1. A Microscopic View of Electric Circuits
Chapter 19
A Microscopic View of Electric Circuits

2. Gustav Robert Kirchhoff
Current at a Node
i1 = i2
i2 = i3 + i4
The current node rule
(Kirchhoff node or junction rule [law #1]):
In the steady state, the electron current entering a node in a circuit is equal to the electron current leaving that node
Gustav Robert Kirchhoff
( )
(consequence of conservation of charge)
Can also be expressed in terms of conventional current
Also called Kirchhoff's first law, Kirchhoff's point rule, Kirchhoff's junction rule, and Kirchhoff's first rule.The principle of conservation of electric charge implies that:
At any point in an electrical circuit where charge density is not changing in time, the sum of currents flowing towards that point is equal to the sum of currents flowing away from that point.

3. 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 + … = along any closed path in a circuit
V= U/q  energy per unit charge

4. Clicker Pick right statement: i1 = i4 and i2 = i3 i1  i4
Make complex circuit, one wire in, one out, ask what I is – circle all elements so that only one wire goes in, one out – charge conservation.

5. Question Write the node equation for this circuit.
What is the value of I2?
1 A
2 A
3 A
4 A
Can also be expressed in terms of conventional current

6. Exercise Write the node equation for this circuit.
What is the value of I2?
I1 + I4 = I2 + I3
I2 = I1 + I4 - I3 = 3A
What is the value of I2 if I4 is 1A?
I1 + I4 = I2 + I3
Can also be expressed in terms of conventional current
I2 = I1 + I4 - I3 = -2A
Charge conservation: 1A
Ii > 0 for incoming
Ii < 0 for outgoing

7. Motion of Electrons in a Wire
In a current-carrying wire there must be an electric field to drive the sea of mobile charges.
What is the relationship between the electric field and the current?
Why is an electric field required?
Interaction between electrons and lattice of atomic cores in metal.
Electrons lose energy to the lattice. Electric field must be present to increase the momentum of the mobile electrons.
Once e is in motion no force is required to keep it moving.

8. The Drude Model Average ‘drift’ speed: - average time between
collisions
For constant temperature
Paul Drude
( )
u – mobility of an electron
N- mobile electron density [electrons/m3]
Electron current:

9. Typical Mobile Electron Drift Speed
Typical electron current in a circuit is ~ 1018 electrons/s.
What is the drift speed of an electron in a 1 mm thick copper wire of circular cross section?

10. Typical Mobile Electron Drift Speed
Typical electron current in a circuit is 1018 electrons/s.
What is the drift speed of an electron in a 1 mm thick copper wire?
How much time would it take for a particular electron to move
through a piece of wire 30 cm long?
How can a lamp light up as soon as you turn it on?

11. Typical E in a Wire
Drift speed in a copper wire in a typical circuit is m/s.
The mobility is u= (m/s)/(N/C). Calculate E.
The drift speed was calculated in one of the earlier exercises. If we use the relation between mobility and mean time between collisions, we find that t is about seconds.
Electric field in a wire in a typical circuit is very small

12. E and Drift Speed
In steady state current is the same everywhere in a series circuit.
Ethick
Ethin i i
What is the drift speed?
Electron current depends on cross section of wire
Since the drift speed must be greater in the thinner wire, the electric field in the thinner wire must be larger
If R_thin is twice smaller than R_thick?
Highway traffic – in cogestion density changes, speed may stay the same
Note: density of electrons n cannot change if same metal
What is E?

13. Clicker Lamps are identical Batteries are identical
In B batteries are connected in series
Lamp shines brighter in A B
Lamps in A and B have the same brightness

14. Two Batteries in Series
Why light bulb is brighter with two batteries?
Two batteries in series can drive more current:
Potential difference across two batteries in series is 2emf  doubles electric field everywhere in the circuit  doubles drift speed  doubles current.
Work per second:
After getting i for both – ask how much brighter? How much more heat dissipate with 2 batteries?
In reality: not 4x because mobility changes with temperature., I.e. current also does not go twice up.

15. Energy in a Circuit Vwire = EL Vbattery = ?
From charge conservation and definition of steady state: Node rule – tells us about relations between currents in different parts. But what about actual current? How large is it.
To answer – turn to energy conservation principle. Consider one electron as it is transported around the circuit.
Energy gained by e as it is transported between terminals in a battery is dissipated in the rest of the circuit.
Use round-trip.
Energy conservation (the Kirchhoff loop rule [2nd law]):
V1 + V2 + V3 + … = along any closed path in a circuit
V= U/q  energy per unit charge

16. Question: Twice the Length1 2 i1 i2
Nichrome wire (resistive)
i1 = i2
i1 = 2*i2
i1 = ½ i2
Electric field is half -> half the drift speed -> half the current. Note that we are considering the wire to be a “resistive” wire.

17. Question:Two Light Bulbs
Parallel
In series
Identical light bulbs are first connected in series then parallel
Will there be a difference in light brightness?
There will be no difference
In series connection will give brighter light
Parallel connection will give brighter light

18. Two Light Bulbs in Parallel
L … length of bulb filament
1. Path ABDFA:
2. Path ACDFA: F
iB = iC
ibatt = 2iB
3. Path ABDCA:
What happens if one light bulb is removed?
We can think of the two bulbs in parallel as equivalent to increasing the cross-sectional area of one of the bulb filaments.

19. Two Identical Light Bulbs in Series
Two identical light bulbs in series are the same as one light bulb with twice as long a filament.
Identical light bulbs
The two bulbs emit “redder” light than the single bulb, indicating that the filaments are not has hot. Here, we consider delta_V to represent the fixed emf of the battery.
Electron mobility in metals decreases as temperature increases!
Conversely, electron mobility in metals increases as temperature
decreases. Thus, the current in the 2-bulb circuit is slightly
more than half of the one-bulb circuit.

20. Long and Round Bulb in Series
only round bulb:
Why round bulb does not glow? The calculation in the box shows the electric field required to light the round bulb.
The electric field doesn’t produce sufficient current through the filament to make it glow.
The round bulb doesn’t light up!

21. How Do the Currents Know How to Divide?
After connection there is a larger current through the battery and a larger gradient of surface charge along the wires to drive this larger current.