THE UNIVERSITY OF BRITISH COLUMBIA

Physics 108 FIRST MIDTERM - 4 February 2005

SOLUTIONS

Jess H. Brewer

time: 50 min

1. QUICKIES  [10 marks each - 60 total]
   1. Under what circumstances would the entropy of a system decrease with the addition of energy, and what could you say about the temperature of such a system? ¹

   2. Charges of $+2Q$ and $-Q$ are located in the plane of the page as shown. Sketch the region in the same plane (if any) where the resultant electric field is zero. ²
       
      
      ${\displaystyle {+2Q \atop \bullet}}$ ${\displaystyle {-Q \atop \bullet}}$ ${\displaystyle {\Downarrow \atop \bullet}}$    
      
       
      

   3. In broad general terms, explain why the thermal distribution of particle speeds is not the same in a 1-dimensional ideal gas as it is in a 3-dimensional ideal gas of the same particles at the same temperature. ³

   4. A positive point charge $Q$ is fixed at an arbitrary location (not on the axis) inside an uncharged, thin-walled copper tube whose length $L$ is much larger than its radius $R$. The charge is not located near either end. Define $r$ as the perpendicular distance from the axis of the tube. Match up all the left and right side phrases that make up true sentences: ⁴
       
      $\parbox{2.75in}{\raggedright ~\\ The electric field inside the tube ($r<R$) \\ ~\\ The electric field outside the tube ($r>R$) }$ $\parbox{3.25in}{\raggedright is zero. \\ is a complicated functio . . . . . . the ends. \\ is in the $\hat{\mbox{\boldmath$r$\unboldmath}}$\ direction. }$
      

   5. The diagram shows an edge-on view of an electrically neutral, semi-infinite, flat conducting slab with a parallel sheet of uniformly distributed positive charge (charge per unit area $+\sigma_\circ$) on the left and a parallel sheet of uniformly distributed negative charge (charge per unit area $-\sigma_\circ$) on the right. What is the direction and magnitude of the electric field . . .
      1. . . . to the left of the positive sheet of charge? ⁵
      2. . . . between the positive sheet of charge and the slab? ⁶
      3. . . . inside the slab? ⁷
      4. . . . between the slab and the negative sheet of charge?
      5. . . . to the right of the negative sheet of charge?
      

   6. Referring to the previous diagram, calculate the induced surface charge $\sigma$ per unit area on each side of the slab in terms of $\sigma_\circ$. ⁸

2. CHARGED COAXIAL CONDUCTORS  [40 marks]
   A long copper cylinder of radius $a$ is surrounded by a coaxial copper tube whose inner radius is $b$, as shown. The inner cylinder carries a uniform charge per unit length ($\lambda$) and the outer shell has an equal and opposite charge per unit length ($-\lambda$) so that the system as a whole is electrically neutral.
   
   1. [5 marks] If $r$ is the distance from the axis, what is the electric field for $r < a$?   Explain. ⁹
   2. [5 marks] What is the electric field $\vec{\mbox{\boldmath$E$\unboldmath }}(r)$ for $r > b$?   Explain. ¹⁰
   3. [10 marks] What is the electric field $\vec{\mbox{\boldmath$E$\unboldmath }}(r)$ between the two cylinders $(a < r < b)$? ¹¹
   Now consider the case where $a = 1$ m, $b = 1.01$ m and $\lambda = +10^{-10}$ C/m. Since $(b - a) \ll a$, you can treat the electric field between the cylinders as approximately constant in magnitude. The 1 cm gap between the inner cylinder and the outer tube is evacuated except for 100 tiny beads, each of which contains a single excess electron fixed at its centre so that its net charge is $-e$. The beads stick to the copper surfaces, but are occasionally shaken loose by thermal motion. The whole system is in thermal equilibrium at 300 K.
   1. [5 marks] What is the difference $\varepsilon = U(b) - U(a)$ between the electrostatic potential energy $U(b)$ of a bead stuck to the surface of the outer shell and that of a bead stuck to the surface of the inner cylinder, $U(a)$? ¹²
   2. [15 marks] On average, how many beads are stuck to each surface? ¹³