Electricity & Magnetism Lecture 8: apacitors Today s oncept: apacitors (apacitors in a circuits, Dielectrics, Energy in capacitors) Alternate terms: condensors, capacitators,... Electricity & Magne/sm Lecture 8, Slide 1
Stuff you asked about:! the formula for calcula/ng capacitance in parallel and series are the opposite of the formula for resistance! Please go over the apac/tors with and without a dielectric from checkpoint.! this stuff is a lot harder than what we ve been doing before, one thing i don t understand is when you disconnect a bahery from a capacitor what exactly happens, how come it s/ll has a charge?? Electricity & Magne/sm Lecture 7, Slide 2
apacitors, connected and unconnected A capacitor that is unconnected to a baaery has constant charge: = Q/ ( is determined by Q) apacitors in parallel have the same voltage. harge may redistribute among them: Q 1 / 1 = Q 2 / 2 = Q 2 / 2 A capacitor connected to a baaery has a constant voltage. Q = (Q is determined by )
Dielectric harged capacitor, not connected to baaery Dielectric makes Δ smaller air capacitor dielectric capacitor = 1 > 1 E = ε0a/l E = ε0a/l E inside dielectric is smaller than E outside. The charge on the surface of the dielectric partially cancels the E field from the charge on the plates. Therefore the Δ between the plates is less.
Simple apacitor ircuit Q Q = Q This Q really means that the baaery has moved charge Q from one plate to the other, so that one plate holds Q and the other Q. Electricity & Magne/sm Lecture 8, Slide 3
Dielectrics 0 Q 0 = 0 1 = 0 Q 1 = 1 By adding a dielectric you are just making a new capacitor with larger capacitance (factor of ) Electricity & Magne/sm Lecture 8, Slide 4
Parallel apacitor ircuit Q total 1 2 Q 1 = 1 Q 2 = 2 Q total Key point: is the same for both capacitors Key Point: Q total = Q 1 Q 2 = 1 2 = ( 1 2 ) total = 1 2 Electricity & Magne/sm Lecture 8, Slide 5
Series apacitor ircuit Q Q = total 1 Q 1 2 2 Q Key point: Q is the same for both capacitors Key point: Q = total = 1 1 = 2 2 Also: = 1 2 Q/ total = Q/ 1 Q/ 2 1 1 1 = total 1 2 Electricity & Magne/sm Lecture 8, Slide 6
heckpoint: Three apacitor onfigurations The three configura/ons shown below are constructed using iden/cal capacitors. Which of these configura/ons has lowest total capacitance? A B total = 1/ total = 1/ 1/ = 2/ total = 2 D: All 3 are the same total = /2 Electricity & Magne/sm Lecture 8, Slide 7
heckpoint: Two apacitor onfigurations The two configura/ons shown below are constructed using iden/cal capacitors. Which of these configura/ons has the lowest overall capacitance? A B total = left = /2 right = /2 total = left right total = o A o B o Both configura/ons have the same capacitance Electricity & Magne/sm Lecture 8, Slide 8
licker Question: apacitor Network Q 2 2 0 1 2 3 1 Q 1 3 Q 3 Which of the following is NOT necessarily true: A) 0 = 1 B) total > 1 ) 2 = 3 D) Q 2 = Q 3 E) 1 = 2 3 total Electricity & Magne/sm Lecture 8, Slide 9
heckpoint: apacitor Network A circuit consists of three unequal capacitors 1, 2, and 3 which are connected to a bahery of voltage 0. The capacitance of 2 is twice that of 1. The capacitance of 3 is three /mes that of 1. The capacitors obtain charges Q 1, Q 2, and Q 3. Q 2 ompare Q 1, Q 2, and Q 3. 2 A. Q 1 > Q 3 > Q 2 B. Q 1 > Q 2 > Q 3. Q 1 > Q 2 = Q 3 0 1 1 Q 1 2 3 3 Q 3 D. Q 1 = Q 2 = Q 3 E. Q 1 < Q 2 = Q 3 Electricity & Magne/sm Lecture 8, Slide 11
Energy in a apacitor In Prelecture 7 we calculated the work done to move charge Q from one plate to another: Q Q U = 1 / 2 Q = 1 / 2 2 Since Q = = 1 / 2 Q 2 / This is potental energy waitng to be used Electricity & Magne/sm Lecture 8, Slide 12
Messing with apacitors If connected to a baaery stays constant 1 = 1 = Q 1 = 1 1 = = Q If isolated then total Q stays constant Q 1 = Q 1 = 1 = Q 1 / 1 = Q/ = / Electricity & Magne/sm Lecture 8, Slide 13
heckpoint: apacitors and Dielectrics 1 Two iden/cal parallel plate capacitors are given the same charge Q, ader which they are disconnected from the bahery. Ader 2 has been charged and disconnected, it is filled with a dielectric. ompare the voltages of the two capacitors. A. 1 > 2 B. 1 = 2. 1 < 2 The electric field decreases on the second due to the dielectric, so its voltage must decrease. The lihle brown thing is stopping the pikachus from reaching the other side. Dielectrics don't change voltages. apacitance lowers in case 2, meaning increases in case 2. Electricity & Magne/sm Lecture 8, Slide 14
licker Question: Messing with apacitors Two identcal parallel plate capacitors are connected to identcal baaeries. Then a dielectric is inserted between the plates of capacitor 1. ompare the energy stored in the two capacitors. 0 1 A) U 1 < U 0 B) U 0 = U 1 ) U 1 > U 0 ompare using U = 1 / 2 2 U 1 /U 0 = PotenTal Energy goes UP Electricity & Magne/sm Lecture 8, Slide 15
heckpoint: apacitors and Dielectrics 2 Two iden/cal parallel plate capacitors are given the same charge Q, ader which they are disconnected from the bahery. Ader 2 has been charged and disconnected, it is filled with a dielectric. ompare the poten/al energy stored by the two capacitors. A. U 1 > U 2 B. U 1 = U 2. U 1 < U 2 If the electric field goes down, then so does the poten/al energy of the second system.. Dielectrics don't affect poten/al. U = Q. Since voltage remains constant and charge increases, poten/al energy must also increase. Electricity & Magne/sm Lecture 8, Slide 16
heckpoint: apacitors and Dielectrics 3 The two capacitors are now connected to each other by wires as shown. How will the charge redistribute itself, if at all? must be the same! o The charges will flow so that the charge on 1 will become equal to the charge on 2. o The charges will flow so that the energy stored in 1 will become equal to the energy stored in 2 o The charges will flow so that the poten/al difference across 1 will become the same as the poten/al difference across 2. o No charges will flow. The charge on the capacitors will remain what it was before they were connected. Q: U: Electricity & Magne/sm Lecture 8, Slide 17
alculation x 0 0 x 0 /4 An air-gap capacitor, having capacitance 0 and width x 0 is connected to a baaery of voltage. onceptual Analysis: A dielectric ( ) of width x 0 /4 is inserted into the gap as shown. What changes when the dielectric added? What is Q f, the final charge on the capacitor? A) Only B) only Q ) only D) and Q E) and Q Adding dielectric changes the physical capacitor does not change and changes changes Q changes Electricity & Magne/sm Lecture 8, Slide 18
alculation x 0 0 x 0 /4 An air-gap capacitor, having capacitance 0 and width x 0 is connected to a baaery of voltage. Strategic Analysis: alculate new capacitance Apply defini/on of capacitance to determine Q To calculate, let s first look at: A dielectric ( ) of width x 0 /4 is inserted into the gap as shown. What is Q f, the final charge on the capacitor? left right A) left < right B) left = right ) left > right The conduc/ng plate is an equipoten/al! Electricity & Magne/sm Lecture 8, Slide 19
alculation x 0 0 An air-gap capacitor, having capacitance 0 and width x 0 is connected to a ba5ery of voltage. an consider capacitor to be two capacitances, 1 and 2, in parallel x 0 /4 A dielectric ( ) of width x 0 /4 is inserted into the gap as shown. What is Q f, the final charge on the capacitor? = 1 2 What is 1? A) 1 = 0 B) 1 = 3 / 4 0 ) 1 = 4 / 3 0 D) 1 = 1 / 4 0 In general. For parallel plate capacitor: = ε 0 A/d A = 3 / 4 A 0 d = d 1 = 3 / 4 (ε 0 A 0 /d 0 ) 1 = 3 / 4 0 0 Electricity & Magne/sm Lecture 8, Slide 20
alculation x 0 0 An air-gap capacitor, having capacitance 0 and width x 0 is connected to a ba5ery of voltage. x 0 /4 A dielectric ( ) of width x 0 /4 is inserted into the gap as shown. = 1 2 1 = 3 / 4 0 What is Q f, the final charge on the capacitor? What is 2? A) 2 = 0 B) 2 = 3 / 4 0 ) 2 = 4 / 3 0 D) 2 = 1 / 4 0 In general. For parallel plate capacitor filled with dielectric: = ε 0 A/d A = 1 / 4 A 0 d = d 0 = ¼(ε 0 A 0 /d 0 ) 2 = 1 / 4 0 Electricity & Magne/sm Lecture 8, Slide 21
x 0 0 alculation An air-gap capacitor, having capacitance 0 and width x 0 is connected to a ba5ery of voltage. x 0 /4 A dielectric ( ) of width x 0 /4 is inserted into the gap as shown. = 1 2 What is Q f, the final charge on the capacitor? 1 = 3 / 4 0 2 = 1 / 4 0 What is? A) B) ) = 1 1 1 2! 1 = parallel combina/on of 1 and 2 : = 1 2 = 0 ( 3 / 4 1 / 4 ) Electricity & Magne/sm Lecture 8, Slide 22
x 0 0 alculation An air-gap capacitor, having capacitance 0 and width x 0 is connected to a ba5ery of voltage. x 0 /4 A dielectric ( ) of width x 0 /4 is inserted into the gap as shown. = 1 2 What is Q f, the final charge on the capacitor? 1 = 3 / 4 0 2 = 1 / 4 0 = 0 ( 3 / 4 1 / 4 ) What is Q? 3 Q = 0 4 1! 4 apple Electricity & Magne/sm Lecture 8, Slide 23
Different Problem Q 0 x 0 0 1/4x 0 An air-gap capacitor, having capacitance 0 and width x 0 is connected to a ba5ery of voltage and then ba5ery is disconnected. A dielectric () of width 1/4x 0 is inserted into the gap as shown. A) f < B) f = ) f > What is f, the final voltage on the capacitor? Q stays same: no way to add or subtract We know : (property of capacitor) Q = Q 0 = 0 = 0 ( 3 / 4 1 / 4 ) f = Q/ = /( 3 / 4 1 / 4 ) Electricity & Magne/sm Lecture 8, Slide 24
Different Problem Q 0 x 0 0 1/4x 0 An air-gap capacitor, having capacitance 0 and width x 0 is connected to a ba5ery of voltage and then ba5ery is disconnected. f = Q/ = /( 3 / 4 1 / 4 ) How did energy stored in capacitor change when dielectric inserted? A dielectric () of width 1/4x 0 is inserted into the gap as shown. What is f, the final voltage on the capacitor? A) U increased B) U stayed same ) U decreased U = ½ Q 2 / Q remained same increased U decreased Electricity & Magne/sm Lecture 8, Slide 25