Download PHY481: Electromagnetism Constant currents Conductors & insulators Lecture 22

PHY481: Electromagnetism
Constant currents
Conductors & insulators
Lecture 22
Carl Bromberg - Prof. of Physics
Exam 2
Lecture 22
Carl Bromberg - Prof. of Physics
1
Current and current densities
No charge can accumulate in a uniform wire carrying
current. Charge going in = charge going out.
!"c
!c (x) = charge density
!t
Volume current density
Current density J is
position dependent
=0
Beware: ρ is also used for
resistivity of a material.
J(x) = nV qv(x) (3 D)
(2 D)
Local current density J for K(x) = nS qv(x)
a differential area element Surface current K
K
ê !
d!
Current is charge
through A per unit time
I = dQ dt (1 D)
Lecture 22
Current through dA
dI = J(x) ! dA (1 D)
Current crossing d!
dI = K ! ê " d! (1 D)
Carl Bromberg - Prof. of Physics
2
Continuity equation -- conservation of charge
Continuity equation:
!"J = #
Remember that ρc and J are
functions of 3D position
$%c
$t
!c (x) & J(x)
Integrate over volume
%&c 3
3
V, and use Gauss’s
! " J d x = $#
d x
#
V
V %t
theorem
Continuity equation, integral
form.
Flux of J through
surface S
"S
J ! dA = #
d
3
$
d
x
"
c
V
dt
Rate of change of
charge inside of V
Rate of change of
charge inside of V
Boundary condition on J at a surface
Discontinuity in normal
"# c Rate of change of
J
!
J
=
!
component of J at a surface
2n
1n
"t surface charge density
with constant
= 0 currents
= 0 and
!t
!t
!"c
Lecture 22
!" c
Reminiscent of
"c
E2n ! E1n =
#0
Carl Bromberg - Prof. of Physics
3
Good conductors & poor conductors (resistors)
Macroscopic
Resistance
1
I= V
R
Microscopic - Local forms of Ohm’s law
Resistivity ρ
Conductivity σ
1
1
!
=
J(x) = ! R E(x)
J(x) =
E(x)
R
"R
!R
ρ and σ are an intrinsic property of a material
“Wire” with area A, and resistivity ρR
z
1
R = ! dR = ! " R ( z # )dz #
A0
With constant resistivity ρR
!R
z
A
Resistance linear in z (length) and ρ .
R=
What is still true about E ?
E = !"V field <--> potential relationship, e.g., E = Ez k̂; V = ! Ez z
!"E=0
violation requires changing magnetic fields
In good conductors, e.g., Cu, Ag, etc., the charge density is negligible
In resistors carrying constant current
!"c
Charge density doesn’t change
= 0 but !c " 0
#
#
!t
free
!"E = c %
$0
$
must specify “free charge density” and permittivity, ε .
Lecture 22
Carl Bromberg - Prof. of Physics
4
Free charge and resistivity
What is free charge, ! free ?
Though currents have moving charge, resistance will allow charge to
accumulate locally.
! free = 0 with constant resistivity (uniform material)
No free charge in resistor volume
Free charge, ! free on
up- & down-stream surfaces
" I # d ! R & with resistivity changing over z
%
(
A $ dz '
Free charge in resistor volume, and
! free =
free charge on
up- & down-stream surfaces
Lecture 22
Carl Bromberg - Prof. of Physics
5
Voltage and charge decay
Two conductors embedded in a resistive matrix
Charge each with a battery and disconnect.
E
Relate Total current I to material
Total current: I = " J(x) ! n̂ dA
S
conductor 1
J(x) = ! RE(x)
Gauss’s Law: I = ! E(x) " n̂ dAI = ! $ " E(x)d 3 x
R#
R#
S
V
!R
!R
3
I=
#(x)d x =
Q
" V$
"
!Q2 (t)
Q1 (t)
Q = CV
conductor 2
Gaussian
surface
! " E(x) =
" R ,#
#(x)
$
V !R
=
CV
R
"
RC =
!
"R
constants with
dimensions of time
dQ(t) ! R
1
=
Q(t) =
Q(t)
dt
"
RC
Lecture 22
Q1 (t) = Q0 e
!t / RC
Exponential decay
of the charge
Carl Bromberg - Prof. of Physics
6
HW-7
7.3 Solid sphere radius a, uniformly distributed Q.
Constant angular velocity. Calculate J(x) .
"S
J(r) ! n̂ dA =
d! = " dt
d
3
#
d
x
dt "V c
7.11 Find surface charge density at discontinuity in resistivity.
"c
E2n ! E1n =
#0
L 2
Lecture 22
L 2
J = J k̂
! R1
++
+
+
! R2
!R2 > ! R1
J = J k̂
! R1
–
––
–
! R2
!R2 < ! R1
Carl Bromberg - Prof. of Physics
7
HW-7
7.9 Cylinders length L , resistive media between radius a and 3a . What is
the resistance? What is the charge density on the interface at 2a ?
General solutions Boundary Conditions
V1 (r) = Aln r + B V1 (a) = V0
V2 (r) = C ln r + D V2 (3a) = 0
V1 (2a) = V2 (2a)
J1 (2a) = J 2 (2a)
Lecture 22
Carl Bromberg - Prof. of Physics
8