In the 1870ʼs Sir William Crookes invented a device consisting of a sealed glass tube containing electrodes in a vacuum. He was able to demonstrate that the electrons flowing between the electrodes exerted a pressure on anyobject placed in their path hence showing that these particles had momentum. This same principle is extended to photons of light in the radiometer, an evacuated glass bulb in which black and white vanes are suspended on a pivot. When light strikes the vanes the photons yield their momentum and the vanes spin. Even in the absence of mass (as with photons) the characteristics of classical momentum are observed.

Differentiation of Functions with One Independent Variable

All of the fundamental characteristics described previously are interrelated. James Maxwell set down these relationships in the form of a set of equations known simply as the “Maxwell Wave Equations”. Weʼll consider the wave equation that relates A to B to convey the types of interrelationships that exist between field characteristics and to continue developing a mental model of how electromagnetic fields behave.

The power force delivered by a moving electron is, in classic electrodynamics, considered to be a function of the acceleration of the particle. Acceleration of charges results in the creation of the

electromagnetic field which then radiates away from an antenna. The field, then, carries energy away from the antenna and the rate of change of the field directly relates to the density of the magnetic flux. There is a specific relationship between the vector potential (A) and the magnetic flux density (B).

Compare an accelerating electric charge to a race car at a drag race. The car is going to go from zero to 300 miles per hour in just under 5 seconds. Assume we could measure the distance that the car had traveled (position) at each moment of the race (time). A graph of the distance versus the time during the time that the car was accelerating might look something like Figure 3.9 below.

Figure 3.9 Position Versus Time and the Rate of Change

Copyright 2003 - Joseph Bardwell

Consider the state of the carʼs motion at the point shown on the curve. If a line is drawn through the point tangent to the curve, it represents the speed of the car at that point. The arrow on the line thatʼs tangent to the curve represents the velocity (miles per hour) at the point shown. The slope of the line (∆y/∆x) is the rate in units of distance over time. A mathematician says “Velocity is the first derivative of position with respect to time” and the equation would be written as follows.

Figure 3.10 The Notation for Differentiation

The equation in Figure 3.10 says that there is a function that operates on a variable “k” and that function is going to be differentiated so that for any value of t the rate at which the value of k is changing in the function can be determined. The “dk/dt” notation is not a fraction or ratio: itʼs a symbol that means “differentiate k with respect to t”. Differentiation means, simply, find the slope of the line tangent to the curve of the function for the variable specified as itʼs varying relative to the “respect to” variable.

Given an equation for time and position the first derivative yields velocity (with the magnitude of velocity being speed). Now, speed itself may be changing and that, of course, is called acceleration. If an equation was presented that gave the function relating time and speed then the first derivative of that equation would be acceleration. This would then be the second derivative of the function relating time and position. The mathematician says “Acceleration is the second derivative of position with respect to time.” The second derivative of a function may be indicated

by putting a small number 2 in a superscript position above the “d” in the top part of the symbol, and above the variable in the bottom, as shown to the right. This does not mean that something is raised to the second power. The superscript 2 is simply a way to indicate that the second derivative of the function is being considered.

This mathematical abstraction continues. The change in the rate of acceleration may also be significant in some circumstances. For example, your coffee will spill out of the cup when youʼre driving if the rate of change of acceleration varies too dramatically and you canʼt tilt the cup accurately enough to compensate for the change in velocity. The ISO (International Standards Organization) Vibration and Shock vocabulary (ISO 2041) refers to this metric as “jerk” and itʼs the third derivative of position with respect to time. Trains are expected, for example, to keep jerk to less than 2 meters per second cubed for passenger comfort. The aerospace industry even has a jerkmeter that measures jerk. A superscript 3 with the “d”s in the derivative symbol would indicate that the third derivative is being considered.

And, not stopping with three, thereʼs even a use for the fourth derivative of position with respect to time. This derivative doesnʼt have an official name, but itʼs been called jounce and snap. Jounce, the rate of change of jerk, is taken into consideration in the design of sophisticated devices like the Hubble space telescope. The fifth and six derivatives have been called crackle and pop respectively although these names are not often taken seriously.

Copyright 2003 - Joseph Bardwell