Physics 1502: Lecture 8 Today s Agenda. Today s Topic :

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1 Physics 1502: Lecture 8 Today s Agenda Announcements: Lectures posted on: HW assignments, solutions etc. Homework #3: On Masterphysics today: due next Friday Go to masteringphysics.com Today s Topic : Review Chapter 23: Definition of Capacitance xample Calculations (1) Parallel Plate Capacitor (2) Cylindrical Capacitor (3) Isolated Sphere nergy stored in capacitors Dielectrics Capacitors in Circuits 1

2 Definitions & xamples a b L d A a a b b a V Capacitors in Parallel Q 1 Q 2 V a Q b b Capacitors in Series Q -Q a b C = C 1 C 2 Q -Q a b 2

3 nergy of a Capacitor The total work W to charge a capacitor to Q is : In terms of the voltage V: The energy is stored in the lectric field itself. The energy density u in the field is : Units: J/m 3 Dielectrics INCRAS the capacitance of a capacitor: κ dielectric constant of the material ε R I ε = R I 3

4 Overview Charges in motion mechanical motion electric current How charges move in a conductor Definition of electric current Text Reference: Chapter 24 4

5 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 =0 inside a conductor If =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 field, then come to rest. Now we consider circuits in which charges can circulate if driven by a force such as a battery. 5

6 Current Definition Consider charges moving down a conductor in which there is an Note: electric This field. definition assumes If I take a the cross current section in of the the direction wire, over of some amount of time Δt I will count a certain number of charges (or total the positive particles, amount of charge) ΔQ moving by. We define current as the ratio of these quantities, NOT in the direction of the electrons! I avg = ΔQ / Δt or I = Q/ t Units for I, Coulombs/Second (C/s) or Amperes (A) How charges move in a conducting material lectric force causes gradual drift of bouncing electrons down the wire in the direction of -. Drift speed of the electrons is VRY 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. 6

7 How charges move in a conducting material ΔQ is the number of carriers in some volume times the charge on each carrier (q). Let n be the carrier density, n = # carriers / volume. The relevant volume is A * (v d Δt). Why??? So, ΔQ = n A v d Δt q And I avg = ΔQ/Δt = n A v d q More on this later Drift speed in a copper wire 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. Volume of copper (1 mol): Because each copper atom contributes one free electron to the current, we have (n = #carriers/volume) 7

8 Drift speed in a copper wire, ctd. We find that the drift speed is with charge / electron q Thus Resistance Resistance Resistance is defined to be the ratio of the applied voltage to the current passing through. UNIT: OHM = Ω I R V I 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? 8

9 Ohm's Law Vary applied voltage V. Measure current I Does ratio ( V/I ) remain constant?? I R V I V slope = R = constant I 9

10 Resistivity Property of bulk matter related to resistance of a sample is the resistivity ρ defined as: where = electric field and j A L j = current density in conductor = I/A. For uniform case: n 0 : carrier density (carriers/volume) q : charge per carrier v : carrier speed η : viscosity material constant material constant Resistivity j A L So, in fact, we can compute the resistance if we know a bit about the device, and YS, the property belongs only to the device! eg, for a copper wire, ρ ~ 10-8 Ω-m, 1mm radius, 1 m long, then R.01Ω 10

11 Make sense? 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 j A L or How thick? How big? What s it made of? 11

12 Alternative Version of Ohm s Law A related empirical observation is that: This is an alternative version of Ohm s Law. It can also be written as, j A L 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 I 1 I 2 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 12

13 Current Idea Current is the flow of charged particles through a path, at circuit. Along a simple path current is conserved, cannot create or destroy the charged particles Closely analogous to fluid flow through a pipe. Charged particles = particles of fluid Circuit = pipes Resistance = friction of fluid against pipe walls, with itself. 13

14 Lecture 8, ACT 2 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? ε 1 - I 4 R x A more detailed model I avg = ΔQ/Δt = n A v d q Difficult to know v d directly. Can calculate it. 14

15 A more detailed model I avg = ΔQ/Δt = n A v d q The force on a charged particle is, If we start from v=0 (on average) after a collision then we reach a speed, τ : average collision-free time Substituting gives, (note j = I/A) or A more detailed model This formula is still true for most materials even for the most detailed quantum mechanical treatment. 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. lectrons 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) 15

16 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) n B) q C) D) τ 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. 16