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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
!E =!Eid!A"" =
qencl
#0
Use it to obtain E field for highlysymmetric charge distributions.
Method: evaluate flux over carefully chosen “Gaussian surface”
spherical cylindrical planar
(point charge, uniform sphere, spherical shell,…)
(infinite uniform line of charge or cylinder…)
(infinite uniform sheetof charge,…)
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Spherical Symmetry: Examples
Spherical symmetry: choose a spherical Gaussian surface, radius r
Then:
Example 1 (last lecture): uniformly charged sphere (radius a, charge Q)
E =qencl
4!"0r2
!Eid!A"! = E4"r2 =
qencl
#0
!Er>a
=Q
4!"0r2r
!Er<a
=Qr
4!"0a3r
Example 2: Thin Spherical ShellFind E field inside/outside a uniformly charged thin
spherical shell, surface charge density σ, total charge q
Gaussian Surfacefor E(outside)
Gaussian Surfacefor E(inside)
Er<a
= 0Er>a
=q
4!"0r2
qencl = 0qencl = q
Trickier examples: set up(holes, conductors,…)22.29-31, 22.54 (board)
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Cylindrical symmetry: ExamplesCylindrical symmetry.
Example: infinite uniform line of charge
!
r
!
E indep of ,
z
z !
Gaussian surface: cylinder of length L
, in radial direction
!Eid!A = E(r)2!rL =
qencl
"0
"#
!E(r) =
!
2"#0rr
Similar examples: infinite uniform cylinder, cylindrical shell
Text examples (board): 22.34-35
E =qencl
2!rL"0
Then:
Planar Symmetry: Examples
Planar symmetry. Example: infinite uniform sheet of charge.
x
y
!
E indep of x,y, in z direction
z
Gaussian surface: pillbox, area of faces=A
!Eid!A = 2EA =
qencl
!0
"" =#A
!0
!E =
!
2"0
z
!
Examples: multiple charged sheets, infinite slab (hmwk)
Text example (board): 22.24 (parallel plate)
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Electric Potential Energy and Electric Potential
Kinetic Energy (K). Potential Energy U: for conservative forces(can be defined since work done by F is path-independent)
If only conservative forces present in system,conservation of mechanical energy: constant
Conservative forces: Springs: elastic potential energy Gravity: gravitational potential energy Electrostatic: electric potential energy (analogy with gravity)
Nonconservative forces Friction, viscous damping (terminal velocity)
Review: Conservation of Energy (particle)
K =1
2mv
2
U(x, y, z)
K +U =
U = kspringx22
Electric Potential EnergyGiven two positive charges q and q0: initially very far apart,
choose
To push particles together requires work (they want to repel).Final potential energy will increase!
If q fixed, what is work needed to move q0 a distance r from q?
Note: if q negative, final potential energy negativeParticles will move to minimize their final potential energy!
q q0U
i= 0
!U =U f "Ui = !W
q q0
!W =!Fus id!l
"
r
# = $!Fe id!l
"
r
# = $kqq
0
%r 2d %r =
"
r
#kqq
0
r
q q0r
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Electric Potential EnergyElectric potential energy between two point charges:
U is a scalar quantity, can be + or - convenient choice: U=0 at r= ∞ SI unit: Joule (J)
Electric potential energy for system of multiple charges:sum over pairs:
r
q0
qU(r) =kq
0q
r
U(r) =kqiqj
rijj
!i< j
!
Text example: 23.54This is the work required to assemble charges.
Electric PotentialCharge q0 is subject to Coulomb force in electric field E.
Work done by electric force:
W =
!Fid!l = q
0i
f
!!Eid!l =
i
f
! " #U
independent of q0
!V "!U
q0
= #!Eid!l
A
B
$ = VB #VA
Electric Potential Difference:
Often called potential V, but meaningful only as potential difference
Customary to choose reference point V=0 at r = ∞ (OK for localized charge distribution)
Units: Volts (1 V = 1 J/C)
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Electric Potential and Point Charges
For point charge q shown below, what is VB-VA ?
A
B
rA
rB
q
Equipotentials:lines of constant potential
VB !VA = ! E(r)dr = !A
B
" kqdr
r2
A
B
" = kq1
rB!1
rA
#
$%&
'(
Many point charges: superposition
V (r) =kq
r
V (r) = kqi
rii
!
Potential of point charge:
independent of path b/w A and B!
Electric Potential: Continuous Distributions
Two methods for calculating V:
1. Brute force integration (next lecture)
2. Obtain from Gauss’s law and definition of V:
V (r) = !!Eid!l
ref
r
"
Examples (today or next lecture) : 23.19, parallel plate
dV = kdq
r,V = dV!
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Obtaining the Electric Field Fromthe Electric Potential
Three ways to calculate the electric field: Coulomb’s Law Gauss’s Law Derive from electric potential
Formalism
dV = !
!Eid!l = !Exdx ! Eydy ! Ezdz
Ex= !
"V
"x , E
y= !
"V
"y ,E
z= !
"V
"z or
!E = !#V
!V = "
!Eid!l
A
B
#
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