$ in each of the following circuits, where $C = 0.1$ F, $R = 10 \; \Omega$, $L = 0.01$ H and ${\cal E} = 10$ V.
      (Graphs are fine as long as the axes are labeled quantitatively.)  
      
   3. The capacitor is initially charged to $Q = 1$ C: ⁶
      

       
       
      

   4. ⁷
      

       
       
      

   5. A loop of limp braided wire lies flat on a frictionless table in an elongated oval shape. Suddenly a uniform magnetic field is turned on normal to the table's surface. What happens to the loop? Why? ⁸
      

    

    

2. Ideal Coaxial Cable  [50 marks]

   An idealized coaxial cable consists of a solid cylindrical wire of length $\ell$ and radius $a$ coated with a thin layer of insulating paint and a second thin layer of metal (outside the paint, so that it does not make electrical contact with the inner wire). The thickness of the paint and that of the outer conductor are both negligible compared with $a$, and we shall treat the wire as "long" ($\ell \gg a$) so that "end effects" can be neglected.

   

   A net current $I_0$ flows down the solid central conductor and back (in the opposite direction) along the thin outer conductor. The current density $\vec{\mbox{\boldmath$J$\unboldmath }}$ is uniform over the cross-sectional area of the central conductor and the returning current is uniformly distributed over the surface of the outer conducting shell.

   1. [10 marks]   In what direction is the vector magnetic field $\vec{B}$ inside and outside the cable? (Indicate on the sketch and/or in words.) ⁹
   2. [10 marks]   Calculate the magnetic field strength $B$ as a function of $r$ (the distance from the central axis), $I_0$, $a$ and any fundamental constants, both inside and outside the cable. ¹⁰
   3. [10 marks] Calculate the cable's inductance in terms of $a$ and $\ell$. ¹¹

   4. [10 marks] Describe what would happen in the cable if the battery driving the current
      were suddenly "shorted out" by a superconducting switch. ¹²

   5. Now assume that the thin outer conductor has no resistance but the solid inner conductor has a resistivity $\rho = 10^{-6}$ $\Omega$m. If $a = 1$ mm and $\ell = 2$ m, what is the resistance of the cable? ¹³

   6. [5 marks]   If $I_0 = 0.5$ A, what is the electric field $\vec{\mbox{\boldmath$E$\unboldmath }}$ in the inner conductor? ¹⁴