8.02 ESG Independent Study

Unit 2: Gauss’s Law

Together, Coulomb’s law, the vector nature of electric forces, and the superposition principle allow us to calculate (in principle) the resultant electric field from any charge distribution. The 1/r^{2} dependence in Coulomb’s law allows an elegant way to relate the electric field to the charge by use of the electric flux. Mathematically, Coulomb’s law and Gauss’s law are equivalent, but which is most readily applicable depends on the circumstances. The field due to a uniformly charged sphere is found immediately from Gauss’s law, whereas use of Coulomb’s law requires a few pages of algebra and nerves of steel. Conversely, Gauss’s law is not readily usable for finding the field due to a dipole (although it can be done with some sneaky mathematics), while Coulomb’s law and a bit of simple calculus give the result quickly.

The concept of flux of a vector through an area is a needed and in some sense intuitive one. The flow of particles across a surface is easily quantified in terms of an integral, and a similar flow of charged particles is a current. In later chapters, we will relate various fields with various fluxes, but for now we’ll deal with fields due to stationary charges and the accompanying fluxes.

Objectives: After completing this unit, you should be able to calculate electric fields from specified charge distributions or find the charge distribution that corresponds to a given electric field, and calculate electric fluxes in simple situations.

Suggested Procedure:

1. Read in UP11, chapter 22, and do problems 2, 11a, 17, 32abc, 36, 39, 40, 45, 57, 63, 66 (optional; the physics is not hard, but the algebra gets involved.), or

2. Read the rest of Purcell, chapter 1 (again, the sections involving electrostatic energy will be used in later units) and do problems 3, 15, 18, 21, 22, 27 and 31 (pp 34–38).

3. Take a unit test.