The graded exams will be returned next Tuesday

The graded exams will be returned next Tuesday, Oct 4. You will have until the next class on Thursday, Oct 6 to rework the problems you got wrong and receive 50% added credit. Make sure you are in class as you will no have another opportunity to rework the exam. I will be going over the answers in class on Thursday. This will also be your only opportunity to ask for corrections/clarifications on any grading mistakes. The homework assignment will be on line this afternoon but will not be due until Tuesday, Oct 11. This will give you the opportunity to start work on the problems so that you will not be overloaded with homework and the exam rework next week.

Energy Storage in Capacitors Electric Field Energy Electric potential energy stored = amount of work done to charge the capacitor i. e. to separate charges and place them onto the opposite plates To transfer charge dq between conductors, work d. W=Vdq Charged capacitor – analog to stretched/compressed spring Capacitor has the ability to hold both charge and energy Density of energy (energy/volume) Energy is conserved in the E-field

In real life we want to store more charge at lower voltage, hence large capacitances are needed Increased area, decreased separations, “stronger” insulators Electronic circuits – like a shock absorber in the car, capacitor smoothes power fluctuations Response on a particular frequency – radio and TV broadcast and receiving Undesirable properties – they limit high-frequency operation

Example: Transferring Charge and Energy Between Capacitors Switch S is initially open 1) What is the initial charge Q 0? 2) What is the energy stored in C 1? 3) After the switch is closed what is the voltage across each capacitor? What is the charge on each? What is the total energy? a) b) c) when switch is closed, conservation of charge Capacitors become connected in parallel d) Where had the difference gone? It was converted into the other forms of energy (EM radiation)

Definitions • Dielectric—an insulating material placed between plates of a capacitor to increase capacitance. • Dielectric constant—a dimensionless factor that determines how much the capacitance is increased by a dielectric. It is a property of the dielectric and varies from one material to another. • Breakdown potential—maximum potential difference before sparking • Dielectric strength—maximum E field before dielectric breaks down and acts as a conductor between the plates (sparks)

Most capacitors have a non-conductive material (dielectric) between the conducting plates. That is used to increase the capacitance and potential across the plates. Dielectrics have no free charges and they do not conduct electricity Faraday first established this behavior

Capacitors with Dielectrics • Advantages of a dielectric include: 1. Increase capacitance 2. Increase in the maximum operating voltage. Since dielectric strength for a dielectric is greater than the dielectric strength for air • 3. Possible mechanical support between the plates which decreases d and increases C. To get the expression for anything in the presence of a dielectric you replace o with o

Field inside the capacitor became smaller – why? We know what happens to the conductor in the electric field Field inside the conductor E=0 outside field did not change Potential difference (which is the integral of field) is, however, smaller. There are polarization (induced) charges – Dielectrics get polarized

Properties of Dielectrics Redistribution of charge – called polarization dielectric constant of a material We assume that the induced charge is directly proportional to the E-field in the material when Q is kept constant In dielectrics, induced charges do not exactly compensate charges on the capacitance plates

Induced charge density Permittivity of the dielectric material E-field, expressed through charge density s on the conductor plates (not the density of induced charges) and permittivity of the dielectric (effect of induced charges is included here) Electric field density in the dielectric Example: A capacitor with and without dielectric Area A=2000 cm 2 d=1 cm; V 0 = 3 k. V; After dielectric is inserted, voltage V=1 k. V Find; a) original C 0 ; b) Q 0 ; c) C d) K e) E-field

Dielectric Breakdown Plexiglas breakdown Dielectric strength is the maximum electric field the insulator can sustain before breaking down

Molecular Model of Induced Charge Electronic polarization of nonpolar molecules

Electronic polarization of polar molecules In the electric field more molecular dipoles are oriented along the field

Polarizability of an Atom d- separation of proton and electron cloud in the applied electric field P- dipole moment per unit volume, N – concentration of atoms

Gauss’s Law in Dielectrics Gauss’s Law in. Dielectrics

Forces Acting on Dielectrics We can either compute force directly (which is quite cumbersome), or use relationship between force and energy Considering parallel-plate capacitor Force acting on the capacitor, is pointed inside, hence, E-field work done is positive and U - decreases x – insertion length Two capacitors in parallel w – width of the plates More charge here constant force