Physics 401 Assignment # 1:
Wed. 04 Jan. 2006 - finish by Wed. 11 Jan.
This first assignment is just review, to make sure you haven't forgotten (or can quickly recall) what you learned in PHYS 301/354 (or earlier) about the E&M covered in the first 7 chapters of our textbook: David Griffiths, "Introduction to Electrodynamics".

    1. Starting with Maxwell's equations in differential form, derive Maxwell's equations in integral form.
    2. Starting with Maxwell's generalization of Ampère's Law,   $\diffAmpere$,  derive the continuity equation,  $\ContinuityEq$,  which is the mathematical expression of charge conservation.
    3. Starting with Maxwell's equations in free space ( $\Vec{J} = 0, \; \rho = 0$), show that $\Vec{E}$ and $\Vec{B}$ each satisfy a wave equation. What is the speed of propagation of the resulting wave in each case?

  2. CHARGED CONDUCTORS: Two spherical cavities, of radii a and b, are hollowed out from the interior of a solid neutral conducting sphere of radius R, as shown in the figure. There are charges qa and qb at the centres of the respective cavities.
    $\textstyle \parbox{4.5in}{
\begin{enumerate}\item What is the electric field i . . . 
 . . . $\ and $b$?
\item What are the forces on $q_a$\ and $q_b$?
\end{enumerate}}$       \epsfbox{images/}
      6.   If a third charge qc were brought near the conductor, which (if any) would change:

       (i) $\sigma_a$?

       (ii) $\sigma_b$?

       (ii) $\sigma_R$?

       (iv) The electric fields inside cavities a and b?

       (v) The electric field outside the conductor?

  3. COAXIAL CAPACITOR: A capacitor is constructed of two very long concentric cylindrical conductors with their common axis horizontal, as shown in the diagram. The space between them is exactly half filled with a linear dielectric liquid with dielectric constant $\kappa$.
    1. Show that the electric field is radial and is the same in the dielectric half as in the vacuum half of the capacitor.
    2. Deduce the capacitance per unit length of this coaxial capacitor.
    3. If the conductors carry free charges per unit length $\pm \lambda$, find the polarization $\Vec{P}$ in the dielectric at any point a distance r from the central axis, in terms of $\epsilon_0, \kappa, \lambda$ and r.


    $\textstyle \parbox{4.0in}{
Two very long parallel wires carry equal currents $\ . . . 
 . . . ) is consistent with
that obtained using Amp{\\lq e}re's Law.
\end{enumerate}}$               \epsfbox{images/}

  5. LAPLACE'S EQUATION: Consider an infinitely long metal pipe, of radius R, which is placed at right angles to an otherwise uniform electric field $\Vec{E}_0 = E_0 \Hat{x}$.
    $\textstyle \parbox{3.5in}{\raggedright
\begin{enumerate}\item What is the \lq\lq un . . . 
 . . . =
E_0 r \left({R^2 \over r^2} - 1 \right) \cos \theta }$~.
\end{enumerate}}$         \epsfbox{images/}
    Hint: Note that this situation has cylindrical symmetry (not spherical!), with no z dependence, and hence simplifies to a 2-D plane polar problem.

See   Solutions to Laplace's Equation $\nabla^2 V = 0$

Jess H. Brewer