PHYS 220, Engineering Physics, Chapter 24 Capacitance an Dielectrics Instructor: TeYu Chien Department of Physics an stronomy University of Wyoming Goal of this chapter is to learn what is Capacitance, its role in electronic circuit, an the role of ielectrics - How to use electric potential to move opposite signe charges to two ifferent conuctors? Using an external electric potential sources (batteries, power supply etc) In general, any two separate conuctors coul be use in above schematics capacitor is compose of a pair of conuctors insulate from each other capacitor coul be use to store opposite signe of electric charges on the two ifferent conuctors Capacitor is represente as the following symbols when raw in circuit - How much charge coul be store? (relate to the efinition of Capacitance) Capacitance is efine as how much charge coul be store in the capacitor per unit volt: C= Q V ab NOTE: The unit use for capacitance is Fara (F) an is efine as F= C V NOTE #2: Please pay attention that on't be confuse with to the notation use for capacitance (C) an the unit for charge (Q): coulomb (C)
- Capacitance calculation for few ifferent types of capacitors Parallel-plate capacitor C= Q =ϵ V 0 ab Spherical capacitor (Do Example 243 (page 792)) Cylinrical capacitor (Do Example 244 (page 792)) - Multiple capacitors in circuit equivalent capacitance When two capacitors are in series, the amount of charge is the same = + C eq C C 2 When two capacitors are in parallel, the electric potential is the same C eq =C +C 2 Equivalent capacitance is the capacitance that you measure from the two terminals regarless how complex the real capacitor collection is In other wors, when analyzing circuit, you can simplify multiple capacitors into one equivalent capacitor Do example 246 (page 796) - How much energy is store in capacitors? W =V q= q C q (Remember: C= Q V ) Q W = W = 0 q C q= Q 2 2 C = 2 CV 2 = 2 QV - Electric-fiel energy The energy store in capacitors coul be consiere as the form of electric fiel
We know that electric fiel is istribute all over the vacuum space between the two conuctors (of the capacitor) Thus, we coul fin out the energy ensity that is store in the form of electric fiel as (take parallel-plate capacitor as example): total energy u=energy per unit volumn= total volumn = 2 CV 2 Note that C= for parallel-plate capacitors Electric-fiel energy ensity in vacuum: u= 2 E 2 (This will be use again in Chapter 32) Do Example 247 (page 798) - How to increase the capacitance without changing the imension of the capacitor? The answer is to a insulating materials between the two conuctors to replace vacuum This insulating material is calle ielectric material How oes that work? () charges in the ielectric material near the conuctors will be inuce; (2) the inuce charges prouce electric fiel with OPPOSITE irection to the original one; (3) to maintain the same electric potential (V), the conuctors nee more charge Q; thus the capacitance increase; (3') or if charge (Q) oes not change, the electric fiel is lowere (as mentione in (2)), thus the electric potential (V) ecreases, which increases the capacitance C The change of the capacitance epens on the selection of the ielectric materials The key property of the ielectric material here is the Dielectric Constant, K (see Table 24 on page 80 for various types of ielectric materials) Quantitatively, the ielectric constant coul be efine as: K = C C 0 Remember, when without the ielectric material, C 0 = Thus, C=K C 0 =K =ϵ Here, we efine K =ϵ, where ϵ is calle the permittivity of the ielectric material
NOTE: Engineers use K more; while scientists use ϵ more NOTE #2: In many cases, K is also enote as ϵ r, means relative permittivity, ue to the relationship: K = ϵ =ϵ r Electric-fiel energy ensity in ielectric material: u= 2 K E 2 = 2 ϵ E 2 - Is there a maximum energy storage (or maximum voltage) in capacitor? Yes, there is a maximum value The threshol comes from the ielectric breakown when the electric fiel is stronger than the ielectric strength Dielectric strength refers to the maximum electric fiel the ielectric material coul withstan without breakown - Free charge vs Boun charge (Let's assume the charge on the conucting plates o not change): Free charge ( σ 0 ): charges that are free to move Boun charge ( σ i ): charges that are boune to molecules an cannot move freely They typically will form electric ipole in electric fiel (See Fig 249 on page 806) Total charge ( σ total ): the combination of free an boun charges: σ total =σ 0 σ i Note here that charge ensity, σ, only take care of the magnitue From Gauss's Law we learne: E 0 = σ 0 ; an E= σ 0 σ i ensity, σ, only take care of the magnitue Remember, capacitance is efine: C= Q V charge: C= Q free V Note here that charge, an the charge Q is actually the free In other wors, for the two situations above We know C=K C 0 =K Q free = Q free V V 2 where V is the voltage ifference in the left conition (without ielectric material), while V 2 is the voltage ifference in the right conition (with ielectric V material) This leas to: K =V 2 n we know in this parallel-plate conition, V = E That leas us to the
conclusion of E 0 K = E, an thus σ i =σ 0 ( K ) - Moification of Gauss's Law with the presence of ielectric materials Originally, E = Q encl Q = Q +Q free boun = Q free K Note here, the sign of charge is in K E = Q encl free or ϵ E =Q encl free Note that when oing the integration, the ϵ is epening on the materials where the Gauss's surfaces locate For a close Gauss's surface, ifferent part of the close surfaces might immerse in ifferent materials Typically, we only care about free charge, that's why in the moifie Gauss's Law, we only relate it to the free charge Math Preview for Chapter 25: Derivative Question to think: Let's focus on a conucting wire that connects to the capacitors During the charging/ischarging process (moving an removing charges in the capacitors, respectively), there is a voltage ifference How to escribe the process of the charge movement?