Compare: relative uncertainty.

One must specify whether the value of Avogadro's constant is expressed for a gram-mole or a kilogram-mole. A few books prefer a kilogram-mole. The unit name for a gram-mole is simply mol. The unit name for a kilogram-mole is kmol. When the kilogram-mole is used, Avogadro's constant should be written: 6.02252 x 10²⁶ kmol^-1. The fact that Avogadro's constant has units further convinces us that it is not "merely a number."

Though it seems inconsistent, the SI base unit is the gram-mole. As Mario Iona reminds me, SI is not an MKS system. Some textbooks still prefer to use use the kilogram-mole, or worse, use it and the gram-mole. This affects their quoted values for the universal gas constant and the Faraday Constant.

Is Avogadro's constant just a number? What about those textbooks which say "You could have a mole of stars, grains of sand, or people." In science we do use entities which are just numbers, such as (pi), e, 3, 100, etc. Though these are used in science, their definitions are independent of science. No experiment of science can ever determine their value, except approximately. Avogadro’s constant, however, must be determined experimentally, for example by counting the number of atoms in a crystal. The value of Avogadro's number found in handbooks is an experimentally determined number. You won't discover its value experimentally by counting stars, grains of sand, or people. You find it only by counting atoms or molecules in something of known relative molecular mass. And you won't find it playing any role in any equation or theory about stars, sand, or people.

The reciprocal of Avogadro's constant is numerically equal to the unified atomic mass unit, u, that is, 1/12 the mass of the carbon 12 atom.

1 u = 1.66043 x 10^-27 kg = 1/6.02252 x 10²³ mole^-1.

Capacitors for use in circuits consist of two conductors (plates). We speak of a capacitor as "charged" when it has charge Q on one plate, and -Q on the other. Of course the net charge of the entire object is zero; that is, the charged capacitor hasn't had net charge added to it, but has undergone an internal separation of charge. Unfortunately this process is usually called charging the capacitor, which is misleading because it suggests adding charge to the capacitor. In fact, this process usually consists of moving charge from one plate to the other. The capacity of a single object, say an isolated sphere, is determined by considering the other plate to be an infinite sphere surrounding it. The object is given charge, by moving charge from the infinite sphere, which acts as an infinite charge reservoir ("ground"). The potential of the object is the potential between the object and the infinite sphere.

Capacitance depends only on the geometry of the capacitor's physical structure and the dielectric constant of the material medium in which the capacitor's electric field exists. The size of the capacitor's capacitance is the same whatever the charge and potential (assuming the dielectric constant doesn't change). This is true even if the charge on both plates is reduced to zero, and therefore the capacitor's potential is zero. If a capacitor with charge on its plates has a capacitance of, say, 2 microfarad, then its capacitance is also 2 microfarad when the plates have no charge. This should remind us that C = |Q/V| is not by itself the definition of capacitance, but merely a formula which allows us to relate the capacitance to the charge and potential when the capacitor plates have equal and opposite charge on them.

A common misunderstanding about electrical capacitance is to assume that capacitance represents the maximum amount of charge a capacitor can store. That is misleading because capacitors don't store charge (their total charge being zero) but their plates have equal and opposite charge. It is wrong because the maximum charge one may put on a capacitor plate is determined by the potential at which dielectric breakdown occurs. Compare: capacity.

We probably should avoid the phrase "charged capacitor" or "charging a capacitor". Some have suggested the alternative expression "energizing a capacitor" because the process is one of giving the capacitor electrical potential energy by rearranging charges in it.

cgs. The system of units based upon the fundamental metric units: centimeter, gram and second.

Obviously we could never make measurements on a closed system unless we were in it^†, for no information about it could get out of it! In practice we loosen up the condition a bit, and only insist that there be no interactions with the outside world which would affect those properties of the system which are being studied.

† Besides, when the experimenter is a part of the system, all sorts of other problems arise. This is a dilemma physicists must deal with: the fact that if we take measurements, we are a part of the system, and must be very certain that we carry out experiments so that fact doesn't distort or prejudice the results.

Example: In a closed system, the charge, mass, total energy, linear momentum and angular momentum of the system are conserved. (Relativity theory allows that mass can be converted to energy and vice-versa, so we modify this to say that the mass-energy is conserved.)

Misuse alert. A very common mistake found in textbooks is to speak of "flow of current". Current itself is a flow of charge; what, then, could "flow of current" mean? It is either redundant, misleading, or wrong. This expression should be purged from our vocabulary. Compare a similar mistake: "The velocity moves West."

Example: In the MKSA (meter-kilogram-second-ampere) system of units, length, mass, time and current are the fundamental measurables, symbolically represented by L, M, T, and I. Therefore we say that velocity has the dimensions LT^-1. Energy has the dimensions ML²T^-2.

Misuse alert: Sometimes the word electricity is colloquially misused as if it named a physical quantity, such as "The capacitor stores electricity," or "Electricity in a resistor produces heat." Such usage should be avoided! In all such cases there's available a more specific or precise word, such as "The capacitor stores electrical energy," "The resistor is heated by the electric current," and "The utility company charges me for the electric energy I use." (I am not being charged based on the power, so these companies shouldn't call themselves Power companies. Some already have changed their names to something like "... Energy")

Misuse example: "The earth's auroras—the northern and southern lights—illustrate how energy from the sun travels to our planet." —Science News, 149, June 1, 1996. This sentence blurs understanding of the process by which energetic charged particles from the sun interact with the earth's magnetic field and our atmosphere to result in the aurorae.

Whenever one hears people speaking of "energy fields", "psychic energy", and other expressions treating energy as a "thing" or "substance", you know they aren't talking physics, they are talking moonshine.

In certain quack theories of oriental medicine, such as qi gong (pronounced chee gung) something called qi is believed to circulate through the body on specific, mappable pathways called meridians. This idea pervades the contrived explanations/rationalizations of acupuncture, and the qi is generally translated into English as energy. No one has ever found this so-called "energy", nor confirmed the uniqueness of its meridian pathways, nor verified, through proper double-blind tests, that any therapy or treatment based on the theory actually works. The proponents of qi can't say whether it is a fluid, gas, charge, current, or something else, and their theory requires that it doesn't obey any of the physics of known carriers of energy. But, as soon as we hear someone talking about it as if it were a thing we know they are not talking science, but quackery.

The statement "Energy is a property of a body" needs clarification. As with many things in physics, the size of the energy depends on the coordinate system. A body moving with speed V in one coordinate system has kinetic energy ½mV². The same body has zero kinetic energy in a coordinate system moving along with it at speed V. Since no inertial coordinate system can be considered "special" or "absolute", we shouldn't say "The kinetic energy of the body is ..." but should say "The kinetic energy of the body moving in this reference frame is ..."

The meaning of the the mathematical symbol, "=" depends upon what stands on either side of it. When it stands between vectors it symbolizes that the vectors are equal in both size and direction.

In algebra the equal sign stands between two algebraic expressions and indicates that two expressions are related by a reflexive, symmetric and transitive relation. The mathematical expressions on either side of the "=" sign are mathematically identical and interchangeable in equations.

When the equal sign stands between two mathematical expressions with physical meaning, it means something quite different. In physics we may correctly write 12 inches = 1 foot, but to write 12 = 1 is simply wrong. In the first case, the equation tells us about physically equivalent measurements. It has physical meaning, and the units are an indispensable part of the quantity.

When we write a = dv/dt, we are defining the acceleration in terms of the time rate of change of velocity. One does not verify a definition by experiment. Experiment can, however, show that in certain cases (such as a freely falling body) the acceleration of the body is constant.

The three-lined equal sign, = , is often used to mean "defined equal to". Unfortunately this symbol is not part of the HTML character set, so in this document we use an underlined equal sign instead.

When we write F = ma, we are expressing a relation between measurable quantities, one which holds under specified conditions, qualifications and limitations. There's more to it than the equation. One must, for example, specify that all measurements are made in an inertial frame, for if they aren't, this relation isn't correct as it stands, and must be modified. Many physical laws, including this one, also include definitions. This equation may be considered a definition of force, if m and a are previously defined. But if F was previously defined, this may be taken as a definition of mass. But the fact that this relation can be experimentally tested, and possibly be shown to be false (under certain conditions) demonstrates that it is more than a mere definition.

Additional discussion of these points may be found in Arnold Arons' book A Guide to Introductory Physics Teaching, section 3.23, listed in the references at the end of this document.

Usage note: When reading equations aloud we often say, "F equals m a". This, of course, says that the two things are mathematically equal in equations, and that one may replace the other. It is not saying that F is physically the same thing as ma. Perhaps equations were not meant to be read aloud, for the spoken word does not have the subtleties of meaning necessary for the task. At least we should realize that spoken equations are at best a shorthand approximation to the meaning; a verbal description of the symbols. If we were to try to speak the physical meaning, it would be something like: "Newton's law tells us that the net vector force acting on a body of mass m is mathematically equal to the product of its mass and its vector acceleration." In a textbook, words like that would appear in the text near the equation, at least on the first appearance of the equation.

See: uncertainty

Misuse alert: In elementary lab manuals one often sees: experimental error = |your value - book value| /book value. This should be called the experimental discrepancy. See: discrepancy.

Misuse alert: Be careful that the reader does not confuse this with the colloquial usage: "One factor in the success of this experiment was…"

Misuse alert: Folks who don't pay attention to details of science, are heard to say "Heisenberg showed that you can't be certain about anything." We also hear some folk justifying belief in esp or psychic phenomena by appeal to the Heisenberg principle. This is wrong on several counts. (1) The precision of any measurement is never perfectly certain, and we knew that before Heisenberg. (2) The Heisenberg uncertainty principle tells us we can measure anything with arbitrarily small precision, but in the process some other measurement gets worse. (3) The uncertainties involved here affect only microscopic (atomic and molecular level phenomena) and have no applicability to the macroscopic phenomena of everyday life.

In modern usage, the measure of translational inertia is mass. Newton's first law of motion is sometimes called the "Law of Inertia", a label which adds nothing to the meaning of the first law. Newton's first and second laws together are required for a full description of the consequences of a body's inertia.

The measure of a body's resistance to rotation is its Moment of Inertia.

Two kinds of magnification are useful to describe optical systems and they must not be confused, since they aren't synonymous. Any optical system which produces a real image from a real object is described by its linear magnification. Any system which one looks through to view a virtual image is described by its angular magnification. These have different definitions, and are based on fundamentally different concepts.

Linear Magnification is the ratio of the size of the object to the size of the image.

Angular Magnification is the ratio of the angular size of the object as seen through the instrument to the angular size of the object as seen with the 'naked eye'. The 'naked eye' view is without use of the optical instrument, but under optimal viewing conditions.

Certain 'gotchas' lurk here. What are 'optimal' conditions? Usually this means the conditions in which the object's details can be seen most clearly. For a small object held in the hand, this would be when the object is brought as close as possible and still seen clearly, that it, to the near point of the eye, about 25 cm for normal eyesight. For a distant mountain, one can't bring it close, so when determining the magnification of a telescope, we assume the object is very distant, or at infinity.

And what is the 'optimal' position of the image? For the simple magnifier, in which the magnification depends strongly on the image position, the image is best seen at the near point of the eye, 25 cm. For the telescope, the image size doesn't change much as you fiddle with the focus, so you likely will put the image at infinite distance for relaxed viewing. The microscope is an intermediate case. Always striving for greater resolution, the user may pull the image close, to the near point, even though that doesn't increase its size very much. But usually, users will place the image farther away, at the distance of a meter or two, or even at infinity. But, because the object is very near the focal point, the magnification is only weakly dependent on image position.

Some texts express angular magnification as the ratio of the angles, some express it as the ratio of the tangents of the angles. If all of the angles are small, there's negligible difference between these two definitions. However, if you examine the derivation of the formula these books give for the magnification of a telescope f_o/f_e, you realize that they must have been using the tangents. The tangent form of the definition is the traditionally correct one, the one used in science and industry, for nearly all optical instruments which are designed to produce images which preserve the linear geometry of the object.

Misuse alert: Many books emphasize that the mole is "just a number," a measure of the number of particles in a collection. They say that one can have a mole of any kind of particles, baseballs, atoms, stars, grains of sand, etc. It doesn't have to be molecules. This is misleading.

To say that the mole is "just a number" is simply wrong, from physical, pedagogical, philosophical and historical points of view. There's no physical significance to a mole of stars or a mole of grains of sand, or a mole of people. The physical significance of the mole as a measure of quantity arises only when dealing with physical laws about matter on the molecular scale. The only physical and chemical laws which use the mole are those dealing with gases, or systems behaving like gases.

One dictionary definition of molar is "Pertaining to a body of matter as a whole: contrasted with molecular and atomic." The mole is a measure appropriate for a macroscopic amount of material, as contrasted with a microscopic amount (a few atoms or molecules). See: mole, Avogadro's constant, microscopic, macroscopic.

F is the net (total) force acting on the body of mass m. The individual forces acting on m must be summed vectorially. In the special case where the mass is constant, this becomes F = ma.

Materials are said to be ohmic when V depends linearly on R. Metals are ohmic so long as one holds their temperature constant. But changing the temperature of a metal changes R slightly. Therefore such a device as an electric light bulb increases its temperature as it warms up, which is why it glows slightly brighter for a very brief time just after it is turned on.

For non-ohmic resistors, R is a function of current and the definition R = dV/dI is far more useful. This is sometimes called the dynamic resistance. Solid state devices such as thermistors are non-ohmic, and non-linear. A thermistor's resistance decreases as it warms up, so its dynamic resistance is negative. Tunnel diodes and some electrochemical processes have a complicated I-V curve with a negative resistance region of operation.

The dependence of resistance on current is partly due to the change in the device's temperature with increasing current, but other subtle processes also contribute to change in resistance in solid state devices.

Example: Length is operationally defined by specifying a procedure for subdividing a standard of length into smaller units to make a measuring stick, then laying that stick on the object to be measured, etc....

Very few quantities in physics need to be operationally defined. They are the fundamental quantities, which include length, mass and time. Other quantities are defined from these through mathematical relations.

For refractive surfaces, define the surface radius to be the directed distance from a surface to its center of curvature. Thus a surface convex to the incident light is positive, one concave to the incident light is negative. The surface equation is then n/s + n'/s' = (n'-n)/R where s and s' are the object and image distances, and n and n' the refractive index of the incident and emergent media, respectively.

For mirrors, the equation is usually written 1/s + 1/s' = 2/R = 1/f. A diverging mirror is convex to the incoming light, with negative f. From this fact we conclude that R is also negative. This form of the equation is consistent with that of the lens equation, and the interpretation of sign of focal length is the same also. But violence is done to the definition of R we used above, for refraction. One can say that the mirror folds the length axis at the mirror, so that emergent rays to a real image at the left represent a positive value of s'. We are forced also to declare that the mirror also flips the sign of the surface radius. For reflective surfaces, the radius of curvature is defined to be the directed distance from a surface to its center of curvature, measured with respect to the axis used for the emergent light. With this qualification the convention for the signs of s' and R is the same for mirrors as for refractive surfaces.

In advanced optics courses, a cartesian sign convention is used in which all things to the left of the lens are negative, all those to the right are positive. When this is used, the lens equation must be written 1/p + 1/f = 1/q. (The sign of the 1/p term is opposite that in the other sign convention). This is a particularly meaningful version, for 1/p is the measure of vergence (convergence or divergence) of the rays as they enter the lens, 1/f is the amount the lens changes the vergence, and 1/q is the vergence of the emergent rays.

Pascal 1: The pressure at any point in a liquid exerts force equally in all directions. This means that an infinitessimal surface area placed at that point will experience the same force due to pressure no matter what its orientation.

Pascal 2: When pressure is changed (increased or decreased) at any point in a homogenous, incompressible fluid, all other points experience the same change of pressure.

Except for minor edits and insertion of the words 'homogenous' and 'incompressible', this is the statement of the principle given in John A. Eldridge's textbook College Physics (McGraw-Hill, 1937). Yet over half of the textbooks I've checked, including recent ones, omit the important word 'changed'. Some textbooks add the qualification 'enclosed fluid'. This gives the false impression that the fluid must be in a closed container, which isn't a necessary condition of Pascal's principle at all.

Some of these textbooks do indicate that Pascal's principle applies only to changes in pressure, but do so in the surrounding text, not in the bold, highlighted, and boxed statement of the principle. Students, of course, read the emphasized statement of the principle and not the surrounding text. Few books give any examples of the principle applied to anything other than enclosed liquids. The usual example is the hydraulic press. Too few show that Pascal's principle is derivable in one step from Bernoulli's equation. Therefore students have the false impression that these are independent laws.

Pascal 3. The hydraulic lever. The hydraulic jack is a problem in fluid equilibrium, just as a pulley system is a problem in mechanical equilibrium (no accelerations involved). It's the static situation in which a small force on a small piston balances a large force on a large piston. No change of pressure need be involved here. A constant force on one piston slowly lifts a different piston with a constant force on it. At all times during this process the fluid is in near-equilibrium. This "principle" is no more than an application of the definition of pressure as F/A, the quotient of net force to the area over which the force acts. However, it also uses the principle that pressure in a fluid is uniform throughout the fluid at all points of the same height.

This hydraulic jack lifitng process is done at constant speed. If the two pistons are at different levels, as they usually are in real jacks used for lifting, there's a pressure difference between the two pistons due to height difference (rho)gh. In textbook examples this is generally considered small enough to neglect and may not even be mentioned.

Pascal's own discussion of the principle is not concisely stated and can be misleading if hastily read. See his On the Equilibrium of Liquids, 1663. He inroduces the principle with the example of a piston as part of an enclosed vessel and considers what happens if a force is applied to that piston. He concludes that each portion of the vessel is pressed in proportion to its area. He does mention parenthetically that he is "excluding the weight of the water..., for I am speaking only of the piston's effect."

Related note: Students have the strange idea that results are better when expressed as percents. Some experimental uncertainties must not be expressed as percents. Examples: (1) temperature in Celsius or Fahrenheit measure, (2) index of refraction, (3) dielectric constants. These measurables have arbitrarily chosen ‘fixed points’. Consider a 1 degree uncertainty in a temperature of 99 degrees C. Is the uncertainty 1%? Consider the same error in a measurement of 5 degrees. Is the uncertainty now 20%? Consider how much smaller the percent would be if the temperature were expressed in degrees Kelvin. This shows that percent uncertainty of Celsius and Fahrenheit temperature measurements is meaningless. However, the absolute (Kelvin) temperature scale has a physically meaningful fixed point (absolute zero), rather than an arbitrarily chosen one, and in some situations a percent uncertainty of an absolute temperature is meaningful.

Per is one of those frustrating words in English. The American Heritage Dictionary definition is: "To, for, or by each; for every." Example: "40 cents per gallon." We must put the blame for per unit squarely on the scientists and engineers.

The colloquial meaning of ‘proof’ causes lots of problems in physics discussion and is best avoided. Since mathematics is such an important part of physics, the mathematician’s meaning of proof should be the only one we use. Also, we often ask students in upper level courses to do proofs of certain theorems of mathematical physics, and we are not asking for experimental demonstration!

So, in a laboratory report, we should not say "We proved Newton's law." Rather say, "Today we demonstrated (or verified) the validity of Newton's law in the particular case of…"

Note: Radioactive is least misleading when used as an adjective, not as a noun. It is sometimes used in the noun form as an shortened stand-in for radioactive material, as in the example above.

Misuse alert: Radioactivity is a process, not a thing, and not a substance. It is just as incorrect to say "U-235 emits radioactivity" as it is to say "current flows." A malfunctioning nuclear reactor does not release radioactivity, though it may release radioactive materials into the surrounding environment. A patient being treated by radiation therapy does not absorb radioactivity, but does absorb some of the radiation (alpha, beta, gamma) given off by the radioactive materials being used.

This misuse of the word radioactivity causes many people to incorrectly think of radioactivity as something one can get by being near radioactive materials. There is only one process which behaves anything like that, and it is called artificially induced radioactivity, a process mainly carried out in research laboratories. When some materials are bombarded with protons, neutrons, or other nuclear particles of appropriate energy, their nuclei may be transmuted, creating unstable isotopes which are radioactive.

In physics the comparison is generally made by taking a quotient. Thus speed is defined to be the dx/dt, the ‘time rate of change of position’.

Common misuse: We often hear non-scientists say such things as "The car was going at a high rate of speed." This is redundant at best, since it merely means "The car was moving at high speed." It is the sort of mistake made by people who don't think while they talk.

Misuse alert: One hears some folks with superficial minds say "Einstein showed that everything is relative." In fact, special relativity shows that only certain measurable things are relative, but in a precisely and mathematically specific way, and other things are, not relative, for all observers agree on them.

Confusion can arise with another use of the word, as when one is asked to “Express the result in terms of mass and time.” This means “as a function of mass and time,” obviously it doesn’t mean that mass and time are to be added as terms.

Misuse alert: Do not call an authoritative or ‘book’ value of a physical quantity a theoretical value, as in: "We compared our experimentally determined value of index of refraction with the theoretical value and found they differed by 0.07." The value obtained from index of refraction tables comes not from theory, but from experiment, and therefore should not be called theoretical. The word theoretically suffers the same abuse. Only when a numeric value is a prediction from theory, can one properly refer to it as a "theoretical value".

See: absolute uncertainty and relative uncertainty. Uncertainties are always present; the experimenter’s job is to keep them as small as required for a useful result. We recognize two kinds of uncertainties: indeterminate and determinate. Indeterminate uncertainties are those whose size and sign are unknown, and are sometimes (misleadingly) called random. Determinate uncertainties are those of definite sign, often referring to uncertainties due to instrument miscalibration, bias in reading scales, or some unknown influence on the measurement.

Note: Some dimensionless quantities are assigned unit names, some are not. Specific gravity has no unit name, but density does. Angles are dimensionless, but have unit names: degree, radian, grad. Some quantities which are physically different, and have different unit names, may have the same dimensions, for example, torque and work. Compare: dimensions.

Elementary textbooks almost universally define weight to be "the size of the gravitational force on a body." This would be fine if they would only consistently stick to that definition. But, no, they later speak of weightless astronauts, loss of weight of a body immersed in a liquid, etc. The student who is really thinking about this is confused. Some books then tie themselves in verbal knots trying to explain (and defend) why they use the word inconsistently. Our definition has the virtue of being consistent with all of these uses of the word.

In the special case of a body supported near the earth's surface, where the acceleration due to gravity is g, the weight happens to have size mg. So this definition gives the same size for the weight as the more common definition.

This definition is consistent with the statement: "The astronauts in the orbiting spacecraft were in a weightless condition." This is because they and their spacecraft have the same acceleration, and in their frame of reference (the spacecraft) no force is needed to keep them at the same position relative to their spacecraft. They and their spacecraft are both falling at the same rate. The gravitational force on the astronauts is still mg (though g is about 12% smaller at an altitude of 400 km than it is at the surface of the earth. It is not zero).

This definition is consistent with statements about the "loss of weight" of a body immersed in a liquid (due to the buoyant force). The "weight" meant here is the external force (not counting the buoyant force) required to support the body in equilibrium in the liquid.

This is equivalent to saying that thermal equilibrium obeys a transitive mathematical relation. Since we define equality of temperature as the condition of thermal equilibrium, then this law is necessary for the complete definition of temperature. It ensures that if a thermometer (body B) indicates that body A and C give the same thermometer reading, then they are at the same temperature.