Chapter 25 Gauss’ Law 第二十五章 高斯定律

A new (mathematical) look at Faraday’s electric field lines Faraday: Gauss: define electric field flux as if E is perpendicular to the surface A.

A new (mathematical) look at Faraday’s electric field lines

Flux of an electric field Gaussian surface is an integration over an enclosed suface. is a surface element with its normal direction pointing outward.

Gauss’ law Proof:

Proof of Gauss’ law The flux within a given solid angle is constant. From Coulomb’s law, Thus, we have

Deriving Coulomb’s law from Gauss’ law Assume that space is isotropic and homogeneous.

A charged isolated conductor Electric field near the outer surface of a conductor:

Applications of Gauss’ law

A uniformly charged sphere

Problem solving guide for Gauss’ law Use the symmetry of the charge distribution to determine the pattern of the field lines. Choose a Gaussian surface for which E is either parallel to or perpendicular to dA. If E is parallel to dA, then the magnitude of E should be constant over this part of the surface. The integral then reduces to a sum over area elements.

Applications of Gauss’ law

Given that the linear charge density of a charged air column is C/m, find the radius of the column.

Applications of Gauss’ law

Earnshaw theorem Earnshaw's theorem states that a collection of point charges cannot be maintained in an equilibrium configuration solely by the electrostatic interaction of the charges. This was first stated by Samual Earnshaw in It is usually referenced to magnetic fields, but originally applied to electrostatic fields, and, in fact, applies to any classical inverse square law force or combination of forces (such as magnetic, electric, and gravitational fields).

A simple proof of Earnshaw theorem This follows from Gauss law. The force acting on an object F(x) (as a function of position) due to a combination of inverse-square law forces (forces deriving from a potential which satisfies Laplace’s equation) will always be divergenceless (  ·F = 0) in free space. What this means is that if the electric (or magnetic, or gravitational) field points inwards towards some point, it will always also point outwards. There are no local minima or maxima of the field in free space, only saddle points.

Home work Question ( 問題 ): 7, 12, 18 Exercise ( 練習題 ): 5, 14, 18 Problem ( 習題 ): 14, 24, 31, 32