Section 1.1 Review from Introductory Physics
Electrical devices are powered by electricity, the movement of charges (which we call current) through a circuit. These charges move due to voltage differences applied by batteries or power supplies. Thus, if we intend to understand the behavior of electrical circuits, we must first remember what we learned in our introductory physics course.
In the presence of charged objects, the space around the objects is altered. This alteration can be represented as a vector field which we call the electric field (which has SI units of V/m). The electric field at a position \(\vec{r}\) relative to a point charge source \(Q\) is given by
\begin{equation*}
\vec{E}=\frac{Q}{4\pi\varepsilon_0 r^2}\hat{r}
\end{equation*}
where \(\varepsilon_0\) is the permitivity of free space (a constant of nature). The electric field has SI units of V/m. While the electric field is a useful construct, it can be convenient to represent this space modification using a scalar field called the electric potential (with SI units of Volts). For a point charge \(Q\text{,}\) the electric potential some distance \(r\) away from the source charge is
\begin{equation*}
V(\vec{r})=\frac{Q}{4\pi\varepsilon_0 r}\text{.}
\end{equation*}
When discussing circuits, this scalar field is often called the voltage. The relationship linking these two representations is described by
\begin{equation*}
\vec{E}=-\nabla V = -\left[\frac{\partial V}{\partial x}\hat{x} + \frac{\partial V}{\partial y}\hat{y} +
\frac{\partial V}{\partial z}\hat{z}\right]\text{.}
\end{equation*}
While both representations have their uses, voltage will be much more commonly used and referenced when analyzing the behavior of electrical circuits.
In circuits, voltage differences lead to electric fields that cause charges to move. Current quantifies the movement of charge through circuit components. More precisely, current is the amount of charge passing by some point in a circuit per unit time, or
\begin{equation*}
I=\frac{dQ}{dt}\text{.}
\end{equation*}
In many materials (e.g. metals), positive charges are largely immobile since they are associated with the nuclei of atoms in a material. Thus, electrons are typically the charge carrier in circuits, constituting the current. Unfortunately, there was still much unknown about atomic theory when initial experiments on electricity were conducted, with the consequence that the direction of current is defined to be in the direction that positive charge carriers would be moving if they were responsible for the current. This means that current often has a direction that is opposite to the direction in which the actual charge carriers are moving.
