Download PHY481: Electromagnetism Current & Resistance Lecture 24 Carl Bromberg - Prof. of Physics

PHY481: Electromagnetism
Current & Resistance
Lecture 24
Carl Bromberg - Prof. of Physics
Current and current densities
No charge can accumulate in a uniform wire carrying
current. Charge going in = charge going out.
∂ρc
ρc (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)
K(x) = nS qv(x) (2 D)
Local current density J for
a differential area element Surface current K
K
ê ⊥
d
Current is charge
through A per unit time
I = dQ dt (1 D)
Lecture 24
Current through dA
dI = J(x) ⋅ dA (1 D)
Current crossing d
dI = K ⋅ ê ⊥ d (1 D)
Carl Bromberg - Prof. of Physics
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Continuity equation -- conservation of charge
Continuity equation:
Integrate over volume V,
and use Gauss’s theorem
∇⋅J = −
∫V ∇ ⋅ J d
3
Remember that ρc and J are
functions of 3D position
∂ρc
∂t
x = −∫
V
ρc (x) & J(x)
∂ρc
∂t
3
d x
Rate of change of
charge inside of V
Continuity equation, integral form.
Flux of J through
surface S
d
3
J
⋅
dA
=
−
ρ
d
x
∫S
dt ∫V c
Rate of change of
charge inside of V
Boundary condition on J at a surface
Discontinuity in normal
∂σ
J 2n − J1n = − c Rate of change of
component of J at a surface
∂t surface charge density
∂ρc
∂t
= 0 and
Lecture 24
∂σ c
∂t
=0
with constant
currents
Reminiscent of
σc
E2n − E1n =
ε0
Carl Bromberg - Prof. of Physics
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Good conductors & poor conductors (resistors)
Macroscopic
Resistance
Microscopic - Local forms of Ohm’s law
Resistivity ρ
1
J(x) =
E(x)
ρR
1
I= V
R
Conductivity σ
σR =
J(x) = σ R E(x)
1
ρR
ρ and σ are an intrinsic property of a material
“Wire” with area A, and resistivity ρR
With constant resistivity ρR
z
1
R = ∫ dR = ∫ ρ R ( z ′ )dz ′
A0
ρR
z
A
Resistance linear in z (length) and ρ .
R=
What is still true about E ?
E = −∇V field <--> potential relationship, e.g., E = E k̂; V = − E z
z
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
∇⋅E =
ρ free
ρc
⇒
ε0
ε
Lecture 24
Charge density doesn’t change
∂ρc
∂t
=0
but ρc ≠ 0
must specify “free charge density” and permittivity, ε .
Carl Bromberg - Prof. of Physics
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Free charge and resistivity
What is free charge, ρ free ?
Charge carriers move w.r.t. atomic background, but resistance causes charge to
(instantaneously) collect locally and then remain constant.
ρ free = 0 with constant resistivity (uniform material)
No free charge in resistor volume
Free charge on up- & down-stream
surfaces, due to resistivity discontinuity
ρ free =
ε I ⎛ d ρ R ⎞ with changing resistivity over z
⎜
⎟
A ⎝ dz ⎠
Free charge in resistor volume, and
free charge on up- & down-stream surfaces
Lecture 24
Carl Bromberg - Prof. of Physics
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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) = σ RE(x)
Gauss’s Law: I = σ R ∫ E(x) ⋅ n̂ dAI = σ R ∫ ∇ ⋅ E(x)d 3 x
S
V
σR
σR
3
I=
ρ(x)d x =
Q
ε V∫
ε
−Q2 (t)
Q1 (t)
Q = CV
conductor 2
Gaussian
surface
∇ ⋅ E(x) =
σ R ,ε
ρ(x)
ε
V σR
=
CV
R
ε
RC =
ε
σR
constants with
dimensions of time
dQ(t) σ R
1
=
Q(t) =
Q(t)
dt
ε
RC
Lecture 24
Q1 (t) = Q0 e
−t / RC
Exponential decay
of the charge
Carl Bromberg - Prof. of Physics
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HW
7.3 Solid sphere radius a, uniformly distributed Q.
Constant angular velocity. Calculate J(x) .
∫S
J(r) ⋅ n̂ dA =
dφ = ω dt
d
3
ρ
d
x
dt ∫V c
7.11 Find surface charge density at discontinuity in resistivity.
σc
E2n − E1n =
ε0
L 2
Lecture 24
L 2
J = J k̂
ρ R1
++
+
+
ρ R2
ρR2 > ρ R1
J = J k̂
ρ R1
–
––
–
ρ R2
ρR2 < ρ R1
Carl Bromberg - Prof. of Physics
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HW
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 ?
Use general solution for cylidrical symmetry twice
V1 (r) = Aln r + B
V2 (r) = C ln r + D
Lecture 24
Carl Bromberg - Prof. of Physics
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