\documentclass[11pt,fleqn]{article}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% template that does not use Revtex4
%%% but allows special fonts
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%
%%% Please use this template.
%%% Edit it using e.g. Notepad
%%% Ignore the header (do not change it)
%%% Process the file in the Latex site
%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Page setup
\topmargin -1.5cm
\oddsidemargin -0.04cm
\evensidemargin -0.04cm
\textwidth 16.59cm
\textheight 24cm
\setlength{\parindent}{0cm}
\setlength{\parskip}{0cm}
% Fonts
\usepackage{latexsym}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{bm}
\usepackage{graphicx}
% Math symbols I
\newcommand{\sinc}{\mbox{sinc}}
\newcommand{\const}{\mbox{const}}
\newcommand{\trc}{\mbox{trace}}
\newcommand{\intt}{\int\!\!\!\!\int }
\newcommand{\ointt}{\int\!\!\!\!\int\!\!\!\!\!\circ\ }
\newcommand{\ar}{\mathsf r}
\newcommand{\im}{\mbox{Im}}
\newcommand{\re}{\mbox{Re}}
% Math symbols II
\newcommand{\eexp}{\mbox{e}^}
\newcommand{\bra}{\left\langle}
\newcommand{\ket}{\right\rangle}
% Mass symbol
\newcommand{\mass}{\mathsf{m}}
\newcommand{\Mass}{\mathsf{M}}
% More math commands
\newcommand{\tbox}[1]{\mbox{\tiny #1}}
\newcommand{\bmsf}[1]{\bm{\mathsf{#1}}}
\newcommand{\amatrix}[1]{\begin{matrix} #1 \end{matrix}}
\newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\eps}{\epsilon}
\newcommand{\fd}[2]{\frac{d #1}{d #2}}
\newcommand{\ffd}[2]{\frac{d^2 #1}{d #2^2}}
\newcommand{\fdl}[2]{(d #1/d #2)}
% Other commands
\newcommand{\hide}[1]{}
\newcommand{\drawline}{\begin{picture}(500,1)\line(1,0){500}\end{picture}}
\newcommand{\bitem}{$\bullet$ \ \ \ }
\newcommand{\Cn}[1]{\begin{center} #1 \end{center}}
\newcommand{\mpg}[2][1.0\hsize]{\begin{minipage}[b]{#1}{#2}\end{minipage}}
\newcommand{\Dn}{\vspace*{3mm}}
% Figures
\newcommand{\putgraph}[2][0.30\hsize]{\includegraphics[width=#1]{#2}}
% heading
\newcommand{\heading}[1]{\begin{center} \Large {#1} \end{center}}
\newcommand{\auname}[1]{\begin{center} \bf Submitted by: #1 \end{center}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% DO NOT EDIT BEFORE THIS LINE
%% START YOUR TEXT FROM HERE
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\heading{Current density}
\auname{Ben Yellin}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% LEAVE THE PART BETWEEN THE COMMENTED LINES
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\bf The problem:}
\Dn
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Current density is given by
$\vec{J}=J\cos^{2}\theta \eexp{-\frac{t}{\tau}}\widehat{r}$
(spherical coordinates).
Find the charge inside the sphere with the radius $R$ as a function of time if
the initial charge was $Q_{0}$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% LEAVE THE PART BETWEEN THE COMMENTED LINES
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Dn
{\bf The solution:}
\Dn
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\begin{eqnarray}
I =
\int\limits_{0}^{2\pi} d\varphi
{\int \limits_{0}^{\pi}} R^2 \sin\theta d\theta J\left( \theta,\varphi\right)
= {\int \limits_{0}^{2\pi}} d\varphi {\int \limits_{0}^{\pi}}
R^2 \sin\theta d\theta J\cos^{2}\theta \eexp{-\frac{t}{\tau}}
= \frac{4\pi JR^{2}}{3}\eexp{-\frac{t}{\tau}}
\end{eqnarray}
%
The current is directed along the positive $\widehat{r}$ axis (away from the
charge inside the sphere).
Therefore,
%
\begin{eqnarray}
\frac{dQ}{dt}=-I
\end{eqnarray}
%
From here
%
\begin{eqnarray}
Q\left( t\right) =Q_{0} - {\int\limits_{0}^{t}}I dt^{\prime}
= Q_{0}+ {\int\limits_{0}^{t}}
\left( -\frac{4\pi JR^{2}}{3}\eexp{-\frac{t^{\prime}}{\tau}}\right) dt^{\prime}
=Q_{0}-\frac{4\pi JR^{2}\tau}{3}\left( 1-\eexp{-\frac{t}{\tau}}\right)
\end{eqnarray}
%
Finally,
%
\begin{eqnarray}
\bigskip\bigskip Q\left( t\right) =Q_{0}-\frac{4\pi JR^{2}\tau}{3}\left(
1-\eexp{-\frac{t}{\tau}}\right)
\end{eqnarray}
%
\end{document}