Lecture 9. DC Current, Resistance, Ohm’s Law
CH1 average: 58% .
Let there be current!
(statics is boring…)
1
Charging a Capacitor
equipot. @ V=V
+
+
∆V=V
∆𝑉 = 0
-
equipot. @ V=0
∆V=V
-
𝑒−
𝑒−
𝑡=0
𝑞=0
∆𝑉 = 0
wires are not
equipotential!
𝑞=0
equipot. @ V=V
+
∆V=V
-
𝑒−
𝑞 = +𝐶𝑉 ∗ < 𝑄
∆𝑉 = 𝑉 ∗ < 𝑉
𝑒−
𝑡 = 𝑡1
𝑞 = −𝐶𝑉 ∗
+
∆V=V
-
𝑞 = 𝑄 = +𝐶𝐶
∆𝑉 = 𝑉
𝑞 = −𝑄 = −𝐶𝐶
equipot. @ V=0
𝑡 = "𝑡𝑒𝑒𝑒𝑒𝑒 "
1. Electrons are driven by the electric field in the wires (potential difference).
2. How much time does it take (how long is 𝑡𝑒𝑒𝑒𝑒𝑒 )? – in two lectures.
3. To keep current running, we need to maintain the potential difference along a wire.
2
Electrons in Metals: “Classical” Microscopic Picture
E=0
Electrons in random motion, colliding with static (=
defects) and dynamic (= vibrations) imperfections of
crystal lattice.
Average distance between scattering events = the
mean free path, m.f.p., 𝑙 (depends on temperature,
density of defects, etc.). Typically, 𝑙~10 nm @ 300K
(still, about 100 lattice periods).
Electron speed
Wrong estimate:
1
3
𝑚𝑣 2 = 𝑚 𝑣𝑥
2
2
2
3
= 𝑘𝐵 𝑇
2
𝑣𝑥 =
1.38 ∙ 10−23 𝐽/𝐾 × 300𝐾
≈ 7 ∙ 104 𝑚/𝑠
−31
9.1 ∙ 10 𝑘𝑘
Quantum mechanics : the mobile electrons, being fermions, cannot have the same
energy if their wavefunctions overlap in space. As a result, their average kinetic energies
are much (~102 times) greater than 𝑘𝐵 𝑇! The current carriers move with 𝒗 ≈ 𝟏𝟏𝟔 𝒎/𝒔
that is T-independent! v<<c - we still can apply non-relativistic mechanics.
Time intervals between elastic (energy-conserving) collisions:
𝜏𝑒𝑒
𝑙
10−8 𝑚
≈ = 6
= 10−14 𝑠
𝑣 10 𝑚/𝑠
3
𝑬 ≠ 𝟎: Electron Drift
𝐸≠0
Under a gentle “breeze” of electric field,
the electron “mosquito cloud” drifts
slowly inside the wire.
𝑣⃗𝑑
Why the drift velocity, not constant acceleration???
𝐸=0
Because there are processes of energy dissipation (compare to air friction). The
terminal velocity is reached when the rate of energy gained from the electric
field becomes equal to the rate of energy loss.
…
τel
τin
drift velocity ≡ terminal velocity
4
Current
𝐸≠0
𝑑𝑑
𝐼=
𝑑𝑑
Current: the charge carried by the current
carriers through a wire cross section in unit
time.
𝐸=0
Units: Amperes
disregard
random
motion
𝑛𝑛 𝑣𝑑 ∙ 1𝑠 ∙ 𝐴
𝐼=
1𝑠
𝐼
𝐼 = 𝑛𝑛𝑣𝑑 𝐴
1𝐶
1𝑠
= 1𝐴
n – concentration of charge carriers
e – their charge
𝐴 ∙ 𝑣𝑑 ∙ 1𝑠 – the volume that passes
through cross section in 1s
𝐼
𝑗 = = 𝑛𝑛𝑣𝑑
𝐴
- current density
5
Estimates and Comments
Estimate: a copper wire (n∼1029 m-3, cross section 1 mm2 = 10-6m2) carries current
1 A. Find 𝑣𝑑 .
𝐼
1𝐴
−4 𝑚/𝑠
𝑣𝑑 =
≈ 29 −3
=
10
𝑛𝑛𝐴 10 𝑚 ∙ 10−19 𝐶 ∙ 10−6 𝑚2
The drift velocity:
𝑣𝑑 ≪ 𝑣
𝑣 ≈ 106 𝑚/𝑠
Regardless of the nature of charge carriers (both
positive and negative carriers exist in different
types of conductors), the current is defined as a
directional motion of positive charges, it always
flows from higher potential to lower potential.
DC current resembles a flow of incompressible
fluid (continuity equation, no accumulation of
charges).
6
Ohm’s Law
For metals 𝐼 ∝ ∆𝑉 over a
broad range of ∆𝑉
(experimental observation)
– Ohm’s Law
1
𝐼
=
𝑉 𝑅
or
𝑉
=𝑅
𝐼
Units: Ohms, Ω
the coefficient of proportionality ≡
the resistance
=
Georg Ohm
Of course, the resistance depends on neither V nor I (in
the linear regime, where there is no overheating, etc.)
11
Voltmeters and Ammeters
A good voltmeter should have
𝐯𝐯𝐯𝐯 𝐡𝐡𝐡𝐡 𝑹𝒊𝒊 and should be
connected in parallel with the
circuit element being measured.
A good ammeter should have
𝐯𝐯𝐯𝐯 𝐥𝐥𝐥 𝑹𝒊𝒊 and should be
connected in series with the
circuit element being measured.
12
Ohm’s Law and Resistivity
Ohm’s Law in terms of the resistivity:
𝑉 𝐸𝐿 𝐸
𝐼= =
= 𝐴
𝐿
𝑅 𝜌
𝜌
𝐴
𝐼 𝐸
𝑗≡ =
𝐴 𝜌
- current density
A/m2
Iclicker Question
Consider two wires. Wire A is 10 cm long, and wire B is 5 cm long.
Both wires are otherwise identical, and both have the same
electric field acting in them. How do the currents in these wires
compare?
A. 𝐼𝐴 = 4𝐼𝐵
B. 𝐼𝐴 = 2𝐼𝐵
C. 𝐼𝐴 = 𝐼𝐵
D. 𝐼𝐴 = 0.5𝐼𝐵
E. 𝐼𝐴 = 0.25𝐼𝐵
13
Typical Resistivities
metals
semi-metal
semiconductor
dielectric
Resistance of a copper wire: cross section 1 mm2, length 1 m:
𝐿
1𝑚
−8
𝑅 = 𝜌 ≈ 10 Ω ∙ 𝑚 ∙ −6 2 = 0.01Ω
𝐴
10 𝑚
17
Temperature Dependence of a Metallic Resistance
Non-superconducting metals (e.g., Cu)
ρ
ρres – residual (T=0) resistance, due
to static defects
less pure
ρt ≈ ρ 0 (1 + α T) - high-T approximation
very pure,
no lattice defects
ρres
ρ0
0
100
200
300
T, K
Platinum
resistance
thermometer
Superconducting materials
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Conclusion
DC current: flow of charge carriers, requires E ≠ 0 in the conductor.
Microscopic picture: electron “mosquito cloud” slowly drifting in the field.
Linear regime: Ohm’s Law.
Resistance: the coefficient of proportionality between V and I, depends on
materials parameters.
Next time: Lecture 10. Resistor circuits,
EMF and Batteries.
§§ 24.4 - 24.5
19
Appendix 1. Mobility and Resistance
Ohm’s Law implies that the drift velocity is proportional to the electric field:
the wire length
𝑉 𝐸𝐸
𝐼≡ =
𝑅
𝑅
↔
𝐼 = 𝑛𝑛𝑛𝑛𝑑
Simplistic microscopic model for the mobility:
𝑣𝑑 ≈ 𝑎𝑎 =
𝑒𝑒
𝜏 = 𝜇𝜇
𝑚
after each scattering,
the electron loses its kin. energy
𝑒𝜏
𝜇=
𝑚
1 𝐿
𝑚 𝐿
𝑅=
=
𝑛𝑛𝑛 𝐴 𝑛𝑒 2 𝜏 𝐴
𝑣𝑑 = 𝜇𝜇
µ - the electron mobility
…
τel
a – carrier acceleration
τin
m – carrier mass
τ ≡ τin – time between inelastic collisions
2
10−19 × 10−14
𝑚
𝜇≈
≈ 10−3
−30
10
𝑠∙𝑉
Resistance
Resistivity (resistance of
a cube 1 m3)
𝑅=
𝑚 𝐿
𝐿
=
𝜌
𝑛𝑒 2 𝜏 𝐴
𝐴
𝑚
𝜌= 2
𝑛𝑒 𝜏
𝐼 = 𝑛𝑛𝜇𝐴𝐴
Units: Ohm
Units: Ohm ∙ m
20