Transcript Lecture 13 - Purdue University

Chapter 19
A Microscopic View
of Electric Circuits
E and Drift Speed
In steady state current is the same everywhere in a series circuit.
Ethin
Ethick
i
i
What is the drift speed?
i = nAv
nAthin vthin = nAthick vthick
vthin
Athick
=
vthick
Athin
Note: density of electrons n cannot change if same metal
What is E?
v = uE
uEthin
Athick
=
uEthick
Athin
Ethin
Athick
=
Ethick
Athin
Exercise
i = nAv
0.1 mm
1 mm
vthin = ?
vthick = 410-5 m/s
Ethin= 10-1 N/C
Ethick= ?
vthin = 40010-5 m/s
Ethick= 10-3 N/C
Direction of Electric Field in a Wire
E must be parallel to the wire
E is the same along the wire
Does current fill the wire?
Is E uniform across the wire?
B
C
D
A
A
B
C
D
DVABCDA = - ò E1 ×dl - ò E3 ×dl - ò E2 ×dl - ò E3 ×dl = 0
VAB
0
E1 = E2
VCD
0
Electric Field in a Wire
What charges make the electric field in the wires?
E
Bulb filament and wires are metals –
there cannot be excess charges in the interior
Are excess charges on the battery?
E
E
i = nAuE
A Mechanical Battery
Van de Graaff generator
Electron Current
Field due to the Battery
E
Blue = vdrift
Ebends
In the steady state there must be some other charges
somewhere that contribute to the net electric field in such a
way that the electric field points upstream everywhere.
Field due to the Battery
i = nAuE
Surface charge arranges itself in such a way as to produce a
pattern of electric field that follows the direction of the wire
and has such a magnitude that current is the same along the
wire.
Field due to Battery
E
Smooth transition from + surface charge to – to provide
constant E.
The amount of surface charge is proportional to the voltage.
Amount of Surface Charge
The average magnitude of E in a closed
circuit can vary from ~.01 V/m in
copper wire to more than 100 V/m in
Nichrome wire - due to a much different
electron mobility.
What is easy: to draw E and i
What is complex: to draw surface
charge distribution
Connecting a Circuit
When making the final connection in a circuit, feedback
forces a rapid rearrangement of the surface charges
leading to the steady state.
This period of adjustment before establishing the steady
state is called the initial transient.
The initial transient
Connecting a Circuit
The initial transient
Egap faces is due to charges on gap faces
Eother
Enet
Before the gap is closed, the net field in the wire must be zero,
because the system is in static equilibrium.
Connecting a Circuit
The initial transient
Speed of light: 30 cm/ns
In just a few nanoseconds
the rearrangement of the
surface charges will extend
all the way around the
circuit.
Connecting a Circuit
1. Static equilibrium: nothing moving
(no current)
2. Initial transient: short-time process
leading to the steady state
3. Steady state: constant current
(nonzero)
Surface Charge and Resistors
Just after connection:
E may be the same
everywhere
i = nAv = nAuE
ithin = nAthin uE
ithick = nAthick uE
After steady state is reached:
ithin = ithick
ithin = nAthin uEthin
ithick = nAthick uEthick
Ethin
Athick
=
Ethick
Athin
ithin
Athin
=
ithick
Athick
Energy in a Circuit
Vwire = EL
Vbattery = ?
Energy conservation (the Kirchhoff loop rule [2nd law]):
V1 + V2 + V3 + … = 0
along any closed path in a circuit
V= U/q  energy per unit charge
Potential Difference Across the Battery
Coulomb force on each e
FC
non-Coulomb force on each e
FC
1. FC =eEC
EC =
e
2. FC =FNC
EC
DVbatt
FC s FNC s
= EC s =
=
e
e
Fully charged battery.
Energy input per unit charge
emf – electromotive force
The function of a battery is to produce and maintain a charge separation.
The emf is measured in Volts, but it is not a potential difference!
The emf is the energy input per unit charge.
chemical, nuclear, gravitational…
Analysis of Circuits
The current node rule (Charge conservation)
Kirchhoff node or junction rule [1st law]:
In the steady state, the electron current entering a node in a
circuit is equal to the electron current leaving that node
Conventional current: I = |q|nAuE
The loop rule (Energy conservation)
Kirchhoff loop rule [2nd law]:
V1 + V2 + V3 + … = 0
along any closed path in a circuit
V= U/q  energy per unit charge
Field and Current in a Simple Circuit
Round-trip potential difference:
DVbatt + DVwire = 0
emf + ( - EL) = 0
E=
emf
L
emf
I = enAuE = enAu
L
We will neglect the battery’s internal resistance for the time being.
Field and Current in a Simple Circuit
Round-trip potential difference:
Path 1
DVbatt + DV1 + DV3 = 0
emf + ( - E1L1 ) + (- E3L3 ) = 0
Path 2
E1 L1 = E2 L2
DVbatt + DV2 + DV3 = 0
emf + ( - E2 L2 ) + (- E3L3 ) = 0
Question: Twice the Length
1
2
i1
i2
Nichrome wire (resistive)
A) i1 = i2
B) i1 = 2*i2
C) i1 = ½ i2
Twice the Length
Nichrome wire (resistive)
DV
i = nAuE = nAu
L
i2 L =
1
iL
2
Current is halved when increasing the length of the wire by a
factor of 2.
Doubling the Cross-Sectional Area
Doubling the cross-sectional area of the
wire will
A) not change electron current
B) increase electron current by 2
C) decrease electron current by 2
Nichrome wire
Doubling the Cross-Sectional Area
i = nAuE = nAu
DV
L
Loop: emf - EL = 0
Electron current in the wire increases
by a factor of two if the crosssectional area of the wire doubles.
Can we achieve infinitely large current
using very thick wire?
Nichrome wire
Internal Resistance of a Battery
FNC = FC = eEC fully charged
DV
(emf )
i = nAuE = nAu
= nAu
L
L
Increase mobility – current increases
Real battery cannot provide  current
Internal battery resistance – limits
maximum current
fixed
EC = FC / e
Drift speed in battery: vb = ub Enet = ub (FNC / e - EC )
We will neglect the battery’s internal resistance for the time being.
Approximate Vbatt = emf
V Across Connecting Wires
The number or length of the
connecting wires has little effect on
the amount of current in the circuit.
DV
L
DV =
i
L
nAu
+ DVbattery = 0
i = nAuE = nAu
DVwires + DVfilament
DVbattery » emf
uwires >> ufilament  DVwires << DV filament
DV filament » (emf )
Work done by a battery goes mostly into energy dissipation in the
bulb (heat).