Ph272- General Physics II (Electricity and Magnetsism)

Lecture 3: Ch. 23 - Electric Field, Gauss's Law

Key Topics

Overview

• Charge density function: A macroscopic charged object contains many, many "point" charges closely spaced. It is more appropriate to approximate this charge arrangement by a continuous distribution. The charge distribution is described by the charge density function, rho(r).
• Calculate electric field of a continuous charge distribution using Coulomb's Law.
• Calculate electric field of a continuous charge distribution using Gauss's Law. (Gauss's Law is more general than Coulomb's Law. Gauss's Law gives the same answer as Coulomb's Law in the electrostatics cases, but Gauss's Law remains valid for electrodynamics).

Charge Density Function

• Definition: We define the charge density function a given location, r, by counting the amount of charge inside a "small" volume around point r and divide the charge by the volume, Eq. 19-1. If we choose the volume small enough the ratio is independent of the size of the volume.

• Total charge: Given the density function, rho(r). The amounf of charge inside a small volume d3r is rho(r)d3r. The total charge is the integral of rho(r)d3r.

Electric Field and Field Lines (Coulomb's Law)

• Superposition: The electric field due to a continous charge distribution can be calulated by breaking the object into small volumes of d3r. The contributions to the electric field from the volumes are "added" together (i.e do an integration), Eq. 19-4.
• Learn the concept: Learn how to set up the integral for various charge distribution. Make sure you have the correct integration variables and correct limits of integration. Don't bother to do the integrals; most of them needs to be look up in integral tables.

We will do one simple example in details in class.

-> Q. What is the expression for the electric field in the regions far away from the charge distribution?

-> Do Problems 19-3,19-6.

• Learn to sketch the electric field lines due to a continuous charge distribution (use symmetry to help you).

Gauss's Law

• Gauss's Law: If we draw a closed surface around a charge, the number of field lines passing through that closed surface is constant (independent of the size of the surface); it depends only on the net charge inside the closed surface. The mathematical statement of this property is Gauss's Law, Eq. 19-19 or 19-22.

-> Do Problem 19-11.

• Electric flux. Do Problem 19-10.
• Use symmetry to simply the surface integration.
• Use Gauss's Law to find the electric field of a symmetric charge distribution. (Spherical, cylindrical, planar). (Next Lecture).

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