Electric Fields and Electric Potential
Electric Fields
We now understand that charges exert attractive and repulsive forces on one another and that we can quantify these forces using Coulomb’s law. These charges also generate electric fields (E), physical fields that surround charges and exert forces on all other charges within this field.
Electric fields sound nearly identical to the forces that already exist between two charges. So why do we have them? They are important because they don’t require the existence of another charge. Instead, they allow us to predict how one charge will affect another even if the other charge isn’t present.
Electric Field Lines
We represent these potential forces using field lines and by convention, they always point away from positive charges and towards negative ones.
Field lines can be really close together or really far apart. When comparing the strength of one field to another we can measure the density of these field lines in a given area, this is called flux. The greater the flux the stronger the field. Said another way the closer the individual field lines are to one another the stronger the field and vice versa.
With Coulomb’s law in mind, why do the lines get closer together as we get closer to the charge in the picture above?
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Remember Coulomb’s law states that:
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\[F=k\frac{q_1q_2}{r^2}\] Given this as our distance to the charge get’s smaller the force increases markedly. Therefore our field lines should get closer to represent this.
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Hold on a second though, didn’t you just tell me that the other charge doesn’t need to be present in order to determine the field. That is correct and we end up using a different equation when calculating the strength of an electric field. Here we only need to know the charge that is producing the electric field (Q) and at what distance we want to quantify the electric field (r).
[latexpage] \[E=\frac{kQ}{r^2}\]
Using this equation we can see that our prediction using Coulomb’s law was still correct. As we get closer to the charge the electric field strength also goes up. Why is this so? Electric fields are based on the idea of a test charge a theoretical charge that we place into a field to quantify its strength. Since this test charge is the same regardless of the electric field we are evaluating it doesn’t show up in our equation.
Fields Interact
Fields obviously don’t exist in a vacuum and are surrounded by other charges all the time, each of which is producing its own electric field. These fields can interact with one another in some cases amplifying the strength of their neighbors and in other cases diminishing the strength of one another.
Field lines will amplify one another when their lines point in the same direction and will cancel when pointing in opposite directions. This idea is identical to how vectors worked when we were looking at displacement and velocity because field lines are vectors too. So when they point in opposite direction and we align the field lines tip to tail they cancel and vice versa if they point in opposite directions.
Uniform Electric Fields
If we were to line enough of these charges up in just the right way we can get a pattern of vector cancellation and vector amplification that results in a uniform electric field. In uniform electric fields, all of the field lines are equally spaced (termed equidistant) and point in one direction. Therefore, you experience the same electric field regardless of your location. Parallel plate capacitors are the most common way of creating uniform electrical fields, but any setup that is similar to them will also produce a uniform electric field.
When dealing with a uniform electric field our old electric field equation won’t work. This is because the strength of the electric field is equal in every point within the field. Our charge could be 2mm away from one of the plates or 2 cm and it would still feel the same field. What matters now is the overall distance between the two sides of the plates (d) and the voltage difference between them (V), but more on that later.
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\[E=-\frac{\Delta V}{d}\]
The negative sign here, as many sign conventions in physics, indicates directionality. Here it means that a positive charge placed into the parallel plate will be repelled by the top plate.
Electric Potential
When we move charges through an electric field we change the energy of the particle in the same way that moving an object to a higher point increases its potential energy. This occurs because the charge “wants” to go somewhere within the field and the more you “want” to go somewhere the higher your energy and we call this electric potential (V) or electric potential energy (PE).
For instance, if you place a positive charge next to another positive charge they both want to get away from each other due to the repulsion between them. As the charges move away from one another they will lose potential energy as their “desire” to leave decreases with distance. We can understand this better by thinking about the potential energy of springs.
As we compress a spring the amount of energy it stores increases. When we move like-charged items together it is like compressing a spring. So as we move two positive charges closer and closer together they will end up having more and more potential energy. The same would also hold true of two negative charges being brought closer and closer together.
The opposite occurs with negative charges. The closer a negative charge gets to the positive charge the lower its potential because here charges that attract one another act just like regular old potential energy. For attractive charges the further you get two charges away the more they “want” to be close together and thus they have higher potential energy.
Calculating Potential
There are actually two ways of calculating electric potential energies. Deciding which equation to use can be a bit tricky but if we understand what each one quantifies it is a lot easier to choose. The first quantifies the potential energy between to charges just like Coulomb’s Law, which quantified the force between two charges. This value isn’t theoretical like fields so it requires we know the value of two charges.
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Coulomb’s Law:
\[F_e=\frac{kq_1q_2}{r^2}\]
Electric Potential Energy:
\[PE=\frac{kq_1q_2}{r}\]
This is in contrast to the equation for calculating electric potential V which is analogous to the calculation for an electric field. It doesn’t tell us how much energy actually exists between known charges. Instead, it quantifies how much energy could exist if we were to place another charge near it.
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Electric Field:
\[E=\frac{kQ}{r^2}\]
Electric Potential V:
\[V=\frac{kQ}{r}\]
In all cases, the units reflect this difference. Electric potential energy is measured in J, which is a known quantity of energy. In contrast, electric potential V is measured in V or J/C, which is a theoretical amount of energy that depends on what other charge is nearby hence J/C.
Quick Note About Conventions
This might seem a little backward. Don’t things tend to go from high to low potential and not vice versa? They do and in this case, they still are. It doesn’t seem like it because in the world of electrostatics we are dealing with both attraction and repulsion so they are going to differ in behavior. However, we like to make the conventions clear so everything ends up being based on positive charges. So long as you remember why the potential energies are what they are it isn’t a big deal.
This V is the same one that came up in our uniform electric field equation above. There it was a ∆V and it indicated the change in potential energy as a charge traveled from one side of the plate to the other. We will see this idea come up later when we get to circuits. The idea is the same, but we will be dealing with batteries generating voltages in order to power cool stuff like our computers.
In The Nervous System
Now let’s apply these new topics to the nervous system.
As a quick recap, our nervous system uses charges to send messages, in the form of waves of moving charge called action potentials. These action potentials are generated by neurons when specialized channels in the membrane open to allow positive charges to flow into the negatively charged interior of the neuron. These charges then speed down axons, which are insulated by myelin until they reach their destination. Since the charges flowing here are positive we will treat them similarly to current in a wire although this isn’t 100% accurate.