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Section 1.1 What is voltage?

You've probably seen or heard various voltage measurements in everyday life: a 1.5 volt battery, or a 12-volt charging port, or 120-volt mains power in a wall socket. But what exactly is voltage? To understand this, we're going to back up a bit and examine the electrostatic force that produces voltage.

Subsection 1.1.1 Electrostatic force

The electrostatic force is a fundamental force in nature, somewhat akin to the gravitational force. Whereas gravity is related to mass, the electrostatic force is related to charge. However, unlike mass for gravity, charges can be both positive and negative: Figure 1

This means that the force may either pull the charges together (the way gravity pulls massive objects toward each other) or it may push them apart. A positively-charged particle will repelled by other positive charges and attracted toward negative charges, and negatively-charged particles will likewise be repelled by negative charges and attracted toward positive charges. Figure 2

Now suppose we have an electron (which has a negative charge) experiencing a force toward the right, as shown in the figure below. [question 1: which scenarios could be causing this?]

Since many different charge configurations can have the same effect, it's helpful to stop thinking about the charges themselves and simply quantify the effect they have. The electric field is a vector quantity (direction and magnitude) which describes the force a hypothetical positive charge would experience at each point in space: [ figure 3 showing some scenarios ] To quantify the strength of the electric field independent of the charge, we can divide the force by the amount of charge:

\begin{equation*} E = \frac{F}{q} \end{equation*}

Force is measured in newtons, and electric charge is measured in coulombs 1 , so the electric field is measured in newtons per coloumb. While the electric field is defined in terms of positve charges, the equations work exactly the same for negative charges: since the charge \(q\) is negative the force will be in the opposite direction.

To get an intuitive sense for how fields behave, take a few minutes to play around with this simulation of electric charges and fields 2 .

exercise: try drawing the field lines for this charge configuration TODO Figure 4

Subsection 1.1.2 Defining voltage

Suppose we have an electric field as shown below, and we place a positive charge in that field. TODO Figure 5 Because the charge experiences a force to the right, pushing it to the left will require some physical work. We can quantify exactly how much using traditional mechanics: \(w = \mathbf{f} \cdot \mathbf{x}\) (work equals force times distance). TODO Figure 5 The force will be the strength of the field (denoted as \(E\)) multiplied by the charge (\(q\)), giving

\begin{equation*} w = \mathbf{E}q \cdot \mathbf{x} \end{equation*}

Said another way, the charge has \(w\) joules of potential energy at point A relative to being at point B. Pushing the charge from B to A requires \(w\) joules of energy; letting the charge "fall" from A to B will release \(w\) joules of energy.

But as with the electric field, it would be convenient to quantify this independent of the charge, and simply talk about the potential difference between points A and B rather than the difference in potential energy. We can do this by simply dividing the potential energy difference by the charge to get electrical potential:

\begin{equation*} V = \frac{w}{q} \end{equation*}

Electrical potential is measured in joules per coulomb, which is renamed the volt in honor of Italian physicist Alessandro Volta 3 . More formally, the electric potential between two points can be found by taking the integral of the electric field along a path from one point to the other. 4 

If it helps, you can think of voltage as analogous to height. In the same way that multiplying height by mass gives potential energy, multiplying voltage by charge gives potential energy. Like height, voltage is always a relative measurement between two points. It makes no sense to talk about the potential of a single point; voltage must always be measured relative to some other point.

Subsection 1.1.3 Voltage in a wire

A battery is a chemical charge pump which supplies positive charge to the positive terminal and negative charge to the negative terminal. If we connect some pieces of wire to the battery, the positive end will become positively charged and the negative end will become negatively charged. The charges can flow freely through the wire, so they eventually spread out. [ figure ] If we connect those wires together to make a complete path from the positive to the negative terminal, the positive and negative charges will be attracted toward each other and begin to cancel each other out. The battery continues to produce more positive and negative charges, so that there is eventually an equilibrium. [ figure ] But notice what has happened in the wire: at every point there is a net electric field pointing down the wire. And since we have an electric field, we can calculate the electrical potential of the positive terminal relative to the negative terminal of the battery. For a typical AA alkaline battery, that electrical potential difference is 1.5 Volts, i.e., the "voltage" of the battery.

Checkpoint 1.1.1. Reading check.

What questions do you have?
A full mathematical treatment generally requires vector calculus (aka Calc 3), so we're not going into that here. Go read about it on Wikipedia or wait until you take electromagnetic fields.