Chapter 25 Gauss’ Law

第二十五章 高斯定律

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

Faraday:NEA

Gauss: define electric field flux as

E EA if E is perpendicular to the surface A.

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

ˆcosE EA E An

Flux of an electric field

Gaussian surface

is an integration over an enclosed suface.

is a surface element with its normal direction pointing outward.

AdE

Ad

Gauss’ law

Proof:

0encqAdE

Proof of Gauss’ law

The flux within a given solid angle is constant.

22 1

21 2

E rE r

1 22 21 2

cosA Ar r

From Coulomb’s law,

Thus, we have 1 2

Deriving Coulomb’s law from Gauss’ law

Assume that space is isotropic and homogeneous.

A charged isolated conductor

0

ˆE n

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

0

2ElrlE

02E

r

Applications of Gauss’ law

Applications of Gauss’ law

Applications of Gauss’ law

Applications of Gauss’ law

Applications of Gauss’ law

Applications of Gauss’ lawGiven that the linear charge density of a charged air column is -10-3 C/m, find the radius of the column.

Applications of Gauss’ law

Applications of Gauss’ law

Applications of Gauss’ law

Earnshaw theoremEarnshaw'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 1842. 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