PHYS 272: Electric and Magnetic Interactions
Electric Fields and Circuits
Jonathan Nistor
Monday, July 8th, 2013
Email: jnistor@purdue.edu
Office: Phys 263
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Electric Field and Circuits
19.1 Introduction and Overview
19.2 Current throughout a circuit
19.3 Electric Field and Current
19.4 What charges make the E-field in the wires
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Overview – Electron and Conventional Currents
Recall that the electron current, i , is defined as the number of
electrons per second that enter a section of a conductor. ϸ.1
For a metal with a cross sectional area A, and density of
mobile electrons, n, then:
Units: # of electrons/sec
where
(1)
is the mean (average) drift speed of the electrons ϸ.2
Conventional current, , is define as the amount of charge (in
coulombs) entering a region per second. Therefore,
Units: Coulombs/sec
(2)
Conventional current is assumed to consist of the motion of
positive charges, and therefore flows in the direction of Enet
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Overview – Electron and Conventional Currents
Conventional current is assumed to consist of the motion of
positive charges, and therefore flows in the direction of Enet
Conventional current
E
Electron current
Electron current, i ,points in the
direction of the drift velocity,
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Overview – Equilibrium and Steady-State
A metal is in Equilibrium when there is no current flow:
i.e. No charges are moving.
Does Enet necessarily have to be zero?
ϸ.3
A conductor is in a steady-state if charges are moving, but their
drift velocities at any location do not change with time.
Furthermore, there is no change in the deposits of excess
charge anywhere
This doesn’t mean that the electron drift velocity must be the
same at every location. The drift velocities of charges
may be different at different locations:
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The Current Node Rule
“nodes”
Conservation of charge is a
fundamental physical principle which
guarantees that the total net charge in
a system in conserved (constant) ϸ.4
i1 = i2
i2 = i3 + i 4
As such, if a conductor is in the steady state, where no excess
deposits of charge occur, then the amount of charge entering a
particular region (node), must be equal to the amount of charge
leaving that same region in the same amount of time.
Written as the current node rule (Kirchhoff 1st Law)
In the steady state, the electron current entering a node in a
circuit is equal to the electron current leaving that node
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Current Node Rule: Example
Write the node equation for this circuit.
What is the value of I2?
I1 + I 4 = I 2 + I 3
I2 = I1 + I4 - I3 = 3A
Write the node equations for this circuit…
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Current Node Rule: Generalized nodes
How many ‘non-trivial’ nodes are there?
(1)
(2)
(3)
node #1
node #2
node #3
In general, a ‘node’ can be any boundary which contains portions of a
circuit in steady-state. ϸ.5
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Electric Field and Current
In a current-carrying wire, there must be an electric field to
drive the sea of mobile charges
What is the relationship between current and the electric field?
Why is an electric field required to maintain a flow of charge (current)?
i.e. Once current is flowing, why is an electric force required to keep the
electrons moving at a constant drift speed
?
Do the electrons push each other?
Can there be excess charges inside a conductor in the steady state?
We already know that there cannot be excess charges inside a conductor
in equilibrium!
What charges produce the electric field inside the wire?
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The Drude Model
What is the relationship between current and the electric field?
Why is an electric field require to maintain a flow of charge (current)?
E
Start From the Momentum Principle:
p  p  0  eEt
p eEt
The speed of the electron is: v 

me
me
The average ‘drift’ speed is: v  eE t where t is the average time
me
between collisions
p  Fnet t  eEt
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The Drude Model
The average ‘drift’ speed is: v  eE t
me
where t is the average time
between collisions
t is a property of the conductor. Dependent on:
Lattice arrangement of atomic cores
Density of metal
Temperature of metal, why?
t is NOT dependent on the applied electric field
Therefore, v ~ E (for constant temperatures)
We can write
v  uE , where
u is called the electron mobility
The electron current is therefore:
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Paul Drude
(1863 - 1906)
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Typical E-Field in a wire
Drift speed in a copper wire in a typical circuit is 5.10-5 m/s.
The mobility of copper is u=4.5.10-3 (m/s)/(N/C). Calculate E.
v
5  10 5 m/s
2
E 

1.1

10
N/C
3
u 4.5  10 (m/s)/(N/C)
Electric field in a wire in a typical circuit is very small
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Electric Field and Drift Speed
In steady state current is the same everywhere in a series circuit.
Ethick
Ethin
i
i
What is the relationship between the drift speeds in the thin and thick wires?
i  nAv
nAthinvthin  nAthick vthick
Athick
vthin 
vthick
Athin
Note: density of electrons n cannot change if same metal
What about E? v  uE
Athick
uEthin 
uEthick
Athin
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Athick
Ethin 
Ethick
Athin
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Direction of Electric Field in a Wire
E must be parallel to the wire, why?
E is the same along the wire, how do
we know this?
Is E uniform across the wire?
  C   D  A 
   E1 dl   E3 dl   E2 dl   E3 dl  0
A
B
D

 
 C 

VAB
VCD
0
0 
E1  E2
B
VABCDA
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