Physics 1402: Lecture 8 Today’s Agenda Announcements: –Lectures posted on: –HW assignments, solutions etc. Homework #3:Homework #3: –On Masterphysics today: due next Friday –Go to masteringphysics.com

Today’s Topic : Review Chapter 24: –Definition of Capacitance – Example Calculations (1) Parallel Plate Capacitor (2) Cylindrical Capacitor (3) Isolated Sphere –Energy stored in capacitors –Dielectrics –Capacitors in Circuits

Definitions & Examples d A a b L a b ab 

Capacitors in Parallel V a b Q2Q2 Q1Q1  V a b Q  C = C 1 + C 2 Capacitors in Series ab  +Q-Q ab +Q-Q 

Energy of a Capacitor The total work W to charge a capacitor to Q is : In terms of the voltage V: The energy density u in the field is : Units: J/m 3 The energy is stored in the Electric field itself. Dielectrics INCREASE the capacitance of a capacitor:  dielectric constant of the material

 R I  = R I

Overview Charges in motion –mechanical motion –electric current How charges move in a conductor Definition of electric current Text Reference: Chapter 25

Charges in Motion Up to now we have considered –fixed charges on isolated bodies –motion under simple forces (e.g. a single charge moving in a constant electric field) We have also considered conductors –charges are free to move –we also said that E=0 inside a conductor If E=0 and there is any friction (resistance) present no charge will move!

Charges in motion We know from experience that charges do move inside conductors - this is the definition of a conductor Is there a contradiction? no Up to now we have considered isolated conductors in equilibrium. –Charge has nowhere to go except shift around on the body. –Charges shift until they cancel the E field, then come to rest. Now we consider circuits in which charges can circulate if driven by a force such as a battery.

Current Definition l Consider charges moving down a conductor in which there is an electric field. If I take a cross section of the wire, over some amount of time  t I will count a certain number of charges (or total amount of charge)  Q moving by. l We define current as the ratio of these quantities, I avg =  Q /  t or I =  Q/  t l Units for I, Coulombs/Second (C/s) or Amperes (A) E Note: This definition assumes the current in the direction of the positive particles, NOT in the direction of the electrons!

How charges move in a conducting material l Electric force causes gradual drift of bouncing electrons down the wire in the direction of -E. l Drift speed of the electrons is VERY slow compared to the speed of their bouncing motion, roughly 1 m / h ! (see example later) Good conductors are those with LOTS of mobile electrons. E

How charges move in a conducting material  Q is the number of carriers in some volume times the charge on each carrier (q). l Let n be the carrier density, n = # carriers / volume. The relevant volume is A * (v d  t). Why ??? So,  Q = n A v d  t q And I avg =  Q/  t = n A v d q l More on this later … E

Drift speed in a copper wire Because each copper atom contributes one free electron to the current, we have (n = #carriers/volume) Volume of copper (1 mol): The copper wire in a typical residential building has a cross-section area of 3.31e-6 m 2. If it carries a current of 10.0 A, what is the drift speed of the electrons? (Assume that each copper atom contributes one free electron to the current.) The density of copper is 8.95 g/cm 3, its molar mass 63.5 g/mol.

Drift speed in a copper wire, ctd. We find that the drift speed is with charge / electron q Thus

Resistance Resistance is defined to be the ratio of the applied voltage to the current passing through. Is this a good definition? i.e. does the resistance belong only to the resistor? Recall the case of capacitance: (C=Q/V) depended on the geometry, not on Q or V individually Does R depend on V or I ? V I I R UNIT: OHM = 

Ohm's Law Vary applied voltage V. Measure current I Does ratio ( V / I ) remain constant?? V I slope = R = constant V I I R

Resistivity L A E j Property of bulk matter related to resistance of a sample is the resistivity  defined as: where E = electric field and j = current density in conductor = I/A. For uniform case: n 0 : carrier density (carriers/volume) material constant q : charge per carrier v : carrier speed  : viscosity material constant

Resistivity L A E j eg, for a copper wire,  ~  -m, 1mm radius, 1 m long, then R .01  So, in fact, we can compute the resistance if we know a bit about the device, and YES, the property belongs only to the device !  

Make sense? L A E j Increase the Length, flow of electrons impeded Increase the cross sectional Area, flow facilitated The structure of this relation is identical to heat flow through materials … think of a window for an intuitive example How thick? How big? What’s it made of? or

Alternative Version of Ohm’s Law L A E j A related empirical observation is that: This is an alternative version of Ohm’s Law. It can also be written as, with We can show this is also Ohm’s Law using the relations, and

Lecture 8, ACT 1 Two cylindrical resistors, R 1 and R 2, are made of identical material. R 2 has twice the length of R 1 but half the radius of R 1. –These resistors are then connected to a battery V as shown: V I1I1 I2I2 –What is the relation between I 1, the current flowing in R 1, and I 2, the current flowing in R 2 ? (a) I 1 < I 2 (b) I 1 = I 2 (c) I 1 > I 2

Current Idea l Current is the flow of charged particles through a path, at circuit. l Along a simple path current is conserved, cannot create or destroy the charged particles l Closely analogous to fluid flow through a pipe.  Charged particles = particles of fluid  Circuit = pipes  Resistance = friction of fluid against pipe walls, with itself. E

Lecture 8, ACT 2 R  I x Consider a circuit consisting of a single loop containing a battery and a resistor. Which of the graphs represents the current I around the loop?

A more detailed model I avg =  Q/  t = n A v d q l Difficult to know v d directly. l Can calculate it. E

A more detailed model I avg =  Q/  t = n A v d q l The force on a charged particle is, l If we start from v=0 (on average) after a collision then we reach a speed, or l Substituting gives, (note j = I/A)  : average collision-free time E

A more detailed model l This formula is still true for most materials even for the most detailed quantum mechanical treatment. l In quantum mechanics the electron can be described as a wave. Because of this the electron will not scatter off of atoms that are perfectly in place in a crystal. l Electrons will scatter off of 1. Vibrating atoms (proportional to temperature) 2. Other electrons (proportional to temperature squared) 3. Defects in the crystal (independent of temperature) E

Lecture 8, ACT 3 I am operating a circuit with a power supply and a resistor. I crank up the power supply to increase the current. Which of the following properties increases, A) nB) qC) ED)  E

Conductivity versus Temperature In lab you measure the resistance of a light bulb filament versus temperature. You find R  T. This is generally (but not always) true for metals around room temperature. For insulators R  1/T. At very low temperatures atom vibrations stop. Then what does R vs T look like?? This was a major area of research 100 years ago – and still is today.