Friday, June 02, 2017

Friday Gen Eds MS13: Electromagnetism

This is the thirteenth post in the math/science part of my "Gen Eds in a Nutshell" series. The Gen Ed series consists of ten subjects you might study in a general education or "liberal arts" core at a university or college. I've already done the subject of philosophy, and I'm on the home stretch through the world history subject on Wednesdays. I'm combining the last two on math and science into one series on Fridays.

Thus far in the math/science subjects:
1. In the winter, I will often touch my wife's arm before I kiss her goodnight, because I have given her a shocking kiss more than once. The reason is that I have built up a static charge somehow. It could be my shoes rubbing against the carpet. When it is warmer, the moisture in the air is greater and such charges disperse more easily. But when it is colder, there is less moisture and the charge is vented on my unsuspecting wife serving as a conductor.

A conductor is a material through which electrons move easily. An insulator is a material through which electrons do not move easily. The flow of electrons through a conductor is called current, and this channeling of electrons is electricity. Benjamin Franklin famously demonstrated in 1752 that lightning was a form of electricity, using a kite in a thunderstorm.

Current, Voltage, Resistance
2. The flow of current is measured by how many electrons are passing any given point in a certain amount of time (technically, how much charge is passing that point). The amount of electrons is measured in coulombs. [1] One coulomb is the charge equivalent of 6.2 x 1018 electrons.

One coulomb of charge moving past a given point in a second is called one ampere. It makes sense that as a current moves through things one by one, the same amount of current goes through each thing, if it doesn't get branched off. Let's say there is a circuit that starts with a power source (like a battery) and goes through a wire and several things (e.g., a light bulb, a heater, etc) but it does so one at a time and never branches off until the wire comes back finally to the power source. We call this a series circuit because the current goes "around" (circuit) through all the elements one by one.

In this case, the amount of "amps" going through each element in that circuit will be the same.

3. The moving charge of electrons always emerges from the power source at its "negative" pole. It thus returns into the source at the "positive" side. Electricity thus flows from negative to positive in a circuit. It is pushed out of the source by an "electromotive force" or EMF. We call the amount of work done by that force on each coulomb of charge a volt. Voltage is thus the work done per coulomb in a circuit.

Unlike current, each element in a series circuit absorbs a certain amount of work. So while every element of a circuit will have the same number of amps, each element in a circuit absorbs some of the work from the electrons. The volts used by each element thus add up to the total voltage at the power source in a series circuit. [2]

4. We might say that each element in the circuit thus provides a certain resistance to the voltage. Without any such resistance, we would have a short circuit and the power source might burn up. The amount of current going through an element is proportional to the amount of work done on that element. That is to say, the more work done on the element, the more current going through it.

This relationship is captured in a law called Ohm's law [3] If E is the voltage and I is the current, then E = I x R, where R is the resistance of an individual element in a circuit. The unit of resistance in electricity is called the ohm and it is the amount of resistance afforded by something that only allows one amp of current to flow through when one volt of force is applied.

So since the current is the same throughout a series circuit and the voltage is the total difference in work done between the poles of a power source, the total resistance in a circuit adds up just like the total voltage does. The total resistance of a series circuit is the sum of all the individual resistances.

5. Power is the amount of work done in a certain amount of time. Electrical power is measured in watts. From every day life, we know that a 100 watt light bulb burns brighter than a 60 watt one. Power equals the voltage times the current, or P = E x I (where E is the voltage and I is the current). So the higher the voltage, the higher the power exerted. The higher the current, the higher the power exerted. And of course if both the voltage and the current are higher, the power exerted is all the higher.

6. I mentioned the rules for series circuits above. As series circuit is one in which there is a single flow of the current. It does not branch at any point. Accordingly, the current is the same throughout. The resistance adds up per element to make up the total resistance. Similarly, the voltage drop across each element adds up to the total voltage in the circuit.

In a parallel circuit, the current branches at some point, perhaps at several points. In this case, it is the voltage that is the same across every branch. Meanwhile, it is the current in each branch that adds up to the total. [4]

The current in each branch depends on the resistance in each branch. As far as the total resistance, it goes down when resistance is added to individual branches, meaning that the total amperage will go up. You can find the current or resistance in any branch using some variation of the E = I x R formula.

Finding the total current or resistance in a parallel circuit is more complicated. If you find the reciprocal (one over x) of the resistance in a branch, then add them together, then take the reciprocal of that sum, you will have the total resistance of the circuit. You can then use the voltage and the total resistance to find the total current.

7. I have assumed that the circuits we have been talking about have been direct current (DC). Direct current is the kind of current that might come from a battery. In a battery, a chemical reaction is used to create the electromotive force that pushes current through a circuit. Direct current is current that is constant, constant in its voltage and amperage unless something interferes with the circuit somehow.

However, the current that comes from a wall plate is alternating current (AC). This is current that alternates between a positive and negative voltage (usually 110 volts), usually at about 60 Hertz or 60 back and forth alternations per second. Another way to think of alternating current is that the flow of current alternates directions.

Alternating current is used because it is more easy to manipulate--for example, to change voltages. It is also easier to generate high voltages in AC rather than in DC. Since high voltage is more suitable for transferring electricity over longer distances, modern societies generate high voltages in AC and then distribute that electricity to local communities where transformers then reduce the voltage and distribute it to houses. Individual devices may then convert the AC to DC so that the device can function.

Magnetism and Induction
8. Most of us know what a magnet is. You can use one to pick up metal. You might touch a screw to a magnet and it will temporarily become magnetized. Then it will stay on your screwdriver more easily. A compass works because the earth has a certain magnetic field. The metal on the compass will turn toward the north pole.

In the early 1800s, a relationship was discovered between electricity and magnetism. When current runs through something, a magnetic field is created around it. Similarly, if a magnet is moved around a conductor, it can generate a current. The rotation of a magnetic field generated by electricity is the cornerstone of the electric motor.

9. Here are some basic facts about magnetism. Magnetic lines form closed loops, the smallest loops possible. Magnetic lines pass through everything, although they can be directed. Like magnetic polarity repels; opposites attract.

10. By coiling a wire around a conductor, you can create an electromagnetic field by running current through the wire. The "left hand rule" expresses how this works. If you picture wrapping your left hand in the way that the wire is wrapped around, then your thumb will point toward what will become the north pole of the artificial magnet (magnetic fields go from north pole out in a closed loop only to re-enter the magnet at its south pole).

By the same token, if you point your left thumb in the direction of the current in a wire (current moves from negative to positive), your fingers will curl in the direction the magnetic field created around the wire moves.

11. Inductance is the property of a circuit that opposes any change of current. When current changes, creating a magnetic field, a counter EMF is created that opposes that change. This is Lenz's Law: "The voltage induced in a circuit by changing current always opposes the change causing it." This is a parallel to Newton's third law. When current is decreasing, inductance wants to maintain it. When current is increasing, inductance wants to resist it.

The unit of inductance is the henry. 1 henry is the amount of inductance that yields one volt when the current is changing at the rate of 1 ampere per second. The symbol for inductance is L.

Induction is the actual creation of a voltage, which requires motion. Inductance does not require current flow. Induction does. Induction is the action of inducing a voltage when current is changing in a circuit.

Charges and Capacitance
12. Coulomb's Law is the analogy to Newton's Law of Gravitation and tells us how to calculate the force of a charge on another charge. The equation looks quite the same!

Fe = k(q1q2)/r2    [5]

Any one charge (e.g., q1) creates a field that travels through space. Any other charge then reacts wit that field, while the initial charge is similarly acted on by the other charge. The main difference between gravity and the electromagnetic force is that the force of gravity only accumulates, while charges are positive and negative and thus can cancel each other out.

13. Capacitance refers to the capacity of two separated plates to retain a certain charge, almost like a battery. Such plates often have some sort of insulator in between them, called a dielectric. The unit of capacitance is a farad. 1 farad is the capacity to store one coulomb of charge when one volt of potential exists across a capacitor.

So C (capacitance) = Q (charge)/E (voltage).

Capacitance is directly proportional to plate area. The bigger the area, the bigger the capacity for storing charge. It is inversely proportional to plate spacing. The bigger the gap, the less the capacity for storing charge.

A vacuum in between the two plates suggests the least capacitance for two plates, so when a vaccuum or air is the "dialectric" between two plates, we say the dialectric constant is 1. Other insulators between the two plates increase capacitance. Pure water, for example, increases capacitance by 81 times!

The equations for calculating capacitance in a circuit are exactly the same as the formulas for resistance and inductance.

Maxwell's Equations
The most fundamental principles of electromagnetism were captured by James Clerk Maxwell (1831-79) in the mid-1800s. In his famous four equations, he captured and built on the work of others before him like Michael Faraday (1791-1867) and Andre-Marie Ampere (1775-1836).

1. Gauss' law for electric fields is named after Carl Friedrich Gauss (1777-1855) and basically says that an electric charge produces an electric field passing through any closed surface around it and that the amount of electric flux passing through that closed surface is proportional to the total charge contained within that surface.

In another form, this law suggests that electric charge is repelled by positive charge and attracted to negative charge.

2. Gauss' law for magnetic fields is closely related to the previous law. It amounts to saying that the total magnetic flux passing through any closed surface is zero.

3. Maxwell's third equation is Faraday's Law, which we have described above but not named. Faraday's law states that a magnetic field passing through a surface induces an electromagnetic force at the boundary of that surface, and a changing magnetic field induces an electric current.

4. Maxwell's fourth equation is the Ampere-Maxwell equation, which we have also encountered above but did not name. An electric current generates a magnetic field.

Next Post: Math/Science 14: Genetics and Evolution

I intent to read through material from a couple books before the next post, so it may take some time to get to the next one, but I will post some review material in the meantime.

[1] After Charles Augustin de Coulomb (1736-1806).

[2] The idea that the voltage difference across each element in a series circuit add up to the total voltage is called "Kirchhoff's law" for voltage.

[3] Named after Georg Ohm (1789-1854).

[4] Kirchhoff's current law.

[5] I have introduced k as a constant to show the similarity with Newton's Law of Gravitation. The constant is usually expressed as 1/4πɛ0, where ɛ0 is the "constant of permittivity of free space," approximately, 8.85 x 10-12 farads/meter.

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