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Electricity and Magnetism Inductance Transformers Maxwell's Laws Electricity and Magnetism Inductance Transformers Maxwell's Laws Lana Sheridan De Anza College Dec 1, 2015 Last time • Ampere's law • Faraday's law • Lenz's law Overview • induction and energy transfer • induced electric fields • inductance • self-induction • RL Circuits Inductors A capacitor is a device that stores an electric field as a component of a circuit. inductor a device that stores a magnetic field in a circuit It is typically a coil of wire. 782 Chapter 26 Capacitance and Dielectrics ▸ 26.2 continued Categorize Because of the spherical symmetry of the sys- ϪQ tem, we can use results from previous studies of spherical Figure 26.5 (Example 26.2) systems to find the capacitance. A spherical capacitor consists of an inner sphere of radius a sur- ϩQ a Analyze As shown in Chapter 24, the direction of the rounded by a concentric spherical electric field outside a spherically symmetric charge shell of radius b. The electric field b distribution is radial and its magnitude is given by the between the spheres is directed expression E 5 k Q /r 2. In this case, this result applies to radially outward when the inner e sphere is positively charged. 820 the field Cbetweenhapter 27 the C spheresurrent and (a R esistance, r , b). b S Write an expression for the potential difference between 2 52 ? S 782 Chapter 26 Capacitance and DielectricsTable 27.3 CriticalVb TemperaturesVa 3 E d s a the two conductors from Equation 25.3: for Various Superconductors ▸ 26.2 continued Material Tc (K) b b dr 1 b ApplyCategorize the result Because of Example of the spherical 24.3 forsymmetry the electric ofHgBa the sys 2fieldCa- 2Cu 3O8 2 52134 52 Ϫ5Q Vb Va 3 Er dr ke Q 3 2 ke Q Tl—Ba—Ca—Cu—O 125a a r r a outsidetem, wea sphericallycan use results symmetric from previous charge studies distribution of spherical Figure 26.5 (Example 26.2) systems to findS the capacitance. S Bi—Sr—Ca—Cu—O 105 c d and note that E is parallel to d s along a radial line: A spherical capacitor consists of 1 1 a 2 b YBa2Cu3O7 an inner sphere of radius a sur- 92 ϩQ a (1) Vb 2 Va 5 ke Q 2 5 ke Q Analyze As shown in Chapter 24, the directionNb of Gethe rounded by a concentric spherical23.2 782 Chapter 26 Capacitance and Dielectrics 3 b a ab electric field outside a spherically symmetric NbchargeSn shell of radius b. The electric field18.05 3 a b b distribution is radial and its magnitude is givenNb by the between the spheres is directed 9.46 Courtesy of IBM Research Laboratory Q Q ▸ 26.2 continued 2 radially outward when the inner ab Substituteexpression the E 5absolute keQ /r . Invalue this case,of D Vthis into result Equation appliesPb to 26.1: 5 5 7.18 5 sphere is positivelyC charged. Ϫ (26.6) A small permanentCategorize magnet levi- Because of the spherical, symmetry, of the sys- DV V 2 QV k b 2 a the fieldtem, we between can use the results spheres from previous (a studiesr b). of spherical Hg b4.15 a e tated above a disk of the super- Figure 26.5 (Example 26.2) systems to find the capacitance. Sn A spherical capacitor consists of b 3.72 conductor YBa Cu O , which is in S ϩQ a Write2 3 an7 expression for the potential difference betweenAl an inner sphere ofV radius2 Va sur52- E ? d Ss 0 1.19 0 1 2 liquid nitrogenFinalize at 77Analyze TheK. Ascapacitance shown in Chapter depends 24, the direction on aof andthe b roundedas expected. by a concentricb sphericalThea potential3 difference between the spheres in Equation the twoelectric conductors field outside from a spherically Equation symmetric 25.3: charge Zn shell of radius b. The electric field a 0.88 b (1) is negativedistribution because is radial and Q its is magnitude positive is given and by b the. a. betweenTherefore, the spheres isin directed Equation 26.6, when we take the absolute value, we change expression E 5 k Q /r 2. In this case, this result applies to radially outward when the inner b b b e sphere is positively charged. dr 1 a 2Apply b tothe bthe field2 result a.between The of the Example resultspheres ( ais ,24.3 ar ,positive bfor). the electric number field. 2 52 52 5 Vb Va 3 Er dr ke Q 3 2 ke Q Today, thousands of superconductors are known, and as Table 27.3 rillustrates, b a a r a outside a spherically symmetric charge distribution S WHAT WriteIF ? an expressionIf Sthe radius for the potential b of the difference outer between sphere approaches2 52 ?infinity,S what does the capacitance become? and note that E is parallelthe to criticald Ss along temperatures a radial line: V bof Vrecentlya 3 E discoveredd s superconductors are substantiallyc d the two conductors from Equation 25.3: a 1 1 a 2 b higher than initially thought possible.(1) V b Two2 Va 5kindske Q of2 superconductors5 ke Q are recog- Answer In Equation 26.6, we let b `: b b b a ab nized. The Smore recently identified ones are dressentially1 b ceramics with high criti- Apply the result of Example 24.3 for the electric field V 2 V 52 E dr 52k Q 5 k Q b aab 3 r abe 3 r 2 a ae r b outside a spherically symmetriccal temperatures,charge distribution whereas superconductinga Q amaterialsQ sucha ab as those observed by S S D C 5 lim 5 5 5 c4pd P0a 980 SubstituteChapterand note the32 that absolute EInductance is parallel value to d s of along V ainto radial Equation line: S26.1: C 5 5 5 (26.6) Kamerlingh-Onnes areb `metals.ke b 2 If a room-temperature1ke 1b ka e2 b superconductor is ever iden- (1) Vb 2 Va 5 ke Q DV2 5Vbke2Q Va ke b 2 a tified, its effect on technology could beb tremendous.a ab Notice that this expression is the same as Equation 26.2, thea capacitanceb of an isolated spherical conductor. The value of T is sensitive1Q to chemicalQ2 1 2abcomposition,0 0 1pressure,2 and molecular Substitute the absolute value of DV into Equation 26.1:c C 5 5 5 (26.6) ▸ 32.5 continuedFinalize The capacitance depends on a and b as expected.D The 2potential difference2 between the spheres in Equation structure. Copper, silver, andV gold,Vb whichVa k eareb aexcellent conductors, do not exhibit (1) is negative because Q is positive and b . a. Therefore, in Equation 26.6, when we take the absolute value, we change superconductivity. 0 0 1 2 SOLUTI O N a 2 bFinalize to b 2 The a. Thecapacitance result depends is a positive on a and number b as expected.. The potential difference between the spheres in Equation (1) is negative because Q is positiveOne and truly b . a . remarkableTherefore, in Equation feature 26.6, whenof superconductors we take the absolute value, is wethat change once a current is set up WHAT IF ? If the radius b of the outer sphere approaches infinity, what does the capacitance become? Conceptualize Bea sure 2 b to byou 2 a. Thecan result identify isin a positivethem, the number it two 26.3persists. coils without inCombinations the any situation applied andpotential understand ofdifference Capacitors that(because a changing R 5 0). currentSteady cur in -one coil induces a currentAnswerWHA T In inIF Equation? theIf the second radius 26.6, rentsb of coil.we the let outerhave b S sphere been`: approaches observed infinity, to whatpersist does the in capacitance superconducting become? loops for several years with Answer In Equation 26.6,no we letapparent b S `: decay! ab ab a TwoC 5 orlimab more ab capacitorsa5 5often5 4arepP0a combined in electric circuits. We can calculate Categorize We will determine the result using conceptsb S ` discussed in this section, so we categorize this example as a An importantC 5 lim andk 5eusefulb 2 a5 application5k4epPb0a k eof superconductivity is in the development b S ` k b 2 a k b k substitution problem. of superconductingthe equivalente magnets,e capacitancee in which the of magnitudescertain combinations of the magnetic using field methods are described in NoticeNotice that that this this expression expression is theis the same same as Equation as Equation 26.2, the1 capacitance26.2,2 the ofcapacitance an1 2isolated spherical of an conductor.isolated spherical conductor. Capacitor approximatelythis section.ten1 times2 Throughout1 2greater than this those section, produced we assume by the thebest capacitors normal elec to- be combined are Circuit component symbols NB Use Equation 30.17symbol to express the magnetictromagnets. initiallyfield Such in thesuperconductinguncharged. B 5m magnets i are being considered as a means of 0 , interior of the base solenoid: storing energy. In Superconductingstudying electric magnets circuits, are we currently use a simplified used in medical pictorial magnetic representation called a Combinations of Capacitors resonance26.3circuit imaging, diagram. or MRI, Suchunits, awhich diagramF produce uses high-quality circuit symbols images toof internalrepresent various circuit Battery ϩ 26.3 Combinations NofH CapacitorsBH N H BA NBNH Find the mutual inductance, notingorgans thatTwo the withoutor more magnetic capacitors the need often for are excessivecombinedM 5 in exposure electric 5circuits. of patientsWe can5 calculate mto0 x-rays orA other harm- batterysymbol V Ϫ the equivalentelements. capacitance The of certaincircuit combinations symbolsi using are methods connectedi described inby , straight lines that represent the flux F through the handle’s coilful caused radiation.
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