Physics 24100 Electricity & Op-cs Lecture 8 Chapter 24 sec. 1-2 Fall 2017 Semester Professor Kol@ck
How Much Energy? V 1 V 2 Consider two conductors with electric potentials V 1 and V 2 We can always pick V 1 = 0 and then V 2 is just the potential difference between them.
How Much Energy? 0 V Take a small charge Q from the blue conductor and move it to the red conductor Work done is U = V Q How much will this change V? It turns out that V Q
Capacitance Q V The constant of proportionality is the capacitance, C: Q = C V If you transfer a total charge Q, then the potential difference will be: V = Q C The capacitance is defined as: C = Q V Units for Capacitance: Farad = Coulomb Volt ε 0 = 8.85 pf/m
Capacitors A capacitor is a device that stores electrostatic potential energy. These are (almost) always two conductors in close proximity separated by an insulator: Air or vacuum Something that prevents sparks Something that also stores electrostatic potential energy Examples: Two parallel wires Two parallel plates Two coaxial cylinders
Examples General procedure for calculating capacitance: 1. Put charge ±Q on the two conductors 2. Calculate E between the conductors 3. Calculate the electric potential difference V = E dl 4. Use C = Q/V
First Example Parallel plate capacitor: Area, A +Q E Q d
First Example Parallel plate capacitor: Area, A +Q E Q C = ε 0A d d This ignores any edge effects It is a good approximation when d is small compared with the length and width of a plate.
Lecture 8-8 Example: (Ideal) Parallel-Plate Capacitor 1. Put charges +q and q on plates of area A and separation d. 2. Calculate E by Gauss s Law EA 3. Calculate V by q 0 b b b a a V V V E dl Ed 4. Divide q by V q q q A q d 0A 0 C V Ed d a q is indeed prop. to V C is prop. to A C is inversely prop. to d
Lecture 8-9 Ideal vs Real Parallel-Plate Capacitors Ideal Real Uniform E between plates No fields outside Non-uniform E, particularly at the edges Fields leak outside
Coaxial Cylinder
Coaxial Cylinder C = 2πε 0 L log R 2 R 1
Lecture 8-11 Long Cylindrical Capacitor 1. Put charges +q on inner cylinder of radius a, -q on outer cylindrical shell of inner radius b. 2. Calculate E by Gauss Law q E 2 rl E 3. Calculate V a q 2 rl 0 0 b q q V E dl dr ln b/ a 2 rl 2 L b 0 a 0 0 4. Divide q by V q q 20L C C L V q ln ln b/ a b/ a C dep. log. on a, b 2 L L b q a +q -q V
Second Example Two long, parallel wires: +Q 2a E Q d L We will suppose that L d a.
Second Example Two long, parallel wires: +Q 2a E Q d L C = πε 0L log d/a (assuming L d a.)
Storing Energy Tuesday s example with the Van de Graaff Initial charge: Q Initial electric potential at the surface: V R = 1 Q 4πε 0 R Work needed to add an additional charge, Q: U = V R Q = 1 Q Q 4πε 0 R Total work needed to add charge Q total : U = 1 4πε 0 1 R 0 Q total Q dq = 1 4πε 0 1 R 1 2 Q total 2 R
Storing Energy Total work needed to charge sphere: U = 1 1 4πε 0 R 1 2 Q2 Final voltage of sphere: 1 Q V = 4πε 0 R Q = 4πε 0 RV Stored energy: R E = 1 2 4πε 0R V 2 In principle, we can use this energy to do work
Lecture 8-2 Electrostatic Potential Energy of Conductors q,v R bring dq in from q+dq,v+dv kq U W v dq dq R Spherical conductor Total work to build charge from 0 up to Q: v V v kq R 2 Q kq kq 1 U U dq QV 0 R 2R 2 dq Q q U
Lecture 8-13 Energy of a charged capacitor How much energy is stored in a charged capacitor? Calculate the work required (usually provided by a battery) to charge a capacitor to Q Calculate incremental work dw needed to move charge dq from negative plate to the positive plate at voltage V. dw V ( q) dq q / C dq Total work is then 1 1 U dw qdq C 2 Q 0 Q C 2 1 QV 2 Q 2 2 2C 2 U CV
Lecture 8-15 Where is the energy stored? Energy is stored in the electric field itself. Think of the energy needed to charge the capacitor as being the energy needed to create the electric field. To calculate the energy density in the field, first consider the constant field generated by a parallel plate capacitor, where - - - - - - - - - - - - - - ++++++++ +++++++ The electric field is given by: E - Q +Q Q 1 U E 2 0 Ad A 2 0 0 The energy density u in the field is given by: U 2 2 1 Q 1 Q 2 C 2 ( A / d) 0 This is the energy density, u, of the electric field. u U volume U Ad 1 2 2 0E Units: 3 J / m
Lecture 8-5 Capacitance Capacitor plates hold charge Q The capacitance C of a capacitor is a measure of how much charge Q it can store for a given potential difference V between the plates. C The two conductors hold charge +Q and Q, respectively. Q V Expect Q V Q Let C V Capacitance is an intrinsic property of the capacitor. Coulomb F Volt farad (Often we use V to mean V.)
Lecture 8-10 Numerical magnitudes Let s say: Area A= 1 cm², Then C 0A d separation d=1 mm 12 2 2 4 2 8.8510 ( C / Nm ) 10 ( m ) 3 10 ( m) 13 13 8.85 10 ( C/ N m) 8.85 10 ( F) This is on the order of 1 pf (pico farad). Generally the values of typical capacitors are more conveniently measured in µf or pf.
Lecture 8-4 Capacitors (II) A capacitor is a device that is capable of storing electric charges and thus electric potential energy. => charging Its purpose is to release them later in a controlled way. => discharging Capacitors are used in vast majority of electrical and electronic devices. Typically made of two conductors and, when charged, each holds equal and opposite charges. Capacitors come in a variety of sizes and shapes. The world's first integrated circuit included one capacitor
Capacitors Capacitors are devices that store energy in an electric field. Capacitors are used in many every-day applications Heart defibrillators Camera flash units Capacitors are an essential part of electronics. Capacitors can be micro-sized on computer chips or super-sized for high power circuits such as FM radio transmitters. 1 Monday, February 10, 2014
Capacitance Capacitors come in a variety of sizes and shapes. The world's first integrated circuit included one capacitor Capacitor controlling power which enters Fermilab, a particle accelerator laboratory in Batavia, IL 2 Monday, February 10, 2014
Parallel Plate Capacitor (2) The proportionality constant between the charge q and the electric potential difference V is the capacitance C. We will call the electric potential difference V the potential or the voltage across the plates. The capacitance of a device depends on the area of the plates and the distance between the plates, but does not depend on the voltage across the plates or the charge on the plates. The capacitance of a device tells us how much charge is required to produce a given voltage across the plates. 5 Monday, February 10, 2014