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

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 (1824 - 1887) (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.

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

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.

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

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

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.

The Drude Model Average ‘drift’ speed: - average time between collisions For constant temperature Paul Drude (1863 - 1906) u – mobility of an electron N- mobile electron density [electrons/m3] Electron current:

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?

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?

Typical E in a Wire Drift speed in a copper wire in a typical circuit is 5.10-5 m/s. The mobility is u=4.5.10-3 (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 10-14 seconds. Electric field in a wire in a typical circuit is very small

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?

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

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.

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 + … = 0 along any closed path in a circuit V= U/q  energy per unit charge

Question: Twice the Length 1 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.

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

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.

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.

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!

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.