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?

I B = I A 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 electron Friction – energy is lost to heat Battery – chemical energy is used up Closed circuit – energy losses to heat: not an isolated system! Current in Different Parts of a Circuit

What is the relationship between the electric field and the current? Why is an electric field required? Collisions with Core gives frictional force. Motion of Electrons in a Wire In a current-carrying wire, the electric field drives the mobile charges. Electrons lose energy to the lattice of atomic cores. Electric field must be present to increase the momentum of the mobile electrons. In steady state, net force is zero.

Momentum principle: E Speed of the electron: Average ‘drift’ speed: - average time between collisions The Drude Model

Average ‘drift’ speed: - average time between collisions For constant temperature u – mobility of an electron Electron current: The Drude Model Paul Drude ( )

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? E thick E thin E and Drift Speed

2 mm 1 mm 1.5. n 1 = n 2 2. u 1 = u 2 Every second electrons enter the thick wire. How many electrons exit from the thin wire every second? Question A)10 18 B)1.5 x C) 2 x D) 4 x E) 12 x 10 18

2 mm 1 mm 1.5. n 1 = n 2 2. u 1 = u 2 What is the ratio of the electric field, E 1 /E 2 ? Question A)3:1 B)6:1 C)8:1 D)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 00V AB V CD 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 process leading 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) i 1 = i 2 i 2 = i 3 + i 4 Current at a Node Gustav Robert Kirchhoff ( )

I 1 + I 4 = I 2 + I 3 I 2 = I 1 + I 4 - I 3 = 3A I 1 + I 4 = I 2 + I 3 I 2 = I 1 + I 4 - I 3 = -2A 1A Charge conservation: I i > 0 for incoming I i < 0 for outgoing Exercise Write the node equation for this circuit. What is the value of I 2 ? What is the value of I 2 if I 4 is 1A?

Energy conservation (the Kirchhoff loop rule [2 nd law]):  V 1 +  V 2 +  V 3 + … = 0 along any closed path in a circuit  V wire = EL  V battery = ?  V=  U/q  energy per unit charge Energy in a Circuit

non-Coulomb force on each e ECEC FCFC 1. F C =eE C Coulomb force on each e 2. F C =F NC The function of a battery is to produce and maintain a charge separation. Energy input per unit charge emf – 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.

Chemical Battery A battery maintains a potential difference across the 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. u wires >> u filament  Work done by a battery goes mostly into energy dissipation in the bulb (heat).  V Across Connecting Wires

 V 1 +  V 2 +  V 3 +  V batt = 0 (-E 1 L 1 ) + (-E 2 L 2 ) + (-E 3 L 3 ) + emf = 0 An electron traversing the circuit D- C-B-A suffers a decrease in its electric potential energy. Where does this loss of energy appear? Application: Energy Conservation

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

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.

1. Path ABDFA: 2. Path ACDFA: 3. Path ABDCA: i B = i C i batt = 2i B 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 current node rule (Charge conservation) Kirchhoff node or junction rule [1 st 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  V 1 +  V 2 +  V 3 + … = 0 along any closed path in a circuit The loop rule (Energy conservation) Kirchhoff loop rule [2 nd law]:  V=  U/q  energy per unit charge Electron current: i = nAuE Conventional current: I = |q|nAuE