Post on 24-Dec-2015
21. Gauss’s Law
“The Prince of Mathematics”Carl Friedrich Gauss(1777 – 1855)
Wikemedia Commons
2
Topics
Electric Field Lines Electric Flux Gauss’s Law Using Gauss’s Law Gauss’s Law and Conductors
Electric Field Lines
4
Electric Field Lines
A field lineshows the direction ofthe electricforce on apositivepoint charge
5
Electric Field Lines
By using a convention for thenumber of linesper unit charge, one can use field lines to indicate the strength of an electric field.
Electric Flux
7
Electric Flux
Flux is the “flow” of any quantity through a surface.
For example, it could be sunlight through a window, or water through a hole.
In particular, it can be electric field through a surface
8
The electric flux through a small surfaceelement A is defined by
E A
where
ˆA An
The unit vector is normal to the surface element
9
Surface
i ii
E A
E dA
This is an example of a surface integral
The total electric flux through a surface is the sum of the individual fluxes
10
Electric Flux – A Closed Surface
Let’s compute the flux through a spherical surface about a point charge:
+
2
2 22
ˆ
4
4
ˆ ndAq
k rr
q qk
E dA
kr rk
r
q
dA
n̂E
We see that the flux is proportionalto the enclosed charge
Gauss’s Law
12
Gauss’s Law
The electric flux through any closed surface is proportional to the net charge enclosed by the surface:
which is usually written as
4E dA kq
+ -
- -+
+ +
+
enclosed
0
qE dA
+ where 0 = (4k)-1 = 8.85 x 10-12 C2/Nm2
is called the permittivity constant
13
Gauss’s Law
Gauss’s law is always true for any closed surface. However, it is most useful when the
charge distribution and the enclosing surface have a high degree of
symmetry.+ -
- -+
+ +
+
enclosed
0
qE dA
+
a
Using Gauss’s Law
15
A Uniformly Charged Sphere
The enclosed charge is Q. Gauss’s law is
0
QE dA
Because of the spherical symmetry of the chargedistribution, we can infer that the magnitude of the electric field is constant on any spherical surface enclosing the charge
16
A Uniformly Charged Sphere
The symmetry makes is easy to evaluate thesurface integral
2
0
2 20
4
4
QE dA E dA E R
Q QE k
R R
The electric field of a spherically symmetric charge distribution is like thatof a point charge
17
A Uniformly Charged Sphere
3r
q QR
We can apply Gauss’s law within the sphereby drawing a Gaussian surface of radius r.
The charge enclosed within this surface is
32
0 0
3 30
4
4
q Q rE dA E r
R
Q QE r k r
R R
therefore,
18
A Uniformly Charged Sphere
Within the sphere the field varieslinearly with radius
Outside, the fieldlooks like that ofa point charge
19
A Hollow Spherical Shell
The shell contains a net charge Q distributed uniformlyover its surface. Because of the sphericalsymmetry, the field outside the shell is like that of a point charge.
But the field inside is zero!Why? Because the field from Acancels that from B.
20
Recap
By convention, the electric field points away from a positive charge and towards a negative charge.
Electric field lines can be used to visualize an electric field. By convention, the number of field lines is proportional to the charge.
21
Recap
Electric field is additive: the field at any point is the vector sum of the electric fields of all charges.
Gauss’s law: the net electric flux through any closed surface is proportional to the net enclosed charge:
enclosed
0Closed Surface
qE dA
22
An Infinite Line of Charge
By symmetry, the electric field is radial.
Therefore, a suitable Gaussian surface is a
cylinder of length L, radius r
placed symmetrically
about the line charge.
The enclosed charge
is q = L, where is the
charge per unit length
23
An Infinite Line of Charge
From Gauss’s law, we deduce that the electric field of a long (strictly infinite) line charge is
24
A Sheet of Charge
E+
E-
For an infinite sheet of charge the field is perpendicular to the sheet. The flux through a cylindrical Gaussian surface is
EA + EA.
The enclosed charge is
q = A,
where is the charge per unit area. Therefore, the field is
02E
A
25
A Charged Disk
The electric field at a point P along the axis of a disk is closely related to the field of a sheet
of charge:
2 2
2 2
3/ 22 2
2 2
cosdq
kx r
dq
dE
x
x rk
x r
xdqk
x r
P
dq
r
x E
2 2k 0dis
12
xE dE
x R
Gauss’s Law & Conductors
27
Conductors
An applied electric field causes
the free positive and negative
charges to separate until the field
they create exactly cancels the
applied field, at which point the
charge migration stops. The
conductor is then in electrostatic
equilibrium.
28
Charged Conductors
Since like charges repel, all
excess charge must reside on the
surface of a conductor. This
is also consistent with the fact
that, in equilibrium, the
electric field within a conductor
is zero.
29
A Hollow Conductor
A conductor carries a net
charge of 1 C and has
a 2 C charge in the
internal cavity.
The charges must distribute
themselves as shown in order to be
consistent with Gauss’s law.
30
A Hollow Spherical Conductor
Consider a neutral spherical
conductor in equilibrium with a
cavity containing a net charge +q.
1. The charge on the inner surface of the cavity is –q. Why?
2. The charge on the outer surface of the conductor must therefore be +q. Why?
3. And this charge is uniformly distributed. Why?
+qE = 0
31
Field at a Conductor Surface
E+
E- = 0 (inside conductor)
The flux through a cylindrical Gaussian surface is just EA since the field inside the conductor is zero, in equilibrium. The enclosed charge
is q = A, therefore, the field at the surface of a charged conductor is
0
E
Applications
33
Electric Shielding
The tendency of conductors to exclude electric fields from their bulk has many applications. For example:
1. Co-axial cables
2. Lightning Safety
3. Sensitive Compartmented Information Facility (SCIF)
34
Co-axial Cables
Co-axial cables connect, for example, iPods to ear-phones.
If the electric
fields are too strong,
the dielectric
can suffer
dielectric breakdownWikemedia Commons
35
Co-axial Cables and Dielectrics
Some molecules, like H2O, have permanent dipole moments. Others can be distorted by an electric field, and become dipolar; that is, acquire induced dipole moments. These materials are called dielectrics
36
Lightning Safety
http://www.lightningsafety.noaa.gov/lightning_map.htm
37
SCIFs
Wright-Patterson Air Force Base in Dayton, Ohio, is one of the major command posts of the U.S. Air Force (USAF).
It contains a giant Faraday cage that houses a
Sensitive
Compartmented
Information
Facility (SCIF)
+ + + ++ + + +
+
+
+ + + ++ + + + +- - - -- - - - -
- - - -- - - - -
-
+-
+-
+-
+-
+-
+-
+-
38
SCIFs
Any externally generated electric field causes electrons in the Faraday cage to migrate in the direction opposite the field.
+ + + ++ + + +
+
+
+ + + ++ + + + +- - -- -- - --
- - - -
-
-- - -
-
+-
+-
+-
+
-
+
- +
-
+-
39
SCIFs
The induced field exactly cancels the externally generated fields. Consequently, any electronic equipment inside is immune from an electromagnetic attack.
+ + + ++ + + +
+
+
+ + + ++ + + + +- - -- -- - --
- - - -
-
-- - -
-
+-
+-
+-
+
-
+
- +
-
+-
0E
40
Summary
Electric Flux
Gauss’s Law
The electric flux through a closed surface is determined by the enclosed charge
E dA
enclosed 0/E dA q
41
Summary
Conductors
The electric field within a conductor, in electrostatic equilibrium, is zero because the charge rushes to, and distributes itself on, the surface of the conductor