Week 4 - Capacitors and Capacitance

Transcription

1 Week 4 - Cpcitors nd Cpcitnce Sheldon, you don t give your mother enough credit. She s wrm, she s loving, she doesn t glue electrodes to your hed to mesure your brin wves while potty trining. Leonrd Hofstdter in The Big Bng Theory s. 2 ep. 15 Exercise 4.1: Cpcitors A prllel-plte cpcitor is plced in vcuum with plte seprtion of 10.0 mm nd n re of 1.00 m 2. It is connected to potentil difference of 20.0 kv. ) Wht is the cpcitnce? C = ε 0 A d = nf (1) b) Wht is the chrge on ech plte? Q = CV = 17.5 µc (2) 1

2 c) Wht is the mgnitude of the electric field? E = σ ε 0 = Q ε 0 A = N/C (3) d) Wht is the energy stored on the cpcitor? U = Q2 2C (17.5 µc)2 = = J (4) nf Exercise 4.2: Discussion Questions ) A prllel-plte cpcitor is chrged by being connected to bttery nd is then disconnected from the bttery. The seprtion between the pltes is then doubled. How does the electric field chnge? The potentil difference? The totl energy? Explin your resoning. The electric field is constnt. The potentil difference increses. The totl energy increses. (This cn be esily seen from the different reltions between cpcitnce, distnce, voltge, etc. b) The freshness of fish cn be mesured by plcing the fish between the pltes of cpcitor nd mesuring the cpcitnce. How does this work? With time the moisture of the fish decys, cusing the dielectric constnt to chnge. If we know how fst the fish loses its moisture we cn mke reltion between the cpcitnce nd the freshness. c) The chrged pltes of prllel cpcitor ttrct echother, so if you pull the pltes distnce d prt you hve to do work. How much work is required? Equl to the new energy stored. W = U = Q2 2 C = Q2 d 2ε 0A Exercise 4.3: Cmer flsh cpcitor Electronic flsh units for cmers contin cpcitor for storing the energy used to produce the flsh. In one such unit, the flsh lsts for 1/675 s with n verge light power output of 0.27 MW. ) If the conversion of electricl energy to light is 95% efficient (the rest of the energy goes into therml energy), how much energy must be stored in the cpcitor before the flsh? E = 421 J (5) Week 4 September 10, compiled September 11, 2013

3 b) The cpcitor hs potentil difference between its pltes of 125 V when the stored energy equls the vlue clculted in prt (). Wht is the cpcitnce? C = 54 mf (6) Exercise 4.4: How is computer keystroke registered? d Figure 1: A keybord ppliction of plte cpcitor. One ppliction of cpcitor is to use it s sensor. This is used in some computers. Consider the button in Figure 1 with A = 50 mm 2 nd d = 0.6 mm. When you push button, the distnce d decreses nd the cpcitnce goes up. When the cpcitnce get bigger thn C = pf the keystroke is registered. Find the distnce we need to press the button down in order for the keystroke to be registered. A d Aε 0 C Exercise 4.5: Combining Cpcitors You need cpcitor with cpcitnce 0.25 µf, nd it is to be connected cross potentil difference of 960 V supplied by bttery. However, the 0.25 µf cpcitors vilble to you cn only hndle voltge of 600 V cross ech, or they will mlfunction. ) Show in digrm how n equivlent cpcitor with the desired properties cn be obtined. Week 4 September 10, compiled September 11, 2013

4 Figure 2 b) Suppose tht you ignored the dvice of not letting the voltge cross ech of the cpcitors exceed 600 V. Wht would hppen when you connected the the bttery of 960 V cross the cpcitor? Exercise 4.6: Cylindricl cpcitor s fuel sensor A cpcitor cn be used s fluid sensor. This type of sensor is commonly used s fuel sensors in lrge trnsport ircrft. The sensors cn lso mesure the wter level in tnk, which we will tke s our exmple in this exercise. Consider the coxil conducting cylinders in Figure 3 of rdii nd b nd length L. b L Figure 3: Cpcitor consisting of two coxil cylinders. ) Let the inner cylinder hs chrge Q nd the outer hs Q. Ignore the fringing fields nd use Guss lw to find the electric field between the cylinders expressed in terms of the cylinders surfce chrge σ. E = Q 2πε Lrˆr 0 Week 4 September 10, compiled September 11, 2013

5 b) Show tht the cpcitnce of the rrngement is given by C = 2πε 0L c) Find the energy stored in the electric field between the cylinders given tht = 1 mm, b = 2 mm nd Q = 0.2 nc. U = 1 Q 2 2 C d) Suppose now tht we prtilly submerge this cpcitor in wter with K = 80 nd tht the totl chrge on the cpcitor is unffected. See figure 4. Find n expression for V in the two regions (submerged nd not submerged). b z Figure 4: Cylinder cpcitor used s wter height sensor. Hint: The polrized wter will chnge the surfce chrge on the prt of the cpcitor immersed V ir = σ ir ε 0. V wter = σ wter Kε 0 e) You know tht conductor is n equipotentil. Cn you use this to find reltion between the surfce chrge in the two regions? Use this to find the cpcitnce C = C(z). σ wter = Kσ ir C = Q V = 2πε 0 (z(k 1) + L) ln b. Week 4 September 10, compiled September 11, 2013