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\Large\centerline{\textbf{Finding Potentials From Fields}}\normalsize
\medskip

\begin{enumerate}[1)]

\item Consider the vector field in rectangular coordinates:
$$\vec{E} = \frac{q}{4 \pi \epsilon_{0}} [(2 x y^3z+z)\hat{x} + (3x^2 y^2 z) \hat{y}+(x^2 y^3+x)\hat{z}]$$

\begin{enumerate}[a)]
\item Have each member of your group take one of the components of this electric field and use it to find as much information as possible about the potential from which the field might come.

\vfill
\item Compare the answers of your group members. Does this field come from a potential? If it does, what is the potential? Is there any freedom in your answer?

\vfill
\item Could this vector field be the electric field of a static collection of charges? If it could be,
can you say what static collection of charges would produce this field?
\vfill

\end{enumerate}

\newpage

\item Consider the vector field in rectangular coordinates:
$$\vec{E} = \frac{q}{4 \pi \epsilon_{0}} (-y \hat{x} + x \hat{y})$$

\begin{enumerate}[a)]
\item Have each member of your group take one of the components of this electric field and use it to find as much information as possible about the potential from which the field might come.

\vfill
\item Compare the answers of your group members. Does this field come from a potential? If it does, what is the potential? Is there any freedom in your answer?

\vfill
\item Could this vector field be the electric field of a static collection of charges? If it could be,
can you say what static collection of charges would produce this field?
\vfill

\end{enumerate}

\newpage
\item Consider the vector field in cylindrical coordinates:
$$
\vec{E}=\frac{q}{4 \pi \epsilon_{0}} \left(\frac{r \hat{r}}{(r^{2} + z^{2})^{3/2}} + \frac{z \hat{z}}{(r^{2} + z^{2})^{3/2}}\right)
$$

\begin{enumerate}[a)]
\item Have each member of your group take one of the components of this electric field and use
it to find as much information as possible about the potential from which the field might
come.

\vfill
\item Compare the answers of your group members. Does this field come from a potential? If it
does, what is the potential? Is there any freedom in your answer?

\vfill
\item Could this vector field be the electric field of a static collection of charges? If it could be,
can you say what static collection of charges would give this field?

\vfill

\end{enumerate}

\newpage

\item Consider the vector field in cylindrical coordinates:
$$
\vec{E}=\frac{q}{4 \pi \epsilon_{0}} \left(r \hat{\phi}\right)
$$

\begin{enumerate}[a)]
\item Have each member of your group take one of the components of this electric field and use it to find as much information as possible about the potential from which the field might come.

\vfill
\item Compare the answers of your group members. Does this field come from a potential? If it does, what is the potential? Is there any freedom in your answer?

\vfill
\item Could this vector field be the electric field of a static collection of charges? If it could be, can you say what static collection of charges would give this field?

\end{enumerate}

\vfill
\item Sketch the vector fields for each of the problems. Compare the vector field from problem 2
with the vector field from problem 4. What do they have in common?

\end{enumerate}
\vfill
\leftline{\textit{by Corinne Manogue and Katherine Meyer}}
\leftline{\copyright 2011 Corinne Manogue}
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