Now that we properly understand Maxwell's equations, we can begin to exploit them for certain tasks. The most obvious and direct application of Maxwell's equations can be found in the field of circuits. A circuit is typically a closed loop of conductive wire that is connected at two points to something that can maintain an electric potential difference between the two points. The device I described that maintains this potential difference is typically a battery or some other power source. The electric potential difference is also known as voltage. If we recall, the electric field \(\textbf{E}\) gives us the force per unit charge at some point. However, this is a vector field. Instead we might want to know what the energy per unit charge is at some point. This is precisely the concept of the electric potential. However, this introduces a new concept of energy, which then raises the question: what is energy?
In this context, energy refers to the ability to do work. Again, we have introduced another foreign concept. This raises the next natural question: what is work? Work is simply the force applied over a displacement. In a mathematical formulation, it can be represented as
Why have we introduced these concepts? Well, anytime we have a force field (electric, gravitational, etc.), the motion of an object under the influence of that field involves, by definition, the work done by the field.
In the electric field example, we can find the force on a charge \(q\) through the relation
Therefore, the work done by the field to move a charge along a displacement \(d\mathbf{\ell}\) is
Think of this work as energy transferred to the particle. Energy is like money in your bank account, and work is like spending or transferring that money. We can connect work to the electric potential difference between two points in space:
For a small displacement in a uniform field, we can substitute and divide by \(q\):
More generally, for non-uniform fields or curved paths, the potential difference is given by an integral:
This shows that the electric field determines how much work is done per unit charge. In other words, it sets the voltage between points in a circuit. Sometimes it is helpful to consider a gravitational example. Imagine we have a hill, and we are determining the effect of gravity on an object. The electric field would be like the slope of the hill at each point, whereas the electric potential would be like the height of the hill at each point.
Again, imagine our electric field. We can also think of the potential at a certain point as follows. Start at the origin, and walk to the point you want to determine the potential of. Each step you add up the vector of the field, and once you get to the point you will have the value of the potential at that point. Every step the vector tells you to go up or down a certain amount, so by the time you reach your point, you will be at the height that you wished to find. This idea of adding up vector contributions is simply an integral and therefore
Since \(\Delta V = V_{\text{final}} - V_{\text{initial}}\), we can write:
This gives the potential difference between two points, which can also be expressed as:
In static electric fields, this integral is path-independent. Since the field is conservative, the potential difference depends only on the start and end points, not the path taken. However, in more complicated scenarios like time-varying electric fields, the potential difference can depend on the path.
Fortunately, in the majority of small-scale circuits, such as those on a breadboard, we can safely use the concept of potential difference between two points. Even if there are slightly changing electric or magnetic fields, the resulting electromagnetic waves have wavelengths much larger than the circuit, so their effect is negligible.
To connect this to circuits: the potential difference across a component “drives” current. The electric field in the wire exerts a force on charges, transferring energy from the source (like a battery) to the device. Think of potential as the “height of a hill” and the electric field as the slope, the charges “roll down” from high to low potential, converting energy into motion or heat.
For example, if a 9V battery is connected across a resistor, each coulomb of charge loses 9J of potential energy as it moves through the resistor, that energy is dissipated as heat.
Finally, note that in AC or high-frequency circuits, this simple ΔV picture is an approximation. At very high frequencies, the full set of Maxwell's equations governs how electric and magnetic fields interact, and voltage becomes a more subtle concept.
Batteries
It's important to have at least a rough understanding of how batteries work. I currently have a limited understanding of chemistry so I will not be able to go into much detail. This discussion will be primarily about the alkaline battery. Now, we start with the idea of an anode and cathode. These are two different types of electrodes. An electrode is a conductor that lets current enter or leave a material (battery for example). The anode is what releases the electrons into the circuit, and is the negative (-) side of a battery. The cathode receives electrons from the circuit, and is the positive (+) side of a battery. An ion is a charged particle (nonzero charge). A cation is a positively charged ion, and an anion is a negatively charged ion. In an alkaline battery, manganese dioxide is used in the cathode, and zinc is used in the anode. Between the two electrodes is an electrolytic fluid (an electrolyte). The electrodes are overall pretty much electrically neutral despite the (+) and (-) distinction. The (+) and (-) distinction is more related to whether it consumes or supplies electrons. Where producing is (-) and consuming is (+). However, as mentioned the zinc in the anode is electrically neutral (30 protons and 30 electrons). Similarly, manganese oxide has 41 protons and electrons (and therefore electrically neutral). Before connecting the battery to a circuit, there is a tiny imbalance of charge. The anode has a slight excess of electrons (therefore slightly negative), and the cathode has a slight lack of electrons (therefore slightly positive). This slight imbalance is tiny and as mentioned the actual anode and cathode are considered relatively neutral, however, it is what establishes the electric field and is the catalyst for current flow. It's really just that their atomic structure has the tendency to lose electrons in the anode, and gain electrons in the cathode. This tendency is called chemical potential (or redox potential).
Now let's discuss the electrolytic fluid in between the anode and cathode. This is a fluid that contains free ions (i.e. cations and anions). Unlike the ions that make up metals and are bound to the rigid lattice, the ions in the electrolytic fluid can move freely. The electrolytic fluid's main role is to keep the battery overall electrically neutral and to disolve the anode as "fuel". It's essentially the battery's balancing system. Before the circuit has been closed, the fluid will already be arranged to maintain electric charge neutrality. Now, the chemical potential mentioned earlier makes the corresponding electrode "want" to gain or lose an electron to reach a lower energy state. In the anode, the atoms want to lose an electron (oxidation). In the cathode, the atoms want to gain an electron (reduction). For example, in the alkaline battery with a zinc anode, oxidation occurs. Zinc atoms lose electrons, which then go into the wire of the circuit. More specifically, zinc undergoes the reaction,
What this says, is that the zinc will lose two electrons, which will flow into the wire, whereas the \(Zn^{2+}\) now positively charged zinc ion will flow into the electrolyte. In other words, the electrolyte dissolves the zinc metal to power the circuit. At the other end, the cathode made of manganese dioxide undergoes reduction reactions, where electrons leaving the wire are "consumed". The formula for this reaction is,
This reaction consumes two electrons for each reaction. Instead of dissolving the cathode, it accumulates its products.
This process is obviously finite and limited by how many atoms you have that can undergo this reaction. Once there are no atoms that can react, the battery is dead. So the battery life can be thought of as the stored up chemical potential of all the atoms that make up the positive and negative side. The ions in the electrolyte arrange themselves to still be neutral with respect to the other ions inside the fluid and to account for the new ions entering the fluid.
Now we need to revisit the slight charge imbalance we mentioned earlier. We have described how the process is fueld once it starts (i.e. by dissolving the zinc anode). But what starts this process? What's the difference between a battery not connected to a circuit and one that is? It turns out that the charge imbalance mentioned earlier is the initial driving factor of current. When the battery is just sitting unconnected, there will be some reactions between the electrolyte and the anode. This releases the aforementioned zinc ion and 2 electrons. After a small scale of these reactions, the newly charged particles tip the electric neutrality to be slightly negatively charged. This new charge density at the surface of the electrode establishes an electric field. However, this will cause the reactions at the surface of the electrolyte and electrode to stop, as the electric field established by negative charge will oppose further electron release. The point at which the tendency for the electrode to react with the electrolyte is exactly countered by the opposing electric field from the surface charge in a disconnected battery is called the open-circuit voltage. It's also the electrochemical equilibrium. The zinc reaction is thermodynamically favorable, because zinc metal is less stable in the presence of the electrolyte than zinc ions are. There is more electrochemistry to consider, but I am not knowledgable enough on the topic. Once we connect the circuit, the built up charge can now be influenced by the electric field. You might question this as the surface charges are themsevles the electric field, so how would they move themselves? Well the electric field isn't established by one charge but rather a charge density near the surface. Electrons close to the wire will experience the most electric force, essentially peeling them off. While this happens more reactions are free to take place, replenshing the lost electrons.
One observation is that the amount of surface charge that can build is essentially the strength of the electric field and therefore the strength of the battery.
One question that I had, was how can the battery be electrically neutral, but have an electric field? Now this might not be a great question but it still made me think. The reason we have an electric field with an overall electrically neutral battery, is because the charge density is what creates electric fields, not total charge. So we have charge density at both ends of the battery, due to very small surface charges between the electrolyte and electrode.
Electricity and the Flow of Electrons
So what exactly is electricity? What happens in a wire connected to a battery? The answer is surprisingly tricky. Electricity is essentially the flow of electrons in a wire, but why do they flow? The main reason is the electric field. A battery maintains a surface charge distribution between the anode and electrolytic fluid, which sets up an electric field that can push electrons through a circuit. However, this isn't the whole story. Once the circuit is closed, the electric field inside the wire itself also becomes important.
Here's the nuance: the electric field along the wire isn't just produced by the battery. Tiny surface charges accumulate along the wire and dynamically adjust to guide electrons along the path of the conductor. These surface charges ensure that the electrons follow the actual geometry of the wire and help maintain a roughly constant current, even if the wire twists or turns. In other words, the wire isn't a passive channel; its own charge distribution is actively shaping the electric field that drives the flow of electrons.
So, while the battery sets the stage by creating a potential difference, the surface charges along the wire play a crucial role in making sure electricity actually flows in the right direction and at a steady rate. This interplay between the battery and the wire's surface charges is what makes the simple idea of “electrons flow because of voltage” more accurate in real-world circuits.
Energy Dissipation and Heat from Currents
Even at room temperature, an electron moves extremely fast, on the order of \(10^6 m/s\). This is known as the Fermi velocity. It's roughly \(1/300\) the speed of light. However this thermal motion is random and therefore the electrons stay in roughly the same area. The wire is already at thermal equilibrium, and therefore this random motion is exactly that equilibrium state of the system. Therefore we don't see any extra energy fluctuations from electrons colliding with the ion lattice (the positively charged atomic cores, consisting of nuclei plus bound electrons). However, the electric field of a battery (plus the electric field mentioned above) causes the entire sea of electrons to start drifting through the metal lattice. This drift speed is extremely slow, on the order of \(\approx 1-2mm/s\), a few millimeters a second. As they drift, the electrons are continuously colliding with the atomic lattice. However, the electric field is doing work on those electrons, and therefore we are no longer in a thermal equilibrium. The strength of the fields determines the force at which the electrons collide with the lattice. As they collide, energy is transferred from the field, to the electrons, and finally to the lattice (i.e. the metal wire). This kinetic energy that the metal wire receives is precisely heat. Therefore we can deduce that to get a hotter wire, we can increase the force at which the electrons collide with the lattice via a stronger electric field and/or increase the drift speed that they are colliding with the lattice. Both are a result of an increase in the strength of the electric field, i.e. more voltage. This will induce more current, which then dissipates more energy as heat. This concept is also how we have things like stoves, heaters, toasters, etc. Note that increasing the voltage only minimally increases the drift speed, but does increase the force substantially. So the energy we see dissipated is mostly as a result of the force of the electric field, not the speed of the electrons.