PHYSICS 405

Spring 2005

Daniel Finley

Notes on Introduction to Electrodynamics, by David J. Griffiths

Chapter 2: Electrostatics

Many parts are review from freshman physics; however, new material is snuck in here and there, and a review is often good!

Ideas of charge, test charges, and the law of superposition of forces. On the other hand, he does not really discuss the fact that q_₁ may affect q_₂ as well as vice versa, so that one truly must take the limit of the problem as the magnitude of the test charge goes to zero. (This is of course what one really means by the phrase test charge, as opposed to simply something that is making the measurement for us.
Coulomb's Law; superposition law for finite number of point charges, and definition of electric field, E,
- definition of coulomb gives us the constant _₀, related to the square of the speed of light, c² by
  the expression (4 10^-7)_₀c^2 = +1 . (The numerical factor in front of _₀ is often referred to by the name _₀.)
Continuous distributions of charge: throughout a volume, , over a surface, , and along a line, . Done rather well, with good figures. Then some good examples worked out in detail.
These examples are NOT better than the ones that are given in Resnick and Halliday! However, some of the problems are, perhaps, more interesting.
Field lines: very nice comment (p. 65) about the density of field lines as they really are, i.e., in 3 dimensions, as opposed to as they are drawn in the text, in only 2 dimensions! These are the substitute that Faraday invented for the purpose of understanding vector fields over space, which NEED to have some place to be drawn, i.e., to really be drawn in the associated vector space (fiber) over the point in question.
Some good diagrams of field lines for pairs of charges on p. 66, one pair both positive, one pair of opposite signs.
Electric Flux: "the number of field lines passing through a given area," with proper account made of their direction (relative to the normal of the area). One could add here, these days, relative to the comments on areas of spheres at the top of p. 68, some thoughts about how the field must "fall off," if there were more than 3 spatial dimensions. Then we acquire both the integral and differential versions of Gauss' Law. Comments about the use of the Dirac Delta here, near p. 69, would be in order, for both 3-, 2- and 1-dimensional systems.
Footnote on p. 70 is beginning to be some of the very nice comments the text has that are explicitly designed to help build up both geometric and electrostatic intuition!
Zero curl of E---in this static case---and the introduction of the Electric Potential. Very good comments on the idea involved in this on p. 79 and 80. The potential satisfies Poisson's Equation; the solution of Poisson's equation can be written as a (scalar) integral over the charge distribution! Notice that when Poisson's equation is considered in a region with zero charge density it is referred to as Laplace's equation.
Electrostatic Boundary Conditions described in some detail in Section 2.3.5, beginning p. 88.
- Notice that the electric field typically undergoes a discontinuity when one crosses a surface charge. More precisely, a discontinuity in the charge density typically causes a discontinuity in the field.
- discontinuities in the electric field when one passes through a surface charge! p. 88-9, but especially formula (2.33).
  This formula comes from the fact that the curl of the electric field vanishes, which, via Stokes' formula, tells that the value of a line integral about a closed path must vanish. When applied to a region containing a discontinuity in the field, this then tells us that the discontinuity in the (vector) electric field is in the normal direction, from below to above, and equal to the surface charge density, on the surface that divides below from above, divided by _₀.
  Notice that this says that the tangential component of the electric field suffers no discontinuity.
  On the other hand, the electric potential difference, between "above" and "below," is given by the negative of that line integral from above to below, so that the potential is continuous across this interface. (It simply has a "kink" in it, which eventually causes the discontinuity in the field.)
- comment in footnote on p. 89 very helpful
Work and Energy:
- for a point charge distribution,
- for a continuous charge distribution,
- good discussion at the end of the section, on self-energy, energy density, and the superposition principle as it does not apply to energy
  - See comment 2/3 of the page down on p. 92, about the lack of ``self-work."
  - p. 96 introduces the energy density of an electric field, already known, certainly, but an extremely important notion!
- The end of this chapter introduces important concepts for later on, when considering materials in more detail: conductors, and/or insulators, along with the induced charges that may be produced in neutral materials.
  Induced charges move very quickly---a matter of nanoseconds---so that it is fair to treat the problem as if it were an electrostatic one.
  The fact that there is no charge density inside a conductor of course follows from Gauss' Law, and is, again, extremely important!
  This leads us to the notion that the surface of any conductor is an equipotential, and that the exterior electric field is normal to the surface immediately outside that conductor.
  Nice comment about a calculation to show that the energy of a conductor is lower when the charge is all distributed across the exterior surface than when it is uniformly spread out throughout its volume.
  - Also introduces surface charge and external forces on a conductor:
  - Exterior charges cause changes in the distribution of surface charges on a conductor, and, of course, (net) forces on the conductor.
    The force per unit area, f, is given by the surface charge density multiplied by the average of the field, from above and below, in the limit as one approaches that surface.
  - On the other hand, this may be used to calculate the surface charge density as caused by an electric field, giving that density via the formulation for the field of a small patch, assumed small enough to be considered flat.
  - This material is definitely new relative to freshman/sophomore physics.
- It then proceeds to begin the discussion of the use of conductors to create capacitors---that store electrical energy! This section on capacitors is done very well. Good comments on single conductors. More, however, will come in later chapters!

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finley@tagore.phys.unm.edu		Last updated/modified: 16 January, 2005