physics 202, lecture 4 gauss’s law: revie€¦ · physics 202, lecture 4 today’s topics !! more...
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Physics 202, Lecture 4
Today’s Topics
!! More on Gauss’s Law
!! Electric Potential
!! 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 highly
symmetric 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 sheet
of charge,…)
Example: Thin Spherical Shell
Find E field inside/outside a uniformly charged thin
spherical shell, surface charge density !, total charge q
Gaussian Surface
for E(outside)
Gaussian Surface
for E(inside)
Er<a = 0
Er>a =
q
4!"0r2
qencl = 0qencl = q Trickier examples
(holes, conductors,…)
Gauss’s Law: 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
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 Energy Given 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 negative
Particles will move to minimize their final potential energy!
q q0 U
i= 0
q q0
q q0 !!
!
Ur "U# = "! F d! l
#
r
$ = " kq0q
% r 2
d % r =#
r
$ kq0q
r
!
"U =Uf #Ui = #"W
Electric Potential Energy
Electric 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
q
U(r) =kq
0q
r
U(r) =kqiqj
rijj
!i< j
!
This is the work required to assemble charges.
Electric Potential
Charge q0 is subject to Coulomb force in electric field E.
Work done by electric force:
independent of q0
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)
!
"U = #"W = #! F d! l = #q
0i
f
$! E d! l
i
f
$
!
"V #"U
q0
= $! E d! l
i
f
%
Electric Potential and Point Charges
For point charge q shown below, what is VB-VA ?
A
B
rA
rB
q
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!
Equipotential Lines
Equipotentials:lines of constant potential
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) = !!E(r)id
!l
ref
r
"
dV = kdq
r,V = dV!
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
Ex= !
"V
"x , E
y= !
"V
"y ,E
z= !
"V
"z or
!E = !#V
!
"V = #! E
A
B
$ d! l
!
dV = "! E d! l = "Exdx " Eydy " Ezdz
A Picture to Remember
$! Field lines always point towards lower electric potential
$! In an electric field:
"! positive charges are always subject to a force in the
direction of field lines, towards lower V
"! negative charge is always subject a force in the
opposite direction of field lines, towards higher V
E
higher V lower V
+q
-q