# Coulomb Force

With the study of electricity, we begin a qualitatively different phase of our study of physics. Up to now, we have for the most part dealt with topics which are macroscopic, and of which we have some intuitive appreciation. For the next few chapters, we will be concerned with the microscopic world, which is largely hidden from our senses and our common sense.

So far, most of the forces, pressures and stresses we have studied have been communicated by direct contact: I lean on the wall and the wall pushes on me (otherwise I would fall through it!). The sole exception has been gravity, which we have unquestioningly accepted as a given part of our experimental environment. As we will see below, there is a more causal way to deal with gravity, and it is intimately related to the way which we treat electricity.

The basic problem which electricity presents to our intuition is: how can the electric force act at a distance, without direct contact? When you pet your cat on a dry day, the fur will rise to meet your hand, defying gravity with no more explanation than "static". What actually happens is a transfer of "electric charge" from the cat's fur to your hand. Since there are two types of electric charge, which we call "positive" and "negative", we explain the behavior of the cat's fur by saying that negative charge has been rubbed off on to your hand, leaving the cat positively charged and your hand negatively charged. And as we all have heard, opposites attract!

At the heart of this simple phenomenon is one of the most universally applicable ideas in physics: action at a distance is caused by charges, which are the sources of forces. In the case of the electric "Coulomb" force, the magnitude is proportional to the product of the charges, and inversely proportional to the square of the distance between them. Like all forces, it is a vector quantity, whose components can be written

F x = q 1 q 2 R x / 4 p e r 3,

or, in scalar form,

F = q 1 q 2 / 4 p e r 2.

Here, q 1 and q 2 are the magnitudes of the charges, R is a vector which points from one to the other, and r is the distance between them. Since R / r is a unit vector pointing from one charge to the other, we see that the vector equation has the same magnitude as the scalar equation; the unit vector factor gives F a direction, but does not change its magnitude.

Note that the force can be either attractive, when the charges have opposite sign, or repulsive, when the charges have like sign. Since R points from one charge to the other, when the product of the charges is positive, the force which one charge exerts on the other is directed away from the first charge, and so the other charge is repulsed. If the product of the charges is negative, the force one charge exerts on the other is directed toward the first charge, and they are attracted. Think of the force as "between the charges", since both charges feel the same force, either repulsive or attractive, relative to each other. Often (but not always!) we will assume that the two charges are equal in magnitude (|q 1| = |q 2|).

The magnitudes of the charges are measured in "Coulombs" (abbreviated C), which is a new fundamental unit for us. The proportionality constant e is called the "electrical permittivity" of the medium through which the force acts. It is a measure of the "effectiveness" with which the electrical force is felt across the medium. It is often written as the product of k and e 0, where e 0 = 8.85 * 10 - 12 C 2 / N m 2 is the electrical permittivity of the vacuum , and k is the "dielectric constant" of the medium (1 for the vacuum, 1.00059 for air, 80 for water at 20 degrees C).

While the cat and your hand are certainly macroscopic objects, physicists have identified the two objects which account for the vast majority of freely moving electrical charge in the universe as microscopic "electrons" and "protons". For the present, we will simply consider them as VERY small "particles": the electron is negative (by convention ) and the proton is positive. The mass of the electron is 9.109 * 10 - 31 kg, the mass of the proton is 1.673 * 10 - 27 kg, and the magnitude of their electrical charge is

e = 1.6 * 10 - 19 C.

All free electric charge appears to be "quantized" in units of e; free charges less than e have never been observed, and all charges are integer multiples of e. Why this is so, and why e has the value it has, are two of the fundamental mysteries of the universe; and two of the most interesting problems in physics! Coming down to Earth, the charge lost by your cat was therefore a LARGE number of electrons. In addition to electrons and protons, charge is also carried by "ionized" atoms and molecules, especially in biological systems. These "ions" are formed by the addition or removal of electrons; their protons are tightly bound in their nuclei, as we will see in Chapter 7. Often we will express the macroscopic charges observed experimentally using a "number" variable to count the number of particles or ions involved as

q = n e.

The force of gravity is very similar to the Coulomb force; the charges are now the masses of the objects which are gravitationally attracted, and the constant of proportionality ("Newton's Constant", equal to 6.673 x 10 - 11 N m 2 / kg 2) is now in the numerator:

F x = - G m 1 m 2 R x / r 3

or

F = - G m 1 m 2 / r 2.

Note that gravitational forces are always attractive, since mass is always positive. Note also that both the Coulomb and gravitational forces are vectors: therefore the superposition of two or more forces is equal to their linear sum.

We can now see how to compute the value of g, the "constant" acceleration due to gravity. The force due to a spherical source is equivalent to a point source at its center. This can be seen using the field model in the next section. Approximating the earth by a sphere, we set m 2 equal to the earth's mass (m E) and r to its average radius (r E). The acceleration g is then G m E / r E 2, or approximately 9.8 m / s 2!

The notion of mass as gravitational charge is perhaps the best "theoretical" notion of mass we have. Note that this idea of mass is qualitatively different from the idea of "inertial mass": that quantity which makes it difficult to change the velocity of an object. That these two quantities, gravitational charge and inertial mass, are equal, is another of the fundamental mysteries of physics.

The next section is about the electric potential.

If you have stumbled on this page, and the equations look funny (or you just want to know where you are!), see the College Physics for Students of Biology and Chemistry home page.

©1996, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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