# Electrostatic concepts: Keeping them straight

#### Prerequisites

We've now introduced four electrostatic concepts that are closely related but distinct in important ways:

• electric force,
• electric energy,
• electric field, and
• electrostatic potential.

## Electric force

The starting concept is the electric force between two charges. This is given mathematically by Coulomb's law, which says that the magnitude of the the force between two charges is proportional to the product of the magnitudes of the two charges divided by the square of the distance between them. The direction of the forces on the two charges is repulsive if the charges have the same sign, attractive if they have opposite signs. These can both be represented by a vector equation that tells the direction as well as the magnitude (see Reading the content in Coulomb's law):

$$\overrightarrow{F}^E_{Q \rightarrow q} = \frac{k_CqQ}{r^2_{qQ}} \hat{r}_{Q \rightarrow q}$$

## Electric energy

Since the electric force is conservative, we can derive an electrical interaction potential energy using the work-energy theorem:

$$\Delta U^{elec} = -F^{elec}_{||}\Delta r$$

where the subscript "||" means "the part of the electric force in the same direction as the displacement.

## Fields

These two concepts are fine when we have a small number of charges. But when we have a system consisting of a large number of charges, working with individual charges no longer becomes practical. What we do is consider the large number of charges as source charges and consider what their impact is on a single extra charge added to the collection: a test charge. But instead of simply looking at what happens to the test charge at a single location, we look at what the test charge finds for force and energy at every place in space. This yield functions: values at every point in space.  In mathematics, functions of space are referred to as fields (but it really just means a function of position in space).

## Electric field

The electric force on a test charge at each point in space produces a vector field — an arrow at every point in space with magnitude and direction. (A good example of this is a weather map that shows the wind speed and direction at each weather station in the country.) Since we are looking to summarize the effect of the source charges at each point, and since the electric force on a source charge is directly proportional to the charge of the test charge, the force divided by the test charge's charge is independent of the test charge; it depends only on the location of the test charge and on the source charges.

So if we have a test charge $q$ at a point in space labeled $\overrightarrow{r}$, then we define the electric field at the point $\overrightarrow{r}$ by

$$\overrightarrow{E}(\overrightarrow{r}) = \overrightarrow{F_q}/q$$

where $\overrightarrow{F_q}$ is the force felt by the test charge $q$ when it is at the position $\overrightarrow{r}$. It does NOT depend on the sign or the value of the test charge.

## Electric potential

If we now turn to energy, a set of source charges has electric energy of its own. But when we are considering the effect of those source charges on a test charge, we are only interested in the extra electric potential energy that is added to a system when we add a test charge $q$ at a position $\overrightarrow{r}$. The will give us a scalar field (a number — positive or negative) at each position in space: $\Delta U(\overrightarrow{r})$. (A good example of a scalar field is a weather map of the temperature at each weather station in the country. It can be positive or negative, but it does not have a direction.)

Just as the electric force on a test charge $q$ was proportional to $q$, the extra electric potential added when a test charge is added is also proportional to $q$. This means the the quantity

$$V(\overrightarrow{r}) = \Delta U_q/q$$

when a charge $q$ is added at position $\overrightarrow{r}$ is independent of $q$ and only depends on the source charges. This is called the electric potential

Warning: There are two notational conventions that can be confusing here. First, the name "electric potential" (or often just "potential") is dangerously close to "electric potential energy". Be careful! Electric potential is not energy. It needs to be multiplied by the charge of a test charge to turn it into energy. Second: I have put a $\Delta$ on the potential energy, since we are looking at the change in the total energy of the system. But when we look at potential, we are focusing only on a single charge so we don't emphasize the change (the comparison with the total electrical energy) and we leave off the $\Delta$. (A good way to think about $V$ in electrical problems is that it is analogous to $h$ — or really $gh$ — in gravitational problems. In a gravity problem we are focusing on the extra energy added my a mass $m$, $mgh$, and not paying attention to the gravitational energy that holds the earth together.)

## The comparison square: Type and focus

Having these four different quantities can be confusing. The problem is there are two ways that the quantities are different: by character —whether they are vectors or scalars; and by focus —whether they are focusing on individual particles, or whether they are focusing on moving a single (test) particle and seeing its response to many other (source) particles. If we focus on individual particles, the quantities we look at are just (single) values. If we focus on a test particle, the quantities we look at are values at every point in space (fields). These difference are summarized in our "big square" comparison diagram:

The horizontal (purple) arrows connect quantities of the same type: vectors (electric force and E field) vs scalars (electric potential energy and electric potential).

The vertical (blue) arrows connect quantities of the same focus: on individual particles (electric force and electric potential energy) vs focus on electric effects from many charges throughout all space (electric field and electric potential).

## The comparison square: Connecting equations

Each neighboring quantity on the big square comparison diagram is related to its height by a simple equation that summarizes the conceptual connection between them. These are added to the diagram in the figure below:

The upper (vector) concepts are connected to the lower (scalar) concepts through what is basically the work-energy theorem: the change in the scalar energy-like quantity is the negative of the force in the direction of the displacement times the displacement.

The left (value) concepts are related to the right (field) concepts by choosing a test charge, letting it wander all over space to get a field, and dividing out by its charge to get a result that only depends on the source charges. Only the second of these conceptual shifts is displayed in the equations, but the first is critically important to keep in mind!

## The comparison square: Diagrams

Because of the different characters of the four quantities, the way we represent them in diagrams is different. The figure below shows typical diagrams that we might use in constructing and/or thinking about each quantity. (We have left off the labels so as not to overload the image. We hope at this point you can remember what goes on each corner of the square.)

The upper left corner (vector, individual particle focus) is the electric force. These are represented as vectors on each charge coming from each other charge as in a free-body diagram. Remember that each force is part of an interaction between two charges so they come in pairs — equal and opposite forces on the two interacting charges. (See Newton's 3rd law.) The total force on each charge is obtained from adding all the forces on the charge as vectors. (See Adding vectors.)

The lower left corner (scalar, individual particle focus) is the electric potential energy. Each pair of particles contributes an amount to the total energy. It might be positive or negative. The total potential energy of a set of charges is obtained from adding the energy of each pair algebraically as numbers (that is, paying attention to signs).

The upper right corner (vector, all space focus) is the electric field. We represent these by putting an arrow at each point in space representing the electric field that a test charge would measure at that point. The electric field is the collection of all the arrows. The electric field at a point is the value of the arrow found at that point.

The lower right corner (scalar, all space focus) is the electric potential. In 2D, we represent these by drawing lines (equipotentials) that connect all points that have the same value of the potential. These are like the equal height lines in a topographical map. In 3D, these equipotential lines become equipotential surfaces. The electric field vector at any point is always perpendicular to the equipotential line or surface at that point.

One additional potential point of confusion is the fact that when we are introducing the idea of electric field and electric potential, we often consider situations with a small number of charges so we can actually carry out the construction steps. This makes the problems look similar to those we carried out when we were studying the basic force and energy concepts. The difference is that when thinking about electric fields and potential, we select one of our charges to be a "test charge" and use it to measure the effect of the other ("source") charges. Once we have divided our measurements by the amount of the test charge, the resulting electric field and potential map becomes only a property of the source charges. Both the field and potential will be used in situations where there are many source charges (for example, plates of a capacitor, current flow in an electric circuit). In these cases, the source charges drop out (they remain conceptually present and we may refer to them) but we work with the fields rather than the charges.

Joe Redish 3/19/20

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