THE UNIVERSITY OF BRITISH COLUMBIA
 
Science 1 Physics Assignment # 4:
 
Gauss' Law, Capacitance, Resistance & Circuits
 
1 Feb. 1999 - try to finish by 8 Feb. 1999

1.
FIELD WITHIN A UNIFORM CHARGE DISTRIBUTION: You have seen how to use GAUSS' LAW to derive the radial (r) dependence of the electric field E(r>R) outside charge distributions of spherical, cylindrical or planar symmetry, where R is the distance the charge distribution extends from the centre of symmetry - the radius of a charged sphere or cylinder, or half the thickness of an infinite slab of charge, respectively. Use similar arguments to show that, for each of these cases (a sphere, cylinder or a slab of uniform charge density), the electric field E(r<R) inside the charge distribution is given in terms of the field E(R) at the boundary of the charge distribution by $\ds{ E(r<R) = \left( r \over R \right) E(R) . }$

2.
CAPACITOR WITH INSERT: Suppose we have a capacitor made of two large flat parallel plates of the same area A (and the same shape), separated by an air gap of width d. Its capacitance is C. Now we slip another planar conductor of width d/2 (and the same area and shape) between the plates so that it is centred halfway in between. What is the capacitance $C^\prime$ of the new system of three conductors, in terms of the capacitance C of the original pair and the other parameters given? (Neglect ``edge effects'' and any dielectric effect of air.)
\begin{figure}\begin{center}\mbox{ \epsfysize 0.4in \epsfbox{PS/cap_with_cond.ps} } \end{center} \end{figure}

3.
TRIUMF POWER USE: The electromagnet that generates the magnetic field for the world's largest cyclotron at TRIUMF has conductors made of aluminum ( $\rho = 2.8 \times 10^{-8}$ $\Omega$m) wound in a circle of radius 9.5 m. The conductor has a rectangular cross section (2.5 cm $\times$ 42 cm). There are 15 turns in the top half of the magnet and 15 in the bottom half, for a total length of 30 circumferences (the top and bottom coils are connected in series). If we apply 100 V to the coils, what current flows through it? How much power does this require to run?
 

4.
RC CIRCUIT TIME-DEPENDENCE: In the circuit shown, ${\cal E} = 1.2 $ kV, C = 6.5 $\mu$F and R1 = R2 = R3 = R = 0.73 M$\Omega$. With C completely uncharged, switch S is suddenly closed (at t=0).
  (a) Determine the currents through each resistor for t=0 and $t=\infty$.
\begin{figure}\begin{center}\mbox{ \epsfysize 1.333in \epsfbox{PS/29-46.ps} } \end{center} \end{figure}

  (b) Draw a qualitative graph of the potential difference V2 across R2 as a function of time from t=0 and $t=\infty$.
  (c) What are the numerical values of V2 at t=0 and $t=\infty$?
  (d) Give the physical meaning of ``$t=\infty$'' in this case.
  (e) Finally, write down expressions for the currents through R1, R2 and R3 as functions of time, in terms of C and R.


Jess H. Brewer
1999-01-31