Forces and Units

Something About Units

A force of 1 newton is easy to imagine: 1 N is the weight on earth of a mass of 1/9.81 [kg][s²/m] = 0.102 kg = 0.225 lbm = 3.6 oz. That’s a moderately sized apple!

The atmosphere weighs down on us. Over our heads is a column of air reaching into the far ionosphere that has significant mass since it is integrated over such a large distance (approximately 50 miles) and, when the resultant mass is averaged over a finite elemental area at ground level, would produce a large force. Fortunately, this pressure force acts uniformly in all directions so it’s as much on the bottom of a plate experiencing the weight of the column of air as it is on the topside. Were this not the case, our own atmosphere would crush us.

One standard atmosphere is defined as 1 bar. This is equal to 10⁵ N/m² (a.k.a. 10⁵ pascals after an 18th century French scientist and mathematician). So, you can imagine how big this force would be if it were not canceled by a compensating upward force. You would have to hold a tray on your head, 1 meter by 1 meter supporting 100,000 apples. Good luck!

If you move something against a force you will expend energy and do work. The natural SI unit of work (or of energy, which has the same units) is the joule, which is defined as the amount of work to move 1 N through 1 meter.

1 joule º 1 N-m

One joule of energy is enough to heat a gram of water by about ¼°C.

We also experience magnetic and electrical forces, which tend to be small in most everyday experiences. An example might be the Earth’s natural magnetic field. It’s a puny field of ~20 mtesla normally written as 20 mT. Whatever this unit of measurement actually means, the Earth’s natural magnetic field is pretty small and it causes no obvious forces on a body in the magnetic field unless there is a gradient of this field (by which is meant a change of magnetic field with position). The Magnetic Resonance Imaging (MRI) medical imager works using a gradient field superimposed on an underlying field from 0.5 T to a few T – a lot bigger than the geomagnetic field! In comparison, a powerful hand-held magnet might have field strength of 500 mT.

Even if a uniform magnetic field has no force associated with it, it does have some energy density usually represented as 0.5B²/m₀ per unit volume. In the earth’s field this is only ~0.020 J/m³. In a MRI machine, the magnetic energy is significant. If B = 1 T, the field contains 4 × 10⁵ J/m³. So there’s a lot of energy stored in a MRI machine! You want it to be restrained at all times while you are being scanned!

We usually don’t usually perceive electric forces directly. One situation in which we do is in winter when the air is dry and pieces of fluff stick to your clothing or if you rub a comb through your hair it can pick up a modest electric charge. The electric force is the result of electric charges being present, perhaps due to electrons being knocked off surfaces by tribological (i.e., frictional) action leaving positively charged residues. You get to recognize that these charges are present when, for example, you reach and grasp a grounded (i.e., zero potential) door handle causing the charges to flow a lower energy state; indeed you often notice the spark before you actually touch the handle due to the high local electric fields that spark across between hand and handle.

Electric fields can be quite high – dry air will breakdown and spark at a potential of about 3 kV if separated from a grounded surface by about 1 mm^[1]. The electric field strength is then given by:

The electric energy density associated with this field is ½ e₀E² or 40 J/m³ at dry air breakdown conditions. Much higher breakdown voltages can be sustained by insulating liquids and by solids^[2], perhaps as high as 10⁹ volts/m. Of course, at these much higher electric breakdown fields, the electric energy is appreciable ~ 5 × 10⁶ J/m³. Note that electrical energy density tends to be larger in general magnitude than the magnetic energy density.

There is also a “practical” unit that is common in scientific and engineering literature. The eV (read “electron-volt”) has some real advantages: it is based on the energy gained by a single electron (with an electron charge of 1.6 × 10^-19 coulombs as compared to a lightening bolt with about 1 coulomb) as it falls through a voltage difference of one volt. Thus the eV is directly tied back to the standard “volt” we use to specify everyday objects such as batteries.

By the definition of eV, 1 eV = 1.6 × 10^-19 joules. The electron volt is often a preferred unit of energy at the atomic level, but the SI unit of energy is still the joule or some multiple of it such as “atta” meaning 10^-18. Some conversions are: 1 a(tta)J = 6.24 eV, or one eV = 0.160 aJ).

In The Gas Laws we expressed the Ideal Gas law in a somewhat unfamiliar form as in which “k” is known as Boltzmann’s constant. In SI units, k = 1.38 × 10^-23 J/particle/K and in eV units, k = 8.6175 × 10^-5 eV/particle K.

Many processes at the atomic level emerge in reasonable multiples or fractions of an eV. Thus, at room temperature of ~ 300K, the K.E in an ideal gas particle has about 0.026 eV of (kinetic) energy per particle. A really hot combustion gas, say at 2,000°C, has ~ 0.2 eV per molecule of (kinetic) energy. Conversely a gas temperature of 1 eV is about 11,000 K. (This puts into perspective proposed processes to duplicate the Sun’s energy by fusion; there temperatures need to reach more than 10 keV or more than 100 million K!)