Lecture 9 – Conductors & Dielectrics
Chapter 29 - Thursday February 8th
•A conductor in an electric field
•Quasi-static conditions
•Dynamic conditions
•Current density and drift velocity
•Ohm’s Law
•Macroscopic and microscopic views
•Insulators/dielectrics
•Capacitance (if time)
Reading: pages 635 thru 685 (chapter 29 & 30) in HRK
Read and understand the sample problems
WebAssign Homework due Thurs. Feb. 17th at 11:59pm
Graded problems (Ch. 28) – Prob. 6, 10
Practice problems (Ch. 29): Ex. 3, 9, 25, 29; Prob. 9, 15 (hard)
Conductors in E-fields: dynamic conditions
Dynamic
Dynamic
Static
Conductors in E-fields: dynamic conditions
• If the E-field is maintained,
then the dynamics persist,
i.e. charge continues to flow
indefinitely.
E
E
+
i
E
E
E
Electrical current:
SI unit:
• This is no longer strictly the
domain of electrostatics.
• Note the direction of flow of
the charge carriers
(electrons).
dq
i=
dt
1 ampere (A) = 1 coulomb per second (C/s)
Conductors in E-fields: dynamic conditions
• If the E-field is maintained,
then the dynamics persist,
i.e. charge continues to flow
indefinitely.
• This is no longer strictly the
domain of electrostatics.
• Note the direction of flow of
the charge carriers
(electrons).
Current density:
i
j=
A
or
i = ∫ j ⋅ dA
di
dA
Current density and drift speed
ΔL
j = − nev d
Starting point
for Ohm’s Law
Ohmic materials (Ohm’s law)
• We will see in a minute
that vd is proportional
to E. Thus, j ∝ E, i.e.
j = σE
E= ρj
σ is the electrical conductivity
ρ is the electrical resistivity
Ohm’s Law
σ=
1
ρ
SI unit for resistivity is ohm⋅ meter: 1 ohm = 1 volt/ampere
SI unit for conductivity is siemens per meter:
1 siemens = 1 ampere/volt = (1 ohm)-1
Ohmic materials (Ohm’s law)
• We will see in a minute
that vd is proportional
to E. Thus, j ∝ E, i.e.
j = σE
Resistance:
R=
ρL
A
Ohm’s Law
SI unit for resistance is ohm (Ω)
Ohmic materials (Ohm’s law)
1000 Ω
Ohmic
Non-ohmic
Ohm’s law: a microscopic view
•The average speed of an electron in
a metal is about 106 m/s. This is
almost 1% of the speed of light!!
•So, how is this reconciled with the
calculated drift velocities of order
10-4 m/s (for a current of 1A)?
Ohm’s law: a microscopic view
vd
∼ 0.02
v
Repeated collisions average electron
velocities to zero. Upon application
of electric field, electrons
accelerate. However, collisions
quickly dissipate any acquired
momentum. Consequently, the
electrons slowly drift in the
direction opposite to the field.
m Δv
Δp
Force =
= eE =
τ
Δt
τ is the average time between collisions, and m is the electron mass.
The average change in velocity ⟨Δv⟩ turns out to be equivalent to the
resultant drift velocity vd of the ensemble of electrons.
Ohm’s law: a microscopic view
Repeated collisions average electron
velocities to zero. Upon application
of electric field, electrons
accelerate. However, collisions
quickly dissipate any acquired
momentum. Consequently, the
electrons slowly drift in the
direction opposite to the field.
vd
∼ 0.02
v
⎛ eτ ⎞
vd = ⎜ ⎟ E
⎝m⎠
⎛ ne τ
j = envd = ⎜
⎝ m
2
⎞
⎟E =σE
⎠
ne τ
σ=
m
2
A dielectric in an electric field
Electric dipoles in an electric field
Non-polar atoms in
an electric field
A dielectric in an electric field
E = E0 + E'
E = E0 − E'
E' opposes E0
A dielectric in an electric field
Linear materials: E' ∝ E,
1
1
⇒E=
E 0 = E0
κe
1 + χe
⇒ E0 = (1 + χe)E
(κ e = 1 + χ e )
χe is the electric susceptibility (dimensionless)
κe is the dielectric constant (dimensionless)
ε = κeεο is the permittivity
Capacitors
•Used to store energy in electromagnetic fields [in contrast
to batteries (chemical cells) that store chemical energy].
•Capacitors can release electromagnetic energy much, much
faster than chemical cells. They are thus very useful for
applications requiring very rapid responses.
+q
+q
−q
−q
Specifications
Capacitor driven
1.5 megajoules at 10 kV
(.44 magnum slug ≈1 kJ)
Imax = 20 kA
Max field: 70 tesla
63 T, 15mm, 7/35ms
50 T, 24mm, 6/30ms
42 T, 24mm, 100/500ms
1 pulse/hour to 63 tesla
500 - 800 pulses, then...
Capacitors
•The energy really is stored in the electromagnetic fields.
•In fact, these fields possess energy and momentum, so
you might think of the capacitor as a fly-wheel, though it
is more common to think of capacitors as the electrical
analog of springs.
Capacitors
•The transfer of charge from one terminal of the capacitor
to the other creates the electric field.
•Where there is a field, there must be a potential
gradient, i.e. there has to be a potential difference
between the terminals.
•This leads to the definition of capacitance C:
q = C ΔV
•q represents the magnitude of the excess charge on either
plate. Another way of thinking of it is the charge that
was transferred between the plates.
SI unit of capacitance:
1 farad (F) = 1 coulomb/volt
(after Michael Faraday)
Capacitances more often have units of picofarad (pF) and microfarad (μF)