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\centerline{\bf\LARGE Gauss Law Activity}

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\section{Infinitely Long Cylinder} 

Consider the cylinder drawn below (except, consider it to be
infinite!) and an observer of the cylinder at a given location
$x_0,y_0,z_0$.  (Note that this location can also be written in
terms of an angle $\theta_0 = \tan^{-1} ({y_0}/{x_0})$ and a
radius $r_0= \sqrt{x_0^2 + y_0^2}$ and the $z_0$ variable.)

\psfig{figure=cylinder.eps,height=2.6in,width=5.3in}

\begin{enumerate}
\item If you close your eyes and the cylinder is moved along
the $x$-axis, would you notice a difference when you open your
eyes?  (e.g. is it closer, further, at a different angle)?
\vfill

\item If you close your eyes and the cylinder is moved along
the $y$-axis, would you notice a difference when you open your
eyes?  (e.g. is it closer, further, at a different angle)?
\vfill
\item If you close your eyes and the cylinder is moved along
the $z$-axis, would you notice a difference when you open your
eyes?  
\vfill
\item If you close your eyes and the cylinder is rotated (that
is, $\theta$ is changed, would you notice a difference when you
open your eyes?  
\vfill
\item If you close your eyes and the cylinder is moved radially
outward (along any angle), would you notice a difference when
you open your eyes?  (e.g. closer, further, at a different
angle)?
\vfill
  
\clearpage

\item We define a symmetry variable as any of the
cylinder variables that, when changed, does not change
the appearance of the cylinder.  What are the symmetry
variables for the infinite cylinder?
\vfill
\item We define all the other variables as non-symmetry
variables.  What are the non-symmetry variables for this
cylinder?
\vfill


\item Consider two test charges at the following locations $$z=5
{\rm cm}, r=2 {\rm cm},
\theta = 0^o \quad {\rm and} \quad z=5 {\rm cm}, r=2 {\rm
cm}, \theta = 90^o$$  How do the electric fields compare at
those two locations?  Explain.


\vfill

\item Consider two test charges at the following locations:
$$z=5{\rm cm}, r=2 {\rm cm},\theta = 0^o \quad {\rm and} \quad
z=5 {\rm cm}, r=20 {\rm cm}, \theta = 0^o$$  How do the
electric fields compare at those two locations?  Explain.

\vfill
\item In general, would you expect the magnitude of the electric
field  to change as symmetry variables changed?  as
non-symmetry variables change?
Explain.


\vfill

\item  Make a guess of the direction of the electric field for
the cylinder.  Sketch your guess on the drawing on the previous
page, and illustrate it in 3 dimensions by using the styrofoam
and wires.

\item If you change a symmetry variable (e.g. rotate the
cylinder or move it along the $z$ axis) does the direction of
the electric field change?  Should it change?  Explain.
\vfill

\clearpage

\item Based on the work on the last page, describe an imaginary
surface (it can be curved, but it must be a surface and not a
line) on which the cylinder's electric field is constant in
magnitude. You may find it useful to use a sheet of paper to
investigate and describe possible surfaces.
\vfill
\item What
is the angle between $\vec E$ and $\vec A$ on this surface?
\vfill

\item Describe an imaginary surface through which there
is no flux due to the cylinder's electric field.  (Hint:  when
can the electric field and the area be non-zero, and yet there
is no flux).  

\vfill

\item A useful Gaussian surface is closed surface for which we
know that $\vec E$ is constant and the angle between $\vec E$
and $\vec A$ is constant, or on which the flux is zero, or a
combination of the two.  Verify that a round cookie tin
surface, concentric with the cylinder, is a useful Gaussian
surface for the infinite cylinder.

\vfill
\end{enumerate}

\section {Sphere}  Now that you are familiar with the ideas, we
will repeat the same questions (only much more quickly) for a
sphere and a cube.

\begin{enumerate}
\item What are the symmetry variables for a sphere?
\vfill
\item What imaginary surface has $\vec E_{\rm sphere}=$
constant?
\vfill
\item What imaginary surface has a zero flux due to the
sphere's electric field?
\vfill
\item What is a useful gaussian surface for a sphere?
\vfill
\end{enumerate}

\clearpage

\section{Cube}

\psfig{figure=cube.eps,height=2.6in,width=3.4in}


\begin{enumerate}


\item What are the symmetry variables for a cube?
\vfill
\item What imaginary surface has $\vec E_{\rm cube}=$
constant?
\vfill
\item What imaginary surface has a zero flux due to the
cube's electric field?
\vfill
\item What is a useful gaussian surface for a cube?
\vfill
\end{enumerate}









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