| E-M Home Page/ | First Created 2/14/02 |
Unit 1 - Electrostatics |
Observation of phenomena which we now recognize as electromagnetic began with the beginnings of awareness, because almost all the phenomena we encounter in everyday life are electromagnetic in their fundamentals (as understood at the atomic scale). But the study which has become what we call electromagnetism traces its roots to observations made in classical Greek times, or possibly somewhat earlier. Some discussion of these observations, and of steps between them and our modern formulation, is in the page designated History.
The essentials of our modern knowledge are summarized in concepts based
in the work of Ben Franklin:
(a) all electrical effects can be expressed in terms of net amounts
(positive or negative) and locations of a single quantity called
electrical charge;
(b) various amounts of this quantity brought to the same place act on
others as a single quantity given by addition of signed numbers;
(c) net electrical charge cannot be created or destroyed, only moved;
(d) like-sign charges repel, opposite-sign charges attract.
In fact, when one adds the formula for the size of the force, that the force obeys a superposition principle, and the facts of conductors and insulators, one has all the independent facts about electricity that we need for this entire unit. The rest is concepts and techniques that give more-workable patterns to these facts: patterns that make problems workable even when the amounts and locations of the charges are not known.
The first step is the force formula, due substantially to Coulomb. The second step is taken in part to make electromagnetism more manageable, and in part for conceptual simplicity: we introduce the electric field.
I expect you have worked problems with electric (and magnetic) fields; but probably without any time spent on why, nor on what is a field and what other instances you could group with those to increase your comfort level with the concept. There are sections of the Fields page on "What", "Why", and "other examples".
The electric field is defined by its effect in instances (actual or hypothetical) of electric force. Just as necessary, and conceptually quite distict from that effect relationship, are the relationships describing how electric fields are caused.
When the charges are all at rest and their locations are known, it is a simple matter to apply the field definition to the Coulomb's Law force formula and come up with a formula for the electric field due to those charges. A formula, however, is often not the most understandable way to present such a result; often, it is more useful to use electric field lines.
From enough electric field information, presumably one can learn about the charges which produced that field. The standard tool expressing this connection is Gauss's Law, which is useful, depending on the problem, in either integral or differential form.
An approach that simplifies many problems is to express a vector function (such as electric field) in terms of a scalar function. Electricity is an example; the scalar function is called the electric potential.
The differential form of Gauss's Law is
div E = r/eo.
E is related to potential by
E = - grad V.
Put these together and you have
div grad V = del2 V =
- r/eo.
This equation is known as Poisson's Equation; the same name is also used
for any equation whose left side has this "del2 (unknown
function)" form and whose right side is nonzero. The special case when
charge density is zero (throughout a region) is Laplace's Equation;
again, any "del2 (unknown function) = 0" equation is also referred
to by that name. These equations are of both historical and practical
importance in physics; Chapter 3 in Griffiths is about some standard
techniques for finding solutions to these equations. I'm grouping
comments on these in a section on Poisson's and
Laplace's Equations.
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Format for this page adapted with permission from pages
constructed by Dr. Lionel D. Hewett for his course
Modern
Physics 1.