Here’s my crash course of chapter 2. I claim you could do the whole problem set after you read this.

V \(\rightarrow\) E

\[\vec{E} = -\nabla V\]

E \(\rightarrow\) F

\[\vec{F} = q\vec{E}\]

F \(\rightarrow\) W

\[W = \int \vec{F} \cdot d\vec{l}\]

V \(\rightarrow\) W

\[W = q V\]

or if you have a collection of charges then…

\[W = \frac{1}{2}\int \rho V d\tau\]

(That’s same formula as the one above it. The \(\frac{1}{2}\) is annoying but it’s just because of double-counting.)

E \(\rightarrow\) W

This is also just the formula above translated into \(\vec{E}\) using Gauss’ Law.

\[W = \frac{\epsilon_0}{2} \int E^2 d\tau\]

Gauss’s law, \(\rho\) \(\rightarrow\) E

\[\oint \vec{E} \cdot d\vec{a} = \frac{1}{\epsilon_0} Q_{\rm encl}\]

E \(\rightarrow \rho\) (Gauss’ law backwards)

\[\rho = \epsilon_0 \nabla \cdot \vec{E}\]