\documentclass{article}
% Call the style package
\usepackage{fun3style}
\title{Fundamentals of Physics 3 Homework Template}
\author{Erin De Pree, St.~Mary's College of Maryland}
\date{Fall 2019}
\begin{document}
\maketitle
\section{Introduction}
You can leave the automatic numbering for sections as is or you can suppress the numbering by using \verb+\section*+ instead of \verb+\section+.
\section*{Problem 2.7}
Find the electric field a distance $z$ from the center of a sphere of uniform surface charge $\sigma$ with radius $R$.
\paragraph{Solution}
The surface element of the sphere is
\begin{align}
\dd a^\prime &= \qty( R \,\dd\theta^\prime )
\qty( R \sin\theta^\prime \,\dd\phi^\prime )
\end{align}
\begin{align*}
\vec{r} &= z \unitvec{z} \\
\vec{r}^\prime &= R \unitr \\
\brcurs &= \vec{r} - \vec{r}^\prime = z \unitz - R \unitr
\end{align*}
Let's start by breaking $\unitr$ into Cartestian coordinates:
\begin{align}
\unitr &= \cos\theta \unitz + \sin\theta \qty(\cos\phi \unitx + \sin\phi \unity)
\end{align}
\begin{align}
\brcurs &= \vec{r} - \vec{r}^\prime \nonumber \\
&= z\unitz - R \qty[ \cos\theta \unitz + \sin\theta \qty(\cos\phi\unitx + \sin\phi\unity)] \nonumber \\
&= - R \sin\theta \cos\phi \unitx - R \sin\theta \sin\phi \unity + \qty( z - R \cos\theta ) \unitz
\end{align}
Next, we use the law of cosines to the magnitude of $\brcurs$.
\begin{align}
%\rcurs^2 &= \abs{r}^2 + \abs{r^\prime}^2 - 2 \abs{r} \abs{r^\prime} \cos\theta \nonumber \\
%\rcurs^2 &= z^2 + R^2 - 2zR\cos\theta
\end{align}
\begin{align*}
\vec{E} &= \int \frac{1}{4\pi\epsilon_0} \frac{\hrcurs}{\rcurs^2} \sigma \,\dd a^\prime \\
&= \frac{\sigma}{4\pi\epsilon_0} \int \frac{\brcurs}{\rcurs^3} R^2\sin\theta \,\dd\theta \,\dd\phi
\end{align*}
\end{document}
