From last time. Attention. Capacitance. Spherical capacitor. Energy stored in capacitors. How do we charge a capacitor? Today:

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1 Attention From last time More on electric potential an connection to Efiel How to calculate Efiel from V Capacitors an Capacitance switch off computers in the room an be prepare to a very lou noise Toay: More on Capacitors an Capacitance Energy store in Capacitors Current esistance Ohm s aw How o we charge a capacitor? Battery has fixe electric potential ifference across its terminals Conucting plates connecte to battery terminals prouce a charge separation on plates Electrons move from negative battery terminal to right plate an from left plate to positive battery terminal This charge motion requires work. The work is supplie by the battery. Between plates there is an Efiel an a potential ifference is estableshe Vplates Vbattery Infinite plates: we normally mean to neglect borer effets (uniform fiel) Eleft Eright Q Q E E E E E E Capacitance Asurface area Q/A surface charge ensity E left E right " /2# o " /2# o " /# o Work of the electric force to move q from positive to negative plate r W el "#U "q(v " "V ) q $ E r s qe % #V E 0 "V V #V "V # Q # A Q C C " o A "V 1 C Q This is a geometrical factor Spherical capacitor Charge Q move from outer to inner sphere Gauss law says EkQ/r 2 until between secon the sphere inner sphere an the shell Potential ifference Along path shown b kq "V # $kq 1 b r 2 r kq % 1 a $ 1 ( ' b * a & ) a C Q "V k % 1 a $ 1 ( ' * & b) $1 Path to fin "V Gaussian surface to fin E Q Energy store in capacitors During the charging of a capacitor, when a charge q is on the plates, the work neee to transfer further q from one plate to the other is: The total work require to charge the capacitor is Q q Q2 W " q 0 C 2C The energy store in any capacitor is: For a parallel capacitor: U 1/2 o AE 2

2 Quick Quiz Energy ensity q pull pull q A parallel plate capacitor given a charge q. The plates are then pulle a small istance further apart. Which of the following apply to the situation after the plates have been move? The energy store per unit volume is U/(A) 1/2 oe2 This is a funamental relationship for the local energy store in an electric fiel 1)The charge ecreases The charge cannot ecrease because C is isolate A) C 0A/ " C ecreases B)2)The capacitance increases E Q/(0A) " E constant C)3)The electric fiel increases 4)The voltage between the D) V E V increases Not restricte to capacitors: Efiel can be from any source Interpretation: energy is store in the fiel Capacitors are evices to store electric energy an charge (use in raio receivers, filters of power supplies, electronic flashes) plates increases E)5)The energy store in the U Q2/(2C) U increases capacitor increases Moeling a cell membrane Human capacitors: cell membranes K Na Na Extracellular flui 100 m ipi bilayers form a capacitor with charge separation at about 78nm 78 nm Cytoplasm outsie cell Na K Charges are / ions instea of electrons (atoms with ifferent numbers of electrons an protons) Charge motion is through cell membrane (ion channels) rather than through wire 0.07 VV Channels are selective for each kin of ion V~0.07 Across the cell membrane about V70 mv (resting potential) since more Na outsie an gates are close while K ions iffuse out an in at the same rate There is more positive charge outsie the cell Extracellular flui 100 m sphere ~ 3x104 cm2 surface area Capacitance: "o A Cytoplasm insie cell (8.85 #10 $12 F /m) 4 % (50 #10$6 m) $9 8 #10 m #10$11 F 35 pf ~0.1F/cm2 Depolarization in a nerve cell Cell membrane epolarization An influx of Na ions thru Na channel can cause a epolarization of the membrane. The potential ifference becomes V~0.07 "V 0.03 V about 0.03 mv. Potential change about 0.1 V About 100 channels/µm2 (each Cytoplasm has raius of 50 µm) K Extracellular flui 78 nm Na K How many ions flow per channel? Charge xfer require "QC"V(35 pf)(0.1v) 3.5 x 1012 C/ (1.6x1019 C/ion) 2.2 x 107 (35x1012 C/V)(0.1V) 3.5 x 1012 Coulombs ions flow (100 channels/m2)x4(50 m)23.14x106 ion channels Ion flow / channel (2.2x107 ions) / 3.14x106 channels ~ 7 ions/channel

3 Charge motion: current Cell membrane capacitor: Foun ~7 ions flow through ion each ion channel to epolarize membrane Occurs in ~ 1 ms sec. This is an electric current I: amount of charge per unit time flowing through a plane perpenicular to charge motion Units: 1 Coulomb / sec 1 Ampere Average current: Instantaneous value: Conventionally we assume: current in irection of charge particles though electrons in reality flow in a wire Battery prouces Efiel in wire Current in a conuctor wire Average current: E n number of electrons/volume electrons travel istance v t in time t # of electrons: n x volume na na v t I av Q/ t nqa v / Current ensity J I/A (A/m 2 ) J nqv (irection of charge carriers) Current flows in response to Efiel v 10 4 m/s Ohm s law When an electric fiel is applie free electrons experience an acceleration (opposite to E): a F/m qe/m Calling t the mean time between 2 collisions electronatoms electrons achieve a rift velocity: v at qe m t " r j nqv r # nq2 t & r % ( E $ m ' Ohm s aw: for many materials, the current ensity is proportional to the electric fiel proucing the current J # E Ohm s aw Ohm s aw: J # E or E " J " resistivity E "V # I A Define: "/A esistance in ohms (") Ohm s law becomes: V I # conuctivity of the conuctor E ohmic evice: relationship current an voltage is linear Quick Quiz Two cylinrical conuctors are mae from the same material. They are of equal length but one has twice the iameter of the other. A. 1 < 2 B. 1 2 " A Depenence on Temperature $ of a conuctor varies approximately linearly with T $ o is the resistivity at T o 20 C $ %(&'& 0 ) % x ($) TT 0 T 0 T C. 1 > % temperature coefficient in SI units of o C 1 The higher T the greater atomic vibrations that increases collision probability 17 Similarly: o [1 %(T T o )]

4 But other materials esistivity esistivity: Semiconuctors: % is negative ) $ ecreases for $1/# SI units of ". m The resistance epens resistivity an geometry: increasing T " " 0 (1 #$T ) Mercury Superconuctors Below a certain temperature, TC critical temperature $ is zero Once a current is set up in a superconuctor, it persists without any applie voltage since 0 esistors control the current level in circuits esistors can be composite or wirewoun esistors in Series I I 1 esistors in parallel 2 resistors in series: ike summing lengths V V1 V2 eq 12 on conuctor 2 # A V V1 V2 I A A V1 " Neglect resistance of wires in circuits respect to resistance V2 I "V1 "V2 # 1 1& "V % ("V 1 2 $ 1 2 ' eq 2A A areas "V "V1 "V2 1 2 (1 2 )I eq I " A Current Conservation: 1st Kirchoff s law As a charge Q moves from a to b, the electric potential energy of the system increases by QV an the chemical energy in the battery ecreases by this same amount Iin I3 I3 As the charge moves through the resistor (c to ), the system loses this electric potential energy uring collisions of the electrons with the Power rate of energy loss atoms of the resistor I3 Iout Iout Iin I3 Junction ule: Electrical power This energy is transforme into internal energy in the resistor (vibrational motion of the atoms in the resistor) Power in units of Watts J/s ( Iin ( Iout A statement of Conservation of Charge 23 "U "Q "V "t "t "V 2 P I"V I 2 P 24

5 Quick Quiz How oes brightness of bulb B compare to that of A? A. B brighter than A B. B immer than A C.Both the same 25