PHYS 272: Electric and Magnetic Interactions Electric Fields and Circuits

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1 Jonathan Nistor (Purdue University) 7/09/ / 14 PHYS 272: Electric and Magnetic Interactions Electric Fields and Circuits Jonathan Nistor Monday, July 9 th, 2012

2 Jonathan Nistor (Purdue University) 7/09/ / 14 Electric Field and Circuits 19.1 Introduction and Overview 19.2 Current throughout a circuit 19.3 Electric Field and Current 19.4 What charges make the E-field in the wires

3 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: where Units: # of electrons/sec is the mean (average) drift speed of the electrons (1) 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 E net Jonathan Nistor (Purdue University) 7/09/ / 14

4 Jonathan Nistor (Purdue University) 7/09/ / 14 Overview Electron and Conventional Currents Conventional current is assumed to consist of the motion of positive charges, and therefore flows in the direction of E net Conventional current E Electron current Electron current, i,points in the direction of the drift velocity,

5 Jonathan Nistor (Purdue University) 7/09/ / 14 Overview Equilibrium and Steady-State A metal is in Equilibrium when there is no current flow: i.e. No charges are moving. Does E net necessarily have to be zero? 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:

6 Jonathan Nistor (Purdue University) 7/09/ / 14 The Current Node Rule nodes Conservation of charge is a fundamental physical principle which guarantees that the total net charge in a system in conserved (constant) 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 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 i 1 = i 2 i 2 = i 3 + i 4

7 Jonathan Nistor (Purdue University) 7/09/ / 14 Current Node Rule: Example Write the node equation for this circuit. What is the value of I 2? I 1 + I 4 = I 2 + I 3 I 2 = I 1 + I 4 - I 3 = 3A Write the node equations for this circuit

8 Jonathan Nistor (Purdue University) 7/09/ / 14 Current Node Rule: Generalized nodes node #1 node #2 How many non-trivial nodes are there? (1) (2) (3) node #3 In general, a node can be any boundary which contains portions of a circuit in steady-state

9 Jonathan Nistor (Purdue University) 7/09/ / 14 Electric Field and Current 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 require 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?

10 Jonathan Nistor (Purdue University) 7/09/ / 14 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 Fnet t eet p p 0 p eet The speed of the electron is: v m The average drift speed is: v ee t m e e m e eet where t is the average time between collisions

11 Jonathan Nistor (Purdue University) 7/09/ / 14 The Drude Model The average drift speed is: t v ee t m e is a property of the conductor. Dependent on: Lattice arrangement of atomic cores Density of metal Temperature of metal, why? t is NOT dependent on the applied electric field Therefore, v ~ E (for constant temperatures) where t is the average time between collisions We can write, where u is called the electron mobility The electron current is therefore: Paul Drude ( )

12 Jonathan Nistor (Purdue University) 7/09/ / 14 Typical E-Field in a wire Drift speed in a copper wire in a typical circuit is m/s. The mobility of copper is u= (m/s)/(n/c). Calculate E. E v u m/s (m/s)/(n/c) N/C Electric field in a wire in a typical circuit is very small

13 Electric Field and Drift Speed In steady state current is the same everywhere in a series circuit. i E thin E thick i What is the relationship between the drift speeds in the thin and thick wires? i nav Athick nathinv thin nathick v v thick thin vthick A Note: density of electrons n cannot change if same metal What about E? v ue A thin thick thick uethin uethick Ethin Ethick Athin Athin Jonathan Nistor (Purdue University) 7/09/ / 14 A

14 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? V ABCDA B E A C D A 1 dl E3 dl E2 dl E3 dl V AB B E 1 E 2 C D 0 0 Jonathan Nistor (Purdue University) 7/09/ / 14 V CD 0