Fundamentals of the Electromagnetic Force
I just finished my second physics course at my University. It was all about electricity, electric fields, magnets, and magnetic fields. There was some content about optics, so that’s for a different article.
First, electric fields. One of the fundamental properties of all particles is charge. Whether the particle actually has charge (proton, electron, etc.) or not (neutron) factors into how the atoms behave. Two opposite charges will attract and two like charges will repel. The particles follow Coulomb’s law, which is F=K*(q1*q2)/r². If you’re thinking it looks a lot like the equation for the force of gravity, you’d be right. Just like the equation for gravity, there’s a constant, two charges multiplied, and you divide by the distance squared. The distance squared means if you go 2x as far away, the force goes down by a factor of 4. K in Coulomb’s equation is Coulomb’s constant and it has a value of 8.99*10⁹. So unlike gravity, the constant is huge instead of really small because the electromagnetic force is so much stronger than gravity. The formula for the strength of the electric field from a point charge is E=K*q/r². Now the force can be written as F=qE.
One of the fundamentals of electric fields is that we always assume we’re working with positive charges. A positive charge by itself will create a field going radially away, or a field going in all directions away from it. A negative charge alone will produce a field that always points toward itself, so you can think of a planet or star, where everything wants to fall towards the body. Likewise, a positive charge will follow the field created. If you put a positive particle in a positive field, the particle will go away from the source. If you put a positive particles in a negative field, the particle will fall toward the source. But a negative charge does the opposite. A negative charge goes against the flow of the field. If you put a negative particle in a positive field, the field says it should go away from the source, but the negative negates that and it will head towards the source. A negative particle in a negative field will go away from the source.
The equation to find the electric potential energy of a particle is really similar to the force. The electric potential energy is U=K*(q1*q2)/r. It’s the exact same as the electric force, but you don’t square the distance. Another common notation is U=qV. This is because V, the electric potential, is V=K*q/r. If you’re thinking I said the same thing, you’re not the first. The electric potential and the electric potential energy are not the same thing. The electric potential is the amount of work needed to move a particle from really far away to the point where it current is. The electric potential energy is the amount of energy that particle needs in order to move against the flow of the field.
You can switch between the electric field and force by just multiplying the field by the charge, and similarly you can get the electric potential energy by multiplying the potential by the charge.
Once we understood the basics of electric fields, in lab we brought the ideas to life. We played around with capacitors (units Farads), which are two charged slabs of metal very close to each other which can act like a battery if it’s charged up. We also experimented with resistors (units Ohms), which just slow down the current inside them. A common lab practice was to find the voltage across and the current through different resistors in different configurations. Different configurations of parallel and series resistors are shown below.
When resistors are in series, you add up their resistances. So if R1 is 5Ω (5 Ohms) and R2 is 5Ω, the total resistance in that circuit is 5+5=10Ω. In parallel, resistance is calculated by R=(1/R1+1/R2)^-1, or 1/R=1/R1+1/R2. In this case, if we keep the same resistances, R=(1/5+1/5)^-1, or R=5/2. However, capacitors do the opposite. In series, they take on the C=(1/C1+1/C2)^-1 equation and in parallel they are simply added.
While the basics of electricity is important to know for appreciating the circuitry of all the devices we use, magnetism was more fun.
To start, a magnetic field is created by charges. Electrons are always moving because they have a property called spin, which is involved in quantum mechanics. This makes things a little easier, because if a charge is not moving, it doesn’t create a magnetic field. The math for that statement is F=q(vxB). B is the symbol for a magnetic field, v is the velocity of the object, and q is the charge. There is also an x in the equation, meaning a cross product. A cross product gives you a 3-dimensional component which you can find easily.
Take your right hand and hold it flat, palm to the left. Curl your middle, ring, and pinky finger in so that you’re pointing at something with your index finger. Now point your thumb straight up. Finally, extend your middle finger in the direction the your palm is facing. This represents the velocity, magnetic field, and force of a system. The way I was taught, your thumb represents the velocity of the body, your index finger is the magnetic field, and your middle finger is the direction of the force.
The other right-hand-rule is for finding the direction of the magnetic field around a wire. Point your thumb in the direction the current is going, and curl the rest of your fingers like you’re holding something. If the point you want to find is above the wire and the current is to the right, you only curl your fingers slightly so that the magnetic field is coming toward you. below the wire you would curl your fingers more so that the field would be going away from you.
Another equation to find the magnetic force is F=ILB, where L is the length of the wire, I is the current going through the wire, and B is the strength of the magnetic field. But to actually find the strength and not just the direction, you use several different equations. Due to there being so many, I won’t get into what they actually are. Both electric and magnetic field change depending on what the object in question is. Electric field strengths change if you want to find it with respect to a plane, a wire, a ring, a disc, and many other different shapes. Magnetic fields have similar circumstances. A common constant for electricity is ε, which is the greek letter epsilon. It has a value of 8.85*10^-12 C²/Nm². That’s a Coulomb squared per Newton-meter squared. While it’s a really weird unit, it works out to be simple in all the equations. The corresponding constant for magnetism is μ (also called the free-space permeability constant), which is the greek letter mu. It’s value is 1.26*10^-6 Tm/A. That unit is a tesla meter per amp. Once again, weird units that work out in the math.
But an interesting experiment we did was that if you have two parallel wires with current flowing through both, the wires will either attract or repel each other. If the current is flowing in the same direction in both (current going upwards or downward in both), they will attract each other. If they are going opposite directions (one going up, the other down), they will repel each other. If you do the right-hand-rule for the magnetic fields and forces, it becomes obvious why that is.
One of the last topics we covered was induced currents and magnetic fields. This one was a little harder to understand. You can change the value of a magnetic field by introducing a new magnetic field into the system (pretty intuitive). Let’s imagine we have a coil of wire wrapped around a tube. Let’s also say this new field adds to the existing field from the wires. This makes the field stronger overall, but the system doesn’t want that. There will now be an induced field that opposes the field you just put in so that the total field is the same. If it helps to think of it mathematically, say your field has a value of 2. You introduce a field of 1, so the induced field is now -1. That means 2+1+-1=2, so the system is good. This induced field, the -1, will generate its own current. Yes, this means you can generate a current by waving a magnet overtop a wire.
There were other topics also covered such as electric and magnetic flux (which ties into induced fields), electron drift velocity, Gauss’ law, power, and resistivity and conductivity, but I won’t get into any of those. Feel free to keep researching those if you’re interested!
