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Chapter Capacitance PHY2049: 25 1 What You Know: Electric Fields Coulombs Electric law fields Equilibrium Gauss law Electric fields for several charge configurations Point Dipole (along axes) Line Plane (nonconducting) Plane (conducting) Ring (along axis) Disk (along axis) Sphere Cylinder PHY2049: Chapter 25 2 What You Know: Electric Potential Electric potential energy Electric potential Equipotential surfaces Potential of point charge Potential of charge distribution Special cases: dipole, line, ring, disk, sphere Relationship of potential and electric field Calculating the potential from the field Calculating the field from the potential Potential energy from a system of charges PHY2049: Chapter 25 3 Capacitance: Basic Idea Capacitance: Capacity to store charge Like a tank Capacitor is electrically neutral (equal + and charge regions) q = CV (C is a property of the device, independent of q, V) Units: [C] = Farad = Coulomb/Volt PHY2049: Chapter 25 4 Calculating Potential Difference Electric Follow field lines start on + charges, terminate on E field line during integration (note cos = 1) E, ds positive So V+ V is always positive ++++++++++++++++ E d --------------------PHY2049: Chapter 25 5 Parallel Plate Capacitor From Gauss law (conducting sheet) So Therefore ++++++++++++++++ Depends of device E only on geometry d --------------------PHY2049: Chapter 25 6 Parallel Plate Capacitor Example A = 10 cm 10 cm = 102 m2, d = 1m A 102 C = 0 = 8.85 1012 = 8.85 108 F d 106 C = 0.0885 F = 88.5nF Somewhat typical value PHY2049: Chapter 25 7 Example of Capacitor in Circuit In this circuit, C = 5F, V = 120. When switch S is closed current flows and capacitor charges. How much charge flows through capacitor until it is charged? ( )( ) q = CV = 5 106 120 = 600 C Upper plate gets + charge, lower plate gets charge V PHY2049: Chapter 25 8 ConcepTest In this circuit, C = 50 nF, V = 60. When switch is S closed current flows and capacitor charges. How much charge flows through capacitor until it is charged? (1) (2) (3) (4) (5) 60 C 30 nC 60 nC 3 C 50 nC V PHY2049: Chapter 25 9 Cylindrical Capacitor Inner = a, outer = b, length = L Gauss law: Using =q/L q/ L 1 E= 2 0 r (ds = dr) b - ( ) + - - a + - + - + + - 2 L C = 0 ln b / a - + + - - - + - - Capacitance proportional to length, e.g. coaxial cable (RF frequencies) PHY2049: Chapter 25 10 Coaxial Cable Used to transmit high frequency (MHz - GHz) signals PHY2049: Chapter 25 From Wikipedia 11 Coaxial Cable PHY2049: Chapter 25 From Wikipedia 12 Coaxial Cable From TigerDirect.com PHY2049: Chapter 25 13 Coaxial Capacitor Example Inner conducting wire a = 0.1mm Outer conductor b = 3 mm C 2 = 0 = 1.63 1011 F/m = 16.3 pF/m L ln 30 () PHY2049: Chapter 25 14 Special Case for Cylinder Outer Use shell very close to inner shell: b a = d (d small) ln(1+x) x (for x small) b a+d ln = ln = ln 1 + a a d d a a Asurface 2 L 2 aL C = 0 0 = 0 d d ln b / a ( Just ) like parallel plate capacitor: Always true if surfaces are close together PHY2049: Chapter 25 15 Spherical Capacitor Inner radius = a, outer radius = b Coulombs - - b - + + law: + - (ds = dr) - a + - - + - + + - PHY2049: Chapter 25 + - - 16 Two Special Cases Isolated Outer sphere: corresponds to b = shell very close to inner shell: b a = d (d small) Again, just like parallel plate: PHY2049: Chapter 25 17 Capacitors in Parallel V1 = V2 = V3 (same potential top and bottom) Total CeqV charge: Qtot = Q1 + Q2 + Q3 = C1V + C2V + C3V Basic law for combining capacitors in parallel Works for N capacitors PHY2049: Chapter 25 18 Capacitors in Series q1 = q2 = q3 (same current charges all capacitors) Total potential: V = V1 + V2 + V3 q/Ceq = q/C1 + q/C2 + q/C3 Basic law for combining capacitors in series Works for N capacitors PHY2049: Chapter 25 19 ... View Full Document