Physics 272 February 3 Spring 2015 www.phys.hawaii.edu/~philipvd/pvd_15_spring_272_uhm go.hawaii.edu/KO Prof. Philip von Doetinchem philipvd@hawaii.edu PHYS272 - Spring 15 - von Doetinchem - 186 Gauß's law ● Electric flux: ● Gauß's law: PHYS272 - Spring 15 - von Doetinchem - 188 Electric potential energy and electric potential ● Charged particle moving in a field: field exerts work on particle ● Conservative force → required work independent of exact path ● ● Moving with the direction of the electric field means moving in the direction of decreasing potential Moving a charge slowly against an electric field requires an external force, equal and opposite to the electric force. PHYS272 - Spring 15 - von Doetinchem - 189 Capacitance and dielectrics ● ● Capacitor: just insulate two conductors (with same amount of negative and positive charge) Work must be done to move charges through the resulting potential → Electric field can be seen as a store-house of electric potential energy Source: http://de.wikipedia.org/wiki/Kondensator_%28Elektrotechnik%29 ● ● Capacitor has a certain capacitance depending on its properties: size, shape, material Many applications (flashs, filters, microphone, etc.) PHYS272 - Spring 15 - von Doetinchem - 190 Capacitors and capacitance ● ● ● ● Charging capacitor: conductors initially uncharged Transfer electrons from one side to the other Net charge on capacitor is zero Common way of charging: connect sides to different terminals of a battery ● Potential difference is proportional to the stored charge ● Capacitance stays constant: ● Capacitance is a measure of the ability of a capacitor to store energy. PHYS272 - Spring 15 - von Doetinchem - 191 Capacitance of a parallel plate capacitor Gauss' law: ● One farad is a very large amount: typical values: – flash unit in a camera: microfarad (F, 10-6) – radio tuning unit: 10-100 picofarad (pF, 10 -12) PHYS272 - Spring 15 - von Doetinchem - 192 Spherical capacitor ● Outer sphere makes no contribution to the field between the sphere PHYS272 - Spring 15 - von Doetinchem - 193 Capacitors in series ● ● ● ● ● Combining capacities helps you to get the capacitance you need for your application Series connection: capacitors are connected one after the other Charges on all plates have the same magnitude Equivalent capacitance of a series combination of capacitors is always less than any individual capacitance. Charges on plates are the same, but if the dimensions are different → potential for each capacitor different PHYS272 - Spring 15 - von Doetinchem - 196 Capacitors in parallel ● ● ● ● Charges can reach capacitors independently from the source Imagine one big capacitor that you split into multiple smaller capacitors The parallel combination of capacitors always has a higher capacitance than the individual capacitances Charges are generally not the same on each capacitor PHYS272 - Spring 15 - von Doetinchem - 197 Capacitor network PHYS272 - Spring 15 - von Doetinchem - 198 Energy storage in capacitors and electric-field energy ● ● Many important applications of capacitors rely on storing energy Electric potential energy stored in a charged capacitor is equal to the amount of work to separate opposite charges ● Discharging of a capacitor: electric field does work ● Work required for charging a capacitor: ● Potential energy of uncharged capacitor set to zero ● Less work is required to transfer charge if capacitance is higher PHYS272 - Spring 15 - von Doetinchem - 203 Energy considerations ● ● ● Capacitance measures the ability to store energy and charge Charge a capacitor then disconnect battery/source – No more charges can move onto the capacitor: charge is constant – When moving plates (i.e., changing capacitance) potential difference changes Change capacitance of a capacitor while leaving the battery/source connected – charges can move onto or off the capacitor: charge is changing – Potential difference stays constant PHYS272 - Spring 15 - von Doetinchem - 204 Electric field energy ● Charge a capacitor by moving electrons from one plate to the other: work against the electric field between the plates ● Energy is stored in the field in the region between the plates ● Energy per unit volume (energy density): Does not depend on geometry! PHYS272 - Spring 15 - von Doetinchem - 205 Electric field energy ● ● ● Not only valid for parallel-plate capacitors → true for any electric field configuration in vacuum Vacuum can have electric fields and is in this sense not really empty Two interpretations: – energy is property of an electric field – or a shared property of all charges creating the field PHYS272 - Spring 15 - von Doetinchem - 206 Eletric field energy ● Magnitude of electric field to store 1J in a volume of 1m3 ● Relation between electric field and energy density ● Dry air can sustain ~3MV/m without breaking down PHYS272 - Spring 15 - von Doetinchem - 207 Dielectrics ● ● ● Most capacitors have non-conducting material between them: dielectric Helps separating two oppositely charged conductors Prevents ionization of the air and allows for higher potential build up before discharges occur → store higher energies PHYS272 - Spring 15 - von Doetinchem - 208 Dielectrics http://www.youtube.com/watch?v=e0n6xLdwaT0 PHYS272 - Spring 15 - von Doetinchem - 209 Dielectrics ● constant voltage: – separation between plates widened: ● ● – plates closer together: ● ● electrometer shows charge flowing off of the plates electroscope shows no change in voltage charge flows back onto the plates fixed amount of charge onto left plate: – separation is widened ● – Plexiglas inserted between the plates: ● – electroscope shows a rising voltage (charge stays constant) voltage drops When Plexiglas is removed ● voltage rises back up again → charge is still there PHYS272 - Spring 15 - von Doetinchem - 210 Induced charge and polarization ● ● ● When dielectric is inserted and charge is kept constant → potential difference drops → electric field drops → surface charge density drops, but not the charge Redistribution of charges in dielectric occurs: polarization For a given charge the potential difference is reduced by a factor K: K=Cdielectric/Cvacuum K is always larger than unity ● Kair=1.0006 KWater=80.4 poor conductor, but good solvent → ions dissolve in water → current flow PHYS272 - Spring 15 - von Doetinchem - 211 Induced charge and polarization PHYS272 - Spring 15 - von Doetinchem - 212 Induced charge and polarization ● High capacitances can be reached with a dielectric with high K value PHYS272 - Spring 15 - von Doetinchem - 213 Capacitor with and without dielectric ● ● A capacitor is charged and disconnected from the battery After that a dielectric is moved in between the plates PHYS272 - Spring 15 - von Doetinchem - 214 Capacitor with and without dielectric 0 PHYS272 - Spring 15 - von Doetinchem - 215 Capacitor with and without dielectric ● ● Electric potential energy stored in a capacitor is reduced when inserting a dielectric into the capacitor The fringing field at the edges of the capacitor exerts force on the dielectric, effectively trying to pull in the dielectric → the work the electric field does is taken from the stored energy PHYS272 - Spring 15 - von Doetinchem - 216 Additional material PHYS272 - Spring 15 - von Doetinchem - 217 Z machine http://www.youtube.com/watch?v=TVaIvAPMd_g PHYS272 - Spring 15 - von Doetinchem - 218 Cylindrical capacitor ● Important property of parallel-plate, spherical, an cylindrical capacitors: capacitance just depends on dimensions PHYS272 - Spring 15 - von Doetinchem - 219 Energy stored in a capacitor PHYS272 - Spring 15 - von Doetinchem - 220 Energy stored in a capacitor PHYS272 - Spring 15 - von Doetinchem - 221