Friday, August 17, 2012

Electromagnetic Units

I have always found electromagnetic units confusing. There are just so many of them, and it is difficult to keep them all straight. Russ Hobbie and I introduce many of these units in the 4th edition of Intermediate Physics for Medicine and Biology. In mechanics, there are three fundamental units: the meter, second, and kilogram. Many other derived units can be expressed in terms of these three, such as: the unit of force, the newton  (after Isaac Newton), equal to a kilogram meter per second squared (N = kg m/s2); the unit of energy, the joule (after James Joule), equal to a kilogram meter squared per second squared (J = kg m2/s2); the unit of power, the watt (after James Watt), equal to a kilogram meter squared per second cubed (W = kg m2/s3); the pascal (after Blaise Pascal), the kilogram per meter per second squared (Pa = kg/(m s2); and (an easy one) the unit of frequency, the hertz (after Heinrich Hertz), which is the reciprocal of a second (Hz = 1/s).

Once electricity is introduced, a fourth unit is needed. Although purists use the ampere (after André-Marie Ampère) leading to the MKSA system of units, I prefer to take that fourth unit to be the unit of charge, the coulomb (after Charles-Augustin de Coulomb), introduced in Chapter 6 of Intermediate Physics for Medicine and Biology. Let’s call my nonstandard system of units the MKSC system. If you insist on using the ampere, just remember that an ampere is a coulomb per second (A = C/s) and you can easily do all the conversions.

Another important unit in electrostatics is the volt (after Alessandro Volta), which is defined in Chapter 6 as a joule per coulomb. Therefore, a volt is a kilogram meter squared per second squared per coulomb (V = kg m2/(s2 C)). The units of electric field are then a volt per meter, but because the unit of force is a newton, equal to a kilogram meter per second squared, we find that a V/m is the same as a N/C. You see, things are already getting a bit complicated.

Next up is the ohm, Ω (after Georg Ohm), also introduced in Chapter 6. From Ohm’s law, a volt must equal an ampere times an ohm. Using the definitions for the volt and ampere given previously, we find that an ohm is actually a fancy way of saying kilogram meter squared per second per coulomb squared (Ω = kg m2/(s C2)). A siemens (after Werner Siemens) is the reciprocal of the ohm (S = s C2/(kg m2)). It used to be known as a “mho” (ohm spelled backwards). Be careful, because the symbol for the siemens, S, and the symbol for the second, s, are the same letter, one uppercase and one lowercase. It’s easy to confuse them.

Yet another electrostatics unit defined in Chapter 6 is the farad (after Michael Faraday), defined as coulomb per volt (F = C/V). The farad is therefore a second squared coulomb squared per kilogram per meter squared (F = s2 C2/(kg m2)). You probably know that the time constant of a resistor-capacitor circuit is given by RC, implying that an ohm times a farad should be equal to second. You can easily verify, using our definition of ohm and farad in the MKSC units, that this is indeed the case.

So far, the units are a bit complicated, but not too bad. Things get worse when magnetism enters the picture. The magnetic field is measured in tesla (after Nikola Tesla), as introduced in Chapter 8. Can we write a tesla in terms of MKSC, or do we need some new unit for magnetism? Well, the force on a moving charge in a magnetic field is equal to the charge times the speed times the magnetic field strength. I will leave it to the reader to show that this equation implies that a tesla is simply shorthand for a kilogram per coulomb per second (T = kg/(C s)).

In Intermediate Physics for Medicine and Biology, we don’t discuss the unit for inductance, the henry (after Joseph Henry). Problem 38 in Chapter 8 implies that a volt is equal to a henry times an ampere per second. Again, I will leave it to the reader to show that the henry is therefore equal to an ohm second, or in other words a kilogram meter squared per coulomb squared (H = kg m2/C2). The time constant in a resistor-inductor circuit is L/R, implying that a henry per ohm should be a second. If we recall that a henry is equal to an ohm second, this result becomes obvious.

Another magnetic unit sometimes used for magnetic flux is the weber (after Wilhelm Weber), defined as a tesla times meter squared, implying that a weber is a kilogram meter squared per coulomb per second (Wb = kg m2/(C s)).

To summarize, below is a table listing all the units we have discussed.

N = kg m/s2
J = kg m2/s2
W = kg m2/s3
Pa = kg/(m s2)
Hz = 1/s
A = C/s
V = kg m2/(s2 C)
Ω = kg m2/(s C2)
S = s C2/(kg m2)
F = s2 C2/(kg m2)
T = kg/(C s)
H = kg m2/C2
Wb = kg m2/(C s)

1 comment:

1. Any idea if/how the number of fundamental constants in the universe relate to the number of fundamental units?