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23. Electrostatic Energy & Capacitors 靜電能和電容器. Electrostatic Energy 靜電能 Capacitors 電容器 Using Capacitors 使用電容器 Energy in the Electric Field 電場中的能. The lifesaving jolt of a defibrillator requires a large amount of energy delivered in a short time. 除纖顫器 救命的電擊需要在短時間內送出大批能量。
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23. Electrostatic Energy & Capacitors 靜電能和電容器 Electrostatic Energy 靜電能 Capacitors 電容器 Using Capacitors 使用電容器 Energy in the Electric Field 電場中的能
The lifesaving jolt of a defibrillator requires a large amount of energy delivered in a short time. 除纖顫器救命的電擊需要在短時間內送出大批能量。 Where does that energy come from? 這些能量從何而來？ Capacitor 電容器
23.1. Electrostatic Energy 靜電能 Electrostatic Energy = work done to assemble the charge configuration of a system. 靜電能= 裝組系統的電荷配置所需做的功。 Reference 參考 ( 0 energy 能): when all component charges are widely separated. 當所有的組分電荷都相隔甚遠時。 Bringing q1 in place takes no work. 把 q1 歸位不需做功 Bringing in q2 takes 把 q2 歸位需功 Bringing in q3 takes 把 q3 歸位需功 Total electrostatic energy 總靜電能
23.2. Capacitors 電容器 Capacitor: pair of conductors carrying equal but opposite charges. 電容器：帶著等量異性電荷的一對導體。 Usage: store electrical energy 用途：存儲電能 Parallel-Plate Capacitor 平行板電容器: 2 conducting plates of area A separated by a small distance d . 兩塊相隔短距離 d ，面積為 A 的導體板。 Plates are initially neutral. 開始時兩塊板都是中性。 They’re charged by connecting to a battery. 接上電瓶後起電。 Charge transfer plates are equal but oppositely charged. 電荷轉移 兩塊板的電荷量等而性異。 Large A, small d E 0 outside. 大 A ，小 d 器外的 E 0 。 Far from the edges 離邊遠處
Capacitance 電容量 Parallel-plate capacitor: 平行板電容器： C = Q / V = capacitance 電容量 Parallel-plate capacitor 平行板電容器 See Probs 參閱習題 41 & 42 Practical capacitor ~ F ( 106 F) or pF ( 1012 F ) 實用電容器 Charging / Discharging 充 / 放電
Energy Stored in Capacitors 存在電容器內的電能 When potential difference between capacitor plates is V, 電容器兩板之間的電位差為 V 時， work required to move charge dQ from to + plate is 把電荷 dQ 從 移至 + 板需作功 E dr < 0 Work required to charge the capacitor from 0 to V is 把電容器從 0 充電至 V 需作功 = U = energy stored in capacitor 存在電容器內的電能 Note 注意: In a “charged” capacitor, Q is the charge on the + plate. 在一個”帶電”的電容器內，Q 是 + 板上的電荷。 The total charge of the capacitor is always zero. 電容器的總電荷永遠為零。
Example 23.1. Parallel-Plate Capacitor 平行板電容器 • A capacitor consists of two circular metal plates of radius R = 12 cm, separated by d = 5.0 mm. • 一個電容器由兩塊圓形金屬板組成；板的半徑為 R = 12 cm，相距 d = 5.0 mm。 • Find 求 • Its capacitance 其電容量， • the charge on the plates, and 板上電荷，和 • the stored energy when the capacitor is connected to a 12-V battery. • 電容器在接上一個 12-V 電瓶後所存的能量。 (a) (b) (c)
23.3. Using Capacitors 使用電容器 Computer memories: billions of 25 fF capacitors. 電腦記憶體：數十億個 25 fF 電容器。 Rectifiers 整流器： mF Fuel-cells 燃料電池: 102 F 220-mF electrolytic capacitor 電解電容器 1 F 43 pF to 2.2 mF
Practical Capacitors 實用電容器 Inexpensive capacitors 廉價電容器: Thin plastic sandwiched between aluminum foils & rolled into cylinder. 薄塑料夾在兩鋁片間再捲成圓筒。 Electrolytic capacitors 電解電容器 (large capacitance 大電容量): Insulating layer developed by electrolysis. 絕緣層以電解生成。 Capacitors in IC circuits IC電路中的電容器(small capacitance 小電容量): Alternating conductive & insulating layers. 導體層和絕緣層交替。
Dielectrics 介電體 Dielectrics: insulators containing molecular dipoles but no free charges. 介電體：含有分子雙極但沒有自由電荷的絕緣體 Molecular dipoles aligned by E0 . 分子雙極依 E0 排向 Dielectric layer lowers V between capacitor plates by factor 1/ ( > 1). 電容器兩板間的介電質層把 V 降低1/ ( > 1)倍。 = dielectric constant 介電常數 Dipole fields oppose E0. 雙極場與 E0 反向。 Net field reduced to E = E0 / . 淨場減至 E = E0 / 。 Hence V = V0 / . 故 V = V0 / 。 Q is unchanged, so C =C0. Q不變，所以 C =C0。
表23.1 常見介電體的性質 介電常數 崩潰電場 介電材料 氧化鋁 : 2 ~ 10 mostly Working voltage V = Max safe potential < Ebkd d 工作電壓 V = 最大安全電位 < Ebkd d
Example 23.2. Which Capacitor? 那個電容器？ A 100-F capacitor has a working voltage of 20 V, 一個 100-F 電容器的運作電壓為 20 V ， while a 1.0-F capacitor is rated at 300 V. 另 一個 1.0-F 電容器可承受 300 V。 Which can store more charge? More energy? 那一個可以儲存較多電荷？較多能量？
GOT IT? 23.1. • You need to replace a capacitor with one that can store more energy. • 你需要把一個電容器換成一個存能較多的。 • Which will give you greater energy increase: • 下面那一個會給你較多的能量： • a capacitor with twice the capacitance and same working voltage as the old one, • 一個電容量是舊的二倍，運作電壓卻和它一樣的電容器， • or 或 • a capacitor with the same capacitance and twice the working voltage? • 一個電容量和舊的一樣，運作電壓卻是它二倍的電容器？
Connecting Capacitors 連接電容器 Two ways to connect 2 electronic components: parallel & series 兩個方法連接兩個電子元件：並聯和串聯 Parallel : Same V for both components 並聯： 兩元件的 V 一樣 Series : Same I (Q) for both components 串聯： 兩元件的 I (Q)一樣
Conceptual Example 23.1. Parallel & Series Capacitors 並聯與串聯電容 Using parallel-plate capacitors, explain why capacitance should increase with capcitors in parallel an decrease with capacitors in series. 用平板電容器來闡述為何並聯會增大電容量，串聯則減少電容量。 What happens to the working voltage in each case? 工作電壓又會怎樣？ Parallel-plate capacitor : 平板電容器 : in series 串聯 in parallel 並聯 Vworking < Vw1 +Vw2 d = d1 + d2 C decreases 減少 A = A1 + A2 C increases 增大 Vworking = min(Vw1 ,Vw2)
Making the Connection 連起來 You’ve got two 10-F capacitors rated at 15 V. 你有兩個 15 V 級的 10-F 電容量。 What are the capacitances & working voltages of their parallel & series combinations? 它們並聯和串聯後的電容量和工作電壓為何 ? Parallel : 並聯 : Series : 串聯 :
GOT IT? 23.2. • You have 2 identical capacitors with capacitance C. • 你有兩個一樣的電容器，電容量都是 C 。 • How would you connect them to get equivalent capacitances • 你會如何把它們連接來達到以下的電容量？ • 2 C, and 和 • ½ C ? • Which combination would have the higher working voltage? • 那一個組合有較高的運作電壓？ parallel 並聯 series 串聯
Example 23.2. Connecting Capacitors 連接電容器 Find the equivalent capacitance of the combinations shown in the Figure. 求圖中所示組合的相當電容量。 If the maximum voltage to be applied between points A and B is 100 V, 如果加到 A 和 B 之間的最高電壓是 100 V ， what should be the working voltage of C1 ? C1 的運作電壓為何？ ( min. working voltage ) 最小運作電壓
Bursts of Power 爆發性電力 Capacitors deliver higher energy much more quickly than batteries. 電容器送出的能量可以比電瓶的更大和更快。 Flash light 閃光燈: Battery charges capacitor, which then discharges to give flash. 電容器經電池充電後放電以產生閃光。 San Francisco’s BART train 三藩市的灣區捷運: KE of deceleration stored as EE in ultracapacitor. 減速時的 KE 以 EE 存於超級電容器中。 Stored EE is used to accelerate train. 儲存的 EE 再用來加速。 Other examples 其他例子: Defibrillator, controlled nuclear fusion, amusement park rides, hybrid cars, … 除纖顫器，受控的核子融合，遊樂場的設施，混合動力汽車，…
23.4. Energy in the Electric Field 電場中的能 Charging a capacitor rearranges charges energy stored in E 電容器充電時把電荷重組 能量存到 E 中 Energy density = energy per unit volume 能量密度= 單位體積內的能量 Parallel-plate capacitor : 平行板電容器： Energy density : 能量密度： is universal 是通用的
Example 23.4. A Thunderstorm 雷雨 Typical electric fields in thunderstorms average around 105 V/m. 雷雨中的電場一般平均有 105 V/m。 Consider a cylindrical thundercloud with height 10 km and diameter 20 km, 茲有一圓筒型雷雨雲塊，其高為 10 km，其直徑為 20 km ， and assume a uniform electric field of 1105 V/m. 假定其均勻電場為 1105 V/m。 Find the electric energy contained in this storm. 求此風暴所藏電能。 ~ 1400 gallons of gasoline. 加侖汽油
Example 23.5. A Shrinking Sphere 一個縮小的球 A sphere of radius R1 carries charge Q distributed uniformly over its surface. 一半徑為 R1 的球帶有一均勻分佈於其表面的電荷 Q。 How much work does it take to compress the sphere to a smaller radius R2 ? 把球壓至較小半徑 R2 時所需作的功為何？ 表面積為 4r2 … Work need be done to shrink sphere 把球縮小要作功 Extra energy stored here 新增的能量存於止 … 厚度為 dr
GOT IT? 23.3. You’re at point P a distance a from a point charge +q. 你在 P 點與一點電荷 +q 相距 a。 You then place a point charge q a distance a on the opposite side of P as shown. 之後你把一點電荷 q 放在 P 的另一邊，離它 a 處。 What happens to 以下各項會有怎麼變化？ • the electric field strength and 在 P 點的電場強度和 • the electric energy density at P ? 電能密度？ • Does the total electric energy U = ∫uE dV of the entire field increase, decrease, or remain the same? 整個電場的總電能 U = ∫uE dV 會增加，減少，還是不變？ doubles加倍 quadruples 四倍 decrease 減少 Negative work done to bring in –q. 把 –q 加進來要作的功是負的。
