Chapter 19

A Microscopic View of Electric Circuits

Steady state and static equilibrium

Static equilibrium: • no charges are moving

Steady state (Dynamic Equilibrium): • charges are moving• their velocities at any location do not change with time• no change in the deposits of excess charge anywhere

Current in a Circuit

In an electric circuit the system does not reach equilibrium!

A microscopic view of electric circuits:• Are charges used up in a circuit?• How is it possible to create and maintain a nonzero electric

field inside a wire?• What is the role of the battery in a circuit?

IB = IA in a steady state circuit

What is used up in the light bulb?We cannot get something for nothing!

Energy is transformed from one form to another

Electric field – accelerates electronFriction – energy is lost to heatBattery – chemical energy is used up

Closed circuit – energy losses to heat:not an isolated system!

Current in Different Parts of a Circuit

Momentum principle:

E

Speed of the electron:

Average ‘drift’ speed: - average time betweencollisions

The Drude Model

Average ‘drift’ speed: - average time betweencollisions

For constant temperature

u – mobility of an electron

Electron current:

The Drude Model

Paul Drude(1863 - 1906)

In steady state current is the same everywhere in a series circuit.

ii

What is the drift speed?

Note: density of electrons n cannot change if same metal

What is E?

EthickEthin

E and Drift Speed

2 mm1 mm

1.5.n1 = n2

2.u1 = u2

Every second 1018 electrons enter the thick wire. How many electrons exit from the thin wire every second?

1018

Question

A) 1018

B) 1.5 x 1018

C) 2 x 1018

D) 4 x 1018

E) 12 x 1018

2 mm1 mm

1.5.n1 = n2

2.u1 = u2

What is the ratio of the electric field, E1/E2?

1018

Question

A) 3:1B) 6:1C) 8:1D) 12:1

Does current fill the wire? Is E uniform across the wire?

E must be parallel to the wire

E is the same along the wire

0 0VAB VCD

Direction of Electric Field in a Wire

Connecting a Circuit

The initial transient

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.

1. Static equilibrium: nothing moving(no current)

3. Steady state: constant current (nonzero)

2. Initial transient: short-time processleading to the steady state

Connecting a Circuit

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

(consequence of conservation of charge)

i1 = i2

i2 = i3 + i4

Current at a Node

Gustav Robert Kirchhoff (1824 - 1887)

I1 + I4 = I2 + I3

I2 = I1 + I4 - I3 = 3A

I1 + I4 = I2 + I3

I2 = I1 + I4 - I3 = -2A

1ACharge conservation:

Ii > 0 for incomingIi < 0 for outgoing

ExerciseWrite the node equation for this circuit. What is the value of I2?

What is the value of I2 if I4 is 1A?

Energy conservation (the Kirchhoff loop rule [2nd law]):

V1 + V2 + V3 + … = 0 along any closed path in a circuit

Vwire = ELVbattery = ?

V= U/q energy per unit charge

Energy in a Circuit

non-Coulomb force on each e

EC

FC

1. FC =eEC

Coulomb force on each e

2. FC =FNC

The function of a battery is to produce and maintain a charge separation.

Energy input per unit chargeemf – electromotive force

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…

Potential Difference Across the Battery

Fully charged battery.

Round-trip potential difference:

Field and Current in a Simple Circuit

We will neglect the battery’s internal resistance for the time being.

Round-trip potential difference:

Path 1

Path 2

Field and Current in a Simple Circuit

The number or length of the connecting wires has little effect on the amount of current in the circuit.

uwires >> ufilament

Work done by a battery goes mostly into energy dissipation in the bulb (heat).

V Across Connecting Wires

Nichrome wire (resistive)

Twice the Length

Current is halved when increasing the length of the wire by a factor of 2.

Doubling the Cross-Sectional Area

Nichrome wire

Electron current in the wire increases by a factor of two if the cross-sectional area of the wire doubles.

Loop: emf - EL = 0

𝑖=𝑛𝐴𝑢𝐸

Two Identical Light Bulbs in Series

Identical light bulbs

Two identical light bulbs in series are the same as one light bulb with twice as long a filament.

The filament lengths add …

1. Path ABDFA:

2. Path ACDFA:

3. Path ABDCA:

iB = iC

ibatt = 2iB

F

Two Light Bulbs in Parallel

We can think of the two bulbs in parallel as equivalent to increasing the cross-sectional area of one of the bulb filaments.

L … length of bulb filament

The filament areas add …

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

Analysis of Circuits

V1 + V2 + V3 + … = 0 along any closed path in a circuit

The loop rule (Energy conservation)Kirchhoff loop rule [2nd law]:

V= U/q energy per unit charge

Electron current: i = nAuE Conventional current: I = |q|nAuE