Voltage and Power
Table of Contents
1. Voltage
Consider a battery connected to a pair of conductive plates which generate an electric field across them, as well as a positive charge in between them:
The charge that is placed in this electric field experiences a force that is proportional to the magnitude of its charge and the strength of the electric field.
The key here is that if we move a positive charge \(q\) over a distance \(d\), we do some work:
\begin{align} W &= F\cdot d \notag \\ W &= (q \times E) \cdot d \notag \\ \left[\text{J}\right] &= \left[\text{C} \cdot \frac{\text{V}}{\text{m}} \cdot \text{m}\right] \notag \\ [\text{J}] &= [\text{C} \cdot \text{V}] \notag \end{align}Rearranging this to isolate the voltage, we get:
\begin{align} [\text{V}] = \left[\frac{\text{J}}{\text{C}}\right] \notag \end{align}So, the voltage between two points is the amount of energy needed per unit charge to move a charge from one point to the other, or equivalently the difference in potential energy per unit charge between two points. We denote this potential energy change as \(\text{d}w\), and the charge as \(\text{d}q\), so voltage between two points \(a\) and \(b\) is:
\begin{align} \boxed{V_{ab} = \frac{\text{d}w}{\text{d}q}} \end{align}1.1. Ground
When we talk about voltages across or between two points, it is clear what the change in energy is referring to. But oftentimes we also would like to talk about a voltage at a point. Since voltage is always in reference to something, it is reasonable to ask what voltage we are referencing when we refer to a voltage at a single point.
Commonly, we define a reference voltage at a node in a circuit. This node is known as ground, with the symbol "⏚", and we define the voltage at any point in the circuit with respect to that ground point. Usually, this point is placed next to the negative terminal of the power supply.
1.2. Ohm's Law
Ohm's law is an empiric relationship which accurately models the condutivity for a vast majority of conductive materials.° The following are equivalent:
\begin{align} \boxed{V = IR} \quad \text{or} \quad \boxed{I = \frac{V}{R}} \end{align}Example: Circuit analysis
Using what we know about voltage and Ohm's law, we can do some circuit analysis. Consider the following set of resistors:
We can use Ohm's law to obtain equations for each of the resistors:
\begin{align} I &= \frac{3 \text{V}}{R_1} \notag \\ I &= \frac{2 \text{V}}{R_2} \notag \\ I &= \frac{1 \text{V}}{R_3} \notag \end{align}Note that the current through the resistors are all the same.
2. Power
Power is defined as the rate of change of energy:
\begin{align} P &= \frac{\text{d}w}{\text{d}t} \\ \left[\text{W}\right] &= \left[\frac{J}{s}\right] \notag \end{align}Multiplying by \(\frac{\text{d}q}{\text{d}q}\), we get:
\begin{align} P &= \frac{\text{d}w}{\text{d}t} \cdot \frac{\text{d}q}{\text{d}q} \notag \\ &= \frac{\text{d}w}{\text{d}q} \cdot \frac{\text{d}q}{\text{d}t} \notag \end{align}Recall that \(I = \frac{\text{d}q}{\text{d}t}\) and \(V = \frac{\text{d}w}{\text{d}q}\), so we end up with:
\begin{align} \boxed{P = IV} \end{align}