Download PHY481: Electromagnetism Constant currents and magnetic fields Lecture 24

PHY481: Electromagnetism
Constant currents and magnetic fields
Lecture 24
Carl Bromberg - Prof. of Physics
Current, and current density
I is charge flow/unit time
vdrift = drift velocity (~1mm/s) of charge carriers
I = qnL vdrift (1 D)
q = charge of charge carriers (+ or -)
nL = no. of charge carriers per unit length
Ohm’s law: I = V R
nV = no. of charge carriers per unit volume
vth = thermal velocity vth >> vdrift
λ = mean free path
Drude model:
Resistance/L
Conductivity σR
mvth
R
=
L q2n λ
L
nV q λ
L
σR =
=
RA
mvth
2
I = qnL vdrift
q nL λ
=
V
mL vth
Charge density ρc (x)
Current densities:
Current in = current out
∂ρc ∂t = 0
ρR = 1 σ R
No charge buildup
Linear current I = qnL v(x) (1 D)
Surface current density
K(x) = qnS v(x) (2 D)
Current through line
dI = K ⋅ ê ⊥ d (1 D)
Lecture 24
2
Resistivity ρR
Volume current density
J(x) = qnV v(x)
(3 D)
Current through area
dI = J(x) ⋅ dA (1 D)
Carl Bromberg - Prof. of Physics
J(x)
dA
dA
1
Conductors, “free” charge, continuity equation,
Macroscopic
Resistance
Conductors
Microscopic - Local forms of Ohm’s law
Resistivity ρR
Conductivity σR
σ R = 1 ρR
J(x) = σ R E(x)
J(x) = E(x) ρ R
I =V R
ρ and σ are an intrinsic property of a material
“Wire” with area A,
and resistivity ρR
L
1
R = ∫ dR = ∫ ρ R ( z ′ )dz ′
A0
“Free” charge
Uniform resistivity
Surface free charge
resistivity
d ρ R dz = 0
none in volume:
resistor constant
Iz
Cu ++
ρ R ≈ 0 +++
––
–––



With constant
ρR L
R=
resistivity ρR
A
Changing resistivity
Surface & volume
free charge:
Cu
ρR ≈ 0
∫ ρC d x = 0 no net charge
3
-->
ρ free =
ε I ⎛ d ρR ⎞
⎜
⎟
A ⎝ dz ⎠
Continuity equation
∇⋅J = −
Lecture 24
∂ρc
∂t
Integrate over volume V, and use Gauss’s theorem
Rate of change of
Flux of J through
d
3
J ⋅ dA = − ∫ ρc d x charge in d3x
∫
3
S
surface S (of d x)
dt V
-> Charge conservation
Carl Bromberg - Prof. of Physics
2
Resistors
Each resistor has the value R. What is the total resistance, RT, of
this infinite set of resistors?
One resistor: RT = R
Four resistors: RT = R+ (3R in parallel)
N resistors: RT = not obvious
The trick!
Lecture 24
Carl Bromberg - Prof. of Physics
3
Magnetic force and field
Magnetic force: F = qv × B
Circular motion:
v0 ⊥ B
v0 ⊥ B
Crossed E and B: E ⊥ B
Perpendicular to direction of motion
Magnetic forces do no work ΔKE = 0
Centripetal acceleration caused by
magnetic force. ω = v R = qB m
Helical motion. Helix has radius and pitch
Exists an un-deflected velocity v = E × B / B
Currents make magnetic fields: Biot-Savart Law
µ0 Id × r̂
µ0
Id × ( x − x ′ )
dB =
4π
r
z
Id k̂
R
r
Id
x
B(x) =
2
dB = Id φ̂
r̂
R̂
y
4π
∫wire
x − x′
3
Wire carrying
current
Current element
Lecture 24
Carl Bromberg - Prof. of Physics
4
2
Magnetic field of long wire carrying current
µ0
B(x) =
4π
∫wire
Id × ( x − x ′ )
x − x′
(the hard way)
x = R R̂ + z k̂
3
Magnetic field does not depend on z.
Any z will do. Pick z = 0 to simplify
dz ′
R + z′
x ′ = z ′ k̂
k̂ × r̂ = k̂ × R̂ cosψ = φ̂ cosψ
cosψ = R
2
d = dz ′ k̂
z
Id = Idz ′ k̂
r̂ = R̂ cosψ − k̂ sin ψ
µ0 I ∞ dz ′ k̂ × r̂
B(x) =
∫ ( R2 + z′2 )
4π −∞
µ0 IR ∞
B(x) =
∫(
4π −∞
x ′ = z ′ k̂
)
2 32
φ̂
∞
( R2 + z ′ 2 )1 2
dz ′
x
r = ( R + z′
ψ
y
x=R
r̂
2
2
∞
z′
2
⎡
⎤
=
=
∫ ( 2 2 )3 2 ⎢ 2 2 2 ⎥ R2
−∞ R + z ′
⎣ R R + z ′ ⎦ −∞
µ0 I
B(R) =
φ̂
2π R
Lecture 24
Carl Bromberg - Prof. of Physics
5
2
)
Ampere’s Law
∇⋅B = 0
Magnetic fields are “solenoidal” (always true)
∇ × B = µ0 J Ampere’s Law (constant currents only.
Maxwell’s equations have another term)
Integrate over a
surface element:
∫ (∇ × B ) ⋅ n dA = µ0 ∫ J ⋅ dA
S
Apply Stokes’s theorem:
S
∫ B ⋅d = µ0 ∫ J ⋅ dA
C
S
dI = J ⋅ dA
∫ B ⋅d = µ0 Iencl
C
Ampere’s Law and symmetries
Straight wire (the easy way)
∫ B ⋅d = µ0 Iencl
C
Right hand rule #2
Lecture 24
Bφ 2π R = µ0 I
B(R) =
µ0 I
φ̂
2π R
Carl Bromberg - Prof. of Physics
6
More Ampere’s law applications
Infinite current sheet
Infinite current slab
Lecture 24
Carl Bromberg - Prof. of Physics
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General field equations for B(x)
Constant vs. changing currents
Always true
∇⋅B = 0
∇ ⋅ ( ∇ × B ) = 0 = µ0∇ ⋅ J
Constant currents only.
∇⋅J = 0
Maxwell’s equations have another term: proportional to
Rate of change of electric field
∇ × B = µ0 ( J + ε 0 dE dt )
∇ ⋅ ( ∇ × B ) = µ0 ( ∇ ⋅ J + ε 0 d ( ∇ ⋅ E ) dt ) = 0
∇ ⋅ J = − d ρc dt
∇ ⋅ E = ρc ε 0
Continuity equation
General B field equation
µ0
B(x) =
4π
∫wire
⎡µ
B(x) = ∇ × ⎢ 0
⎣ 4π
Lecture 24
Id × ( x − x ′ )
x − x′
3
3
J( x ′ )d x ′ ⎤
∫ x − x′ ⎥⎦
( x − x′ )
x − x′
3
= −∇
1
x − x′
B(x) = ∇ × A(x)
1 ⎞
⎛
⎛ J( x ′ ) ⎞
∇
×
J(
x
)
=
∇
×
′
⎜⎝
⎟
⎜⎝
⎟
x − x′ ⎠
x − x′ ⎠
3
µ J( x ′ )d x ′
A(x) = 0 ∫
4π
x − x′
Carl Bromberg - Prof. of Physics
Vector
potential
8