Transcript Chapter 25 Gauss` Law

Chapter 25 Gauss’ Law
第二十五章 高斯定律
A new (mathematical) look at
Faraday’s electric field lines
Faraday:
N
E
A
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
 E  EA cos  E  Anˆ
Flux of an electric field
Gaussian surface
 
   E  dA
an integration over an
 isenclosed
suface.

dA
is a surface element with
its normal direction
pointing outward.
Gauss’ law
  qenc
   E  dA 
0
Proof:
Proof of Gauss’ law
From Coulomb’s law,
E2 r12
 2
E1 r2
The flux within a given solid angle is constant.
A1 A2 cos 
 2 
r1
r22
Thus, we have 1   2
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:

E  nˆ
0
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
l
E  2 rlE 
0

E
2 r 0
Applications of Gauss’ law
Applications of Gauss’ law
Applications of Gauss’ law
Applications of Gauss’ law
Applications of Gauss’ law
Applications of Gauss’ law
Given 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 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
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