PHYS 272: Electric and Magnetic Interactions Electric Fields and Circuits
Jonathan Nistor
Tuesday, July 15th, 2014
Email: [email protected] Office: Phys 263
Jonathan Nistor (Purdue University) Lecture 16 7/15/2014 1 / 23 Lecture 16
Electric Field and Circuits
19.1 Introduction and Overview 19.2 Current throughout a circuit 19.3 Electric Field and Current 19.4-19.5 What charges make the E-field in the wires
Reading: 15.5, 15.6 (up through the Drude Model) 18.1, 18.5 19.1 through 19.5
Jonathan Nistor (Purdue University) Lecture 16 7/15/2014 2 / 23 Lecture 16 Overview – Electron and Conventional Currents
Recall that the electron current, i , is defined as the number of electrons per second that enter a section of a conductor. For a metal with a cross sectional area A, and density of mobile electrons, n, then: Units: # of electrons/sec (1) where is the mean (average) drift speed of the electrons
Conventional current, , is define as the amount of charge (in
coulombs) entering a region per second. Therefore, Units: Coulombs/sec (2) Conventional current is assumed to consist of the motion of
positive charges, and therefore flows in the direction of Enet
Jonathan Nistor (Purdue University) Lecture 16 7/15/2014 3 / 23 Lecture 16 Overview – Electron and Conventional Currents
Conventional current is assumed to consist of the motion of
positive charges, and therefore flows in the direction of Enet
Conventional current
E
Electron current Electron current, i ,points in the
direction of the drift velocity,
Jonathan Nistor (Purdue University) Lecture 16 7/15/2014 4 / 23 Lecture 16 Overview – Equilibrium vs. Steady-State
A metal is in Equilibrium when there is no current flow: i.e. No charges are moving.
Does Enet necessarily have to be zero? ϸ A conductor is in a steady-state if charges are moving, but their drift velocities at any location do not change with time. Furthermore, there is no change in the deposits of excess charge anywhere This doesn’t mean that the electron drift velocity must be the same at every location. The drift velocities of charges may be different at different locations:
Jonathan Nistor (Purdue University) Lecture 16 7/15/2014 5 / 23 Lecture 16 “nodes” The Current Node Rule
Conservation of charge is a i1 = i2 fundamental physical principle which i2 = i3 + i4 guarantees that the total net charge in a system in conserved (constant) ϸ As such, if a conductor is in the steady state, where no excess deposits of charge occur, then the amount of charge entering a particular region (node), must be equal to the amount of charge leaving that same region in the same amount of time. Written as the current node rule (Kirchhoff 1st Law) In the steady state, the electron current entering a node in a circuit is equal to the electron current leaving that node
Jonathan Nistor (Purdue University) Lecture 16 7/15/2014 6 / 23 Lecture 16 Current Node Rule: Example Write the node equation for this circuit.
What is the value of I2?
I1 + I4 = I2 + I3
I2 = I1 + I4 - I3 = 3A
Write the node equations for this circuit…
Jonathan Nistor (Purdue University) Lecture 16 7/15/2014 7 / 23 Lecture 16 Current Node Rule: Generalized nodes How many ‘non-trivial’ nodes are there? (1) (2) (3) node #1 node #2
node #3 In general, a ‘node’ can be any boundary which contains portions of a circuit in steady-state. ϸ.5
Jonathan Nistor (Purdue University) Lecture 16 7/15/2014 8 / 23 Lecture 16 Electric Field and Current -- What produces currents?
In a current-carrying wire, there must be an electric field to drive the sea of mobile charges
What is the relationship between current and the electric field? Why is an electric field required to maintain a flow of charge (current)? i.e. Once current is flowing, why is an electric force required to keep the electrons moving at a constant drift speed ? Do the electrons push each other? Can there be excess charges inside a conductor in the steady state? We already know that there cannot be excess charges inside a conductor in equilibrium! What charges produce the electric field inside the wire?
Jonathan Nistor (Purdue University) Lecture 16 7/15/2014 9 / 23 Lecture 16 The Drude Model
What is the relationship between current and the electric field? Why is an electric field require to maintain a flow of charge (current)?
E
Start From the Momentum Principle:
p Fnett eEt p p 0 eEt p eEt The speed of the electron is: v me me eE t The average ‘drift’ speed is: v where t is the average time me between collisions
Jonathan Nistor (Purdue University) Lecture 16 7/15/2014 10 / 23 Lecture 16 The Drude Model eE t The average ‘drift’ speed is: v where t is the average time me between collisions is a property of the conductor. Dependent on: Lattice arrangement of atomic cores Density of metal Temperature of metal, why? is NOT dependent on the applied electric field Therefore, v ~ E (for constant temperatures)
We can write v uE , where u is called the electron mobility Paul Drude The electron current is therefore: (1863 - 1906)
Jonathan Nistor (Purdue University) Lecture 16 7/15/2014 11 / 23 Lecture 16 Typical E-Field in a wire
Drift speed in a copper wire in a typical circuit is 5.10-5 m/s. The mobility of copper is u=4.5.10-3 (m/s)/(N/C). Calculate E.
v 5 105 m/s E 1.1102 N/C u 4.5 103 (m/s)/(N/C)
Electric field in a wire in a typical circuit is very small
Jonathan Nistor (Purdue University) Lecture 16 7/15/2014 12 / 23 Lecture 16 Electric Field and Drift Speed
In steady state current is the same everywhere in a series (ϸ.6) circuit.
Ethick Ethin i i
What is the relationship between the drift speeds in the thin and thick wires? A v thick v i nAv nAthinvthin nAthickvthick thin thick Athin Note: density of electrons n cannot change if same metal
What about E? v uE
Athick Athick uEthin uEthick Ethin Ethick Athin Athin Jonathan Nistor (Purdue University) Lecture 16 7/15/2014 13 / 23 Lecture 16 Direction of Electric Field in a Wire
E must be parallel to the wire, why? E is the same along the wire, how do we know this? Is E uniform across the wire?
B C D A V E dl E dl E dl E dl 0 ABCDA 1 3 2 3 A B C D VAB 0 VCD 0 E1 E2 Jonathan Nistor (Purdue University) Lecture 16 7/15/2014 14 / 23 Lecture 16 Electric Field in a Wire
What charges produce the electric field inside the wire? There cannot be any excess charges in the interior of a conductor! The wire and bulb filament are electrically neutral
Are there excess charges on the battery? E E
E
If so, then you might expect the current through the light bulb to change
when you bring the battery closer. Does this happen?
Jonathan Nistor (Purdue University) Lecture 16 7/15/2014 15 / 23 Lecture 16 Electric Field in a Wire – A Mechanical Battery
Battery consists of “conveyer belt” driven by a hand crank or motor Electrons are pulled off of one plate and are deposited on the other plate This action replenishes the electrons that leave the negative plate and move through the wire
Electron Current Van de Graaff generator Jonathan Nistor (Purdue University) Lecture 16 7/15/2014 16 / 23 Lecture 16 Electric Field in a Wire – A Mechanical Battery
E
Drift velocities
E-Field due to Battery
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.
Jonathan Nistor (Purdue University) Lecture 16 7/15/2014 17 / 23 Lecture 16 Electric Field in a Wire – Surface Charge
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.
Jonathan Nistor (Purdue University) Lecture 16 7/15/2014 18 / 23 Lecture 16 Electric Field in a Wire – Surface Charge
E
Smooth transition from + surface charge to – to provide constant E . The gradient of surface charge (change in surface charge) is proportion to E The amount of surface charge is proportional to the applied voltage
Jonathan Nistor (Purdue University) Lecture 16 7/15/2014 19 / 23 Lecture 16 Connecting the 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
Currently, circuit is in static equilibrium
Jonathan Nistor (Purdue University) Lecture 16 7/15/2014 20 / 23 Lecture 16 Connecting the circuit – The Initial Transient
Enet
Before the gap is closed, the net field in the wire must be zero, because the system is in static equilibrium.
Jonathan Nistor (Purdue University) Lecture 16 7/15/2014 21 / 23 Lecture 16 Connecting the circuit – The Initial Transient
In just a few nanoseconds the rearrangement of the surface charges will extend all the way around the circuit. Speed of light: 30 cm/ns = 3 x 108 m/s
Jonathan Nistor (Purdue University) Lecture 16 7/15/2014 22 / 23 Lecture 16 Connecting the circuit – The states of a circuit
Static Equilibrium: Nothing moving, (no current)
Initial Transient: Short-time process leading to the steady state – surface charges rearrange
Steady State: Constant (non-zero) current
Jonathan Nistor (Purdue University) Lecture 16 7/15/2014 23 / 23