Capacitance. IV. Capacitance. 1a. Leyden Jar. Battery of Leyden Jars. A. The Electric Condenser. B. Dielectrics. C. Energy in Electric Field

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Merryl Blankenship
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1 IV. apacitance apacitance A. The Electric ondenser B. Dielectrics Revised: Feb5. Energy in Electric Field Sections.7-9 and 3.8 in book A. The Electric ondenser 3. History of the apacitor 4 ) History of the apacitor a) The Leyden Jar ) alculation from Geometry b) Parallel Plate apacitor 3) apacitors in ircuits c) The Law of apacitance a. Leyden Jar 5 Battery of Leyden Jars 6 Invented in 745 by Pieter van Musschenbroek (7 748) as a device to store electric fluid in a bottle (747?) Daniel Gralath was the first to combine several jars in parallel into a "battery" to increase the total possible stored charge. He demonstrated its effects on a chain of persons

2 b. Aepinus ondenser (c759) 7 The Franklin Square 8 The first parallel capacitor (called a condenser ) was probably developed by Franz Aepinus (around 759). Benjamin Franklin investigated the Leyden jar, and proved that the charge was stored on the glass, not in the water as others had assumed Leyden jars began to be made by coating the inside and outside of jars with metal foil, leaving a space at the mouth to prevent arcing between the foils. The spacing could be changed and materials inserted between the plates. Invented the Franklin Square, a capacitor using a square glass plate. c. Alessandro Volta s law of apacitance 9 Definition of apacitance 776 Law of apacitance: =V 78 called the device a condenser (derived from the Italian condensatore), with reference to the device's ability to store a higher density of electric charge than a normal isolated conductor The apacity to store charge Formal definition: apacitance=(charge stored)/(voltage applied) Units of apacitance: Farads =oulomb/volt Unit is too big! Usually we measure in microfarads.. alculation of apacitance a. Parallel Plate apacitor apacitance is a function only of the geometry of the device a) Parallel Plate apacitor Voltage between plates: V=Ed Electric Field: E = Aε b) ylindrical apacitor c) Spherical apacitors apacitance Formula: ε A = = = V Ed d

3 b. ylindrical apacitor 3 c. Spherical apacitor 4 A coaxial cable is an example of a cylindrical capacitor. The capacitance is given by the formula below where a is the radius of inner conductor, b is the (inner) radius of the outer conductor and the length is L. πε = L Ln a ( b ) Faraday experimented with spherical capacitors. The capacitance is given by the formula below where a is the radius of inner conductor, b is the (inner) radius of the outer conductor. 4πε = a b 3. apacitors in ircuits 5 3a. apacitors in Parallel 6 a) Parallel apacitors in Parallel add: = + b) Series Its like tanks next to each other have more capacity. Elements in parallel have same voltage, hence: c) R ircuits = q + q = V + V = ) V ( + 3b. apacitors in Series 7 3c. R ircuits (incomplete) 8 apacitors in series will have the same charge, but the total voltage is the sum. V V = = So we have: + V = + = + No notes here, as we did this in detail in lab. See lab notes See Section 3.8 in Knight (college Physics) Time constant for decay Hence Ohm Farad=Second τ = R Alternate form: = + 3

4 B. Dielectrics 9. Dielectric onstant ) Dielectric onstant a. 837 Faraday finds inserting an insulator (aka dielectric ) between plates will increase capacitance ) Electric Polarization 3) Dielectric Strength b. Increase by factor K (dielectric constant) i. Air: K=.59 ii. Water: K=8 c. The electric field (and hence voltage) is reduced by a factor of K. Electric Polarization 3. Dielectric Strength a. 833 Faraday shows an electric field induces dipoles in an insulator b. This dipole field opposes the applied field, and so the net field is reduced by a factor of K a. Definition is the maximum electric field before material ionizes (dielectric breakdown) c. Inside of dielectric, Gauss s law is valid if you replace ε ε=kε b. For Air: 3,, volts/meter c. This puts a limit on the maximum charge a capacitor can hold.. Energy in apacitors ) Energy Storage Formula ) Electric Stress 3) Energy stored in E field 3. Energy Storage a) Analogy: Recall Spring: Hooke Force: F=kx Work: W = F Δx = k x Δx Energy in Spring: kx 4 Note that it goes like the square of the displacement! 4

5 b. Hydro Analogy 5 c. apacitive 6 Energy stored in a tank is proportional to square of the volume! Pressure in tank Work done Energy in tank: V P = ρ g x = ρg A ρg W = P ΔV = V ΔV A ρg A V Energy stored is proportional to square of Voltage (or charge) Work done W = V Δ = Energy in apacitor: = V Δ V =. Electric Stress 7 b. Force Between Plates 8 a) harge on surface of conductor (i.e. plate of capacitor) experiences pressure due to the electric field that wants to tear it apart: F E Pressure: P = = A A Recall from Gauss s law the electric field near a conductor with surface charge: E = Hence: P = ε E Aε The opposite charges on the parallel plates of a capacitor attract each other. ΔU Force must be equal to change in energy: F = Δx There are two ways to argue this. The first is to have a charged capacitor, disconnected from the battery, so that the charge is constant. Then since, Aε = = d D A ε ΔU F = = Δx Aε c. Force Between Plates 9 3. Energy Stored in Electric Field 3 We get a different answer however if the capacitor is connected to a battery while the plates are pulled apart. Here the voltage will remain constant, but the charge will change as the plates are pulled apart (charge will be forced out because the capacitance decreases with increase in distance) Ouch, need calculus to derive the force between the plates: ε A V = d V ΔU ε A F = = V Δx d a) Energy in terms of Electric Field between plates: Energy in apacitor: = V = V Voltage: Relate harge to Field (in dielectric): Energy= ε E ( Ad ) U V = Ed E = Aε 5

6 3b. Energy Density 3 3c. Energy in Dielectric 3 Divide by the volume (Ad) to get the energy per unit volume: U u = = ε vol E Interpret that the energy IS stored in the electric field ε E ( Ad ) By inserting a dielectric the capacitance is increased. If capacitor is charged (but not connected to battery), then inserting dielectric will reduce the electric field by a factor of K, reducing the energy. Hence it will suck a dielectric into the plates. If capacitor is connected to a battery however, then the voltage is constant (E field unchanged) and so energy increases (because of increase in permittivity). Hence will push dielectric out! u = E ε 33 References Misc static electricity animations at History at:

Capacitors in Circuits

Capacitors in Circuits apacitors in ircuits apacitors store energy in the electric field E field created by the stored charge In circuit apacitor may be absorbing energy Thus causes circuit current to be reduced Effectively