Download Physics 202, Lecture 4 Gauss`s Law: Review

Physics 202, Lecture 4
Today’s Topics

More on Gauss’s Law

Electric Potential (Ch. 23)
 Electric Potential Energy and Electric Potential
 Electric Potential and Electric Field
Gauss’s Law: Review
! ! q
! E = " E idA = encl
#0
spherical
(point charge, uniform
sphere, spherical shell,…)
Use it to obtain E field for highly
symmetric charge distributions.
cylindrical
(infinite uniform line of
charge or cylinder…)
planar
(infinite uniform sheet
of charge,…)
Method: evaluate flux over carefully chosen “Gaussian surface”
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Spherical Symmetry: Examples
Spherical symmetry: choose a spherical Gaussian surface, radius r
Then:
!
!
"! EidA = E4" r
2
=
qencl
#0
E=
qencl
4!" 0 r 2
Example 1 (last lecture): uniformly charged sphere (radius a, charge Q)
!
Er >a =
Q
r̂
4!" 0 r 2
!
Er <a =
Qr
r̂
4!" 0 a 3
Example 2: Thin Spherical Shell
Find E field inside/outside a uniformly charged thin
spherical shell, surface charge density σ, total charge q
Gaussian Surface
for E(outside)
qencl = q
Er >a =
q
4!" 0 r 2
Gaussian Surface
for E(inside)
qencl = 0
Er <a = 0
Trickier examples: set up
(holes, conductors,…)
22.29-31, 22.54 (board)
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Cylindrical symmetry: Examples
Cylindrical symmetry.
E indep of
z, !, in radial direction
Gaussian surface: cylinder of length L
z
!
! !
q
Eid
"# A = E(r)2! rL = "encl0
qencl
E=
2! rL" 0
Then:
r
!
Example: infinite uniform line of charge
!
E(r) =
!
r̂
2"# 0 r
Similar examples: infinite uniform cylinder, cylindrical shell
Text examples (board): 22.34-35
Planar Symmetry: Examples
Planar symmetry.
y
Example: infinite uniform sheet of charge.
E indep of x,y, in z direction
!
x
Gaussian surface: pillbox, area of faces=A
!
!
" EidA = 2EA =
!
z
qencl # A
=
!0
!0
!
!
E=
ẑ
2" 0
Examples: multiple charged sheets, infinite slab (hmwk)
Text example (board): 22.24 (parallel plate)
3
Electric Potential Energy and Electric Potential
Review: Conservation of Energy (particle)
Kinetic Energy (K). Potential Energy U: for conservative forces
(can be defined since work done by F is path-independent)
K=
1 2
mv
2
U(x, y, z)
If only conservative forces present in system,
conservation of mechanical energy: K + U = constant
Conservative forces:
U = kspring x 2 2
 Springs: elastic potential energy
 Gravity: gravitational potential energy
 Electrostatic: electric potential energy (analogy with gravity)
Nonconservative forces
 Friction, viscous damping (terminal velocity)
Electric Potential Energy
Given two positive charges q and q0: initially very far apart,
q
q0
choose U i = 0
q
q0
To push particles together requires work (they want to repel).
Final potential energy will increase! !U = U " U = !W
f
i
If q fixed, what is work needed to move q0 a distance r from q?
q
r
q0
r
r
! !
! !
kqq
kqq0
!W = # Fus i dl = $ # Fe i dl = $ # 2 0 dr % =
r%
r
"
"
"
r
Note: if q negative, final potential energy negative
Particles will move to minimize their final potential energy!
4
Electric Potential Energy
Electric potential energy between two point charges:
U(r) =
kq0 q
r
q
r
 U is a scalar quantity, can be + or  convenient choice: U=0 at r= ∞
 SI unit: Joule (J)
q0
Electric potential energy for system of multiple charges:
sum over pairs:
kqi q j
U(r) = ! !
rij
i< j j
This is the work required to assemble charges.
Text example: 23.54
Electric Potential
Charge q0 is subject to Coulomb force in electric field E.
Work done by electric force:
W=
!
i
f
!
! !
f !
F i dl = q0 ! E i dl = " #U
i
independent of q0
Electric Potential Difference:
!V "
!
B !
!U
= # $ E i dl = VB # VA
A
q0
Units: Volts
(1 V = 1 J/C)
Often called potential V, but meaningful only as potential difference
Customary to choose reference point V=0 at r = ∞
(OK for localized charge distribution)
5
Electric Potential and Point Charges
For point charge q shown below, what is VB -VA ?
# 1 1&
dr
= kq % ! (
2
r
$ rB rA '
A
B
B
VB ! VA = ! " E(r)dr = !kq "
A
independent of path b/w A and B!
B
rB
A
rA
Potential of point charge:
V (r) =
q
kq
r
Many point charges: superposition
V (r) = k !
i
qi
ri
Equipotentials:lines of constant potential
Electric Potential: Continuous Distributions
Two methods for calculating V:
1. Brute force integration (next lecture)
dV = k
dq
,V = ! dV
r
2. Obtain from Gauss’s law and definition of V:
!
r !
V (r) = ! " E i dl
ref
Examples (today or next lecture) : 23.19, parallel plate
6
Obtaining the Electric Field From
the Electric Potential
Three ways to calculate the electric field:
 Coulomb’s Law
 Gauss’s Law
 Derive from electric potential 
Formalism
!
B !
!V = " # E i dl
A
! !
dV = ! E i dl = !Ex dx ! Ey dy ! Ez dz
"V
Ex = !
"x
"V
, Ey = !
"y
"V
,Ez = !
"z
or
!
E = !#V
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