Short Version 23 Electrostatic Energy Capacitors 23 1

Short Version : 23. Electrostatic Energy & Capacitors

23. 1. Electrostatic Energy = work done to assemble the charge configuration of a system. Reference ( 0 energy): when all component charges are widely separated. Bringing q 1 in place takes no work. Bringing in q 2 takes Bringing in q 3 takes Total electrostatic energy

23. 2. Capacitors Capacitor: pair of conductors carrying equal but opposite charges. Usage: store electrical energy Parallel-Plate Capacitor: 2 conducting plates of area A separated by a small distance d. Plates are initially neutral. They’re charged by connecting to a battery. Charge transfer plates are equal but oppositely charged. Large A, small d E 0 outside. Far from the edges

Capacitance Parallel-plate capacitor: C = Q / V = capacitance Parallel-plate capacitor See Probs 41 & 42 Practical capacitor ~ F ( 10 6 F) or p. F ( 10 12 F ) Charging / Discharging

Energy Stored in Capacitors When potential difference between capacitor plates is V, work required to move charge d. Q from to + plate is E dr < 0 Work required to charge the capacitor from 0 to V is = U = energy stored in capacitor Note: In a “charged” capacitor, Q is the charge on the + plate. The total charge of the capacitor is always zero.

Example 23. 1. Parallel-Plate Capacitor A capacitor consists of two circular metal plates of radius R = 12 cm, separated by d = 5. 0 mm. Find (a) Its capacitance, (b) the charge on the plates, and (c) the stored energy when the capacitor is connected to a 12 -V battery. (a) (b) (c)

Practical Capacitors Inexpensive capacitors: Thin plastic sandwiched between aluminum foils & rolled into cylinder. Electrolytic capacitors (large capacitance): Insulating layer developed by electrolysis. Capacitors in IC circuits (small capacitance): Alternating conductive & insulating layers.

Dielectrics: insulators containing molecular dipoles but no free charges. Molecular dipoles aligned by E 0. Dielectric layer lowers V between capacitor plates by factor 1/ ( > 1). = dielectric constant Dipole fields oppose E 0. Net field reduced to E = E 0 / . Hence V = V 0 / . Q is unchanged, so C = C 0.

: 2 ~ 10 mostly Working voltage V = Max safe potential < Ebkd d

Example 23. 2. Which Capacitor? A 100 - F capacitor has a working voltage of 20 V, while a 1. 0 - F capacitor is rated at 300 V. Which can store more charge? More energy?

Connecting Capacitors Two ways to connect 2 electronic components: parallel & series Parallel: Same V for both components Series: Same I (Q) for both components

Bursts of Power Capacitors deliver higher energy much more quickly than batteries. Flash light: Battery charges capacitor, which then discharges to give flash. San Francisco’s BART train: KE of deceleration stored as EE in ultracapacitor. Stored EE is used to accelerate train. Other examples: Defibrillator, controlled nuclear fusion, amusement park rides, hybrid cars, …

23. 4. Energy in the Electric Field Charging a capacitor rearranges charges energy stored in E Energy density = energy per unit volume Parallel-plate capacitor: Energy density : is universal

Example 23. 4. A Thunderstorm Typical electric fields in thunderstorms average around 105 V/m. Consider a cylindrical thundercloud with height 10 km and diameter 20 km, and assume a uniform electric field of 1 105 V/m. Find the electric energy contained in this storm. ~ 1400 gallons of gasoline.

Example 23. 5. A Shrinking Sphere A sphere of radius R 1 carries charge Q distributed uniformly over its surface. How much work does it take to compress the sphere to a smaller radius R 2 ? Work need be done to shrink sphere Extra energy stored here