( ) Energy storage in CAPACITORs. q C

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Abner Dalton
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1 Energy storage in CAPACITORs Charge capacitor by transferring bits of charge q at a time from bottom to top plate. Can use a battery to o this. Battery oes work which increase potential energy of capacitor. q is magnitue of charge on plates V q/c V across plates +q -q V q V q increase in potential energy Q q C q Q ( ) CV C C CV two ways to write

2 Question! Suppose the capacitor shown here is charge to Q an then the battery is isconnecte. A Now suppose I pull the plates further apart so that the final separation is. How o the quantities Q, C, E, V, change? Q: C: E: V: : remains the same.. no way for charge to leave. ecreases.. since capacitance epens on geometry remains the same... epens only on charge ensity increases.. since C, but Q remains same (or but E the same) increases.. a energy to system by separating How much o these quantities change?.. exercise for stuent!! Answers: C C V V

3 Suppose the battery (V) is kept attache to the capacitor. Again pull the plates apart from to. Now what changes? Relate Question V A C: V: Q: E: : ecreases (capacitance epens only on geometry) must stay the same - the battery forces it to be V must ecrease, QCV charge flows off the plate must ecrease ( CV ) How much o these quantities change?.. exercise for stuent!! Answers: C V D must ecrease ( E, E ) C E E σ E

4 Preflight 3: Two ientical parallel plate capacitors are connecte to a battery, as shown in the figure. C is then isconnecte from the battery, an the separation between the plates of both capacitors is ouble. ) What is the relation between the charges on the two capacitors? a) Q > Q b) Q Q c) Q < Q 3) How oes the electric fiel between the plates of C change as separation between the plates is increase? The electric fiel: a) increases b) ecreases c) oesn t change

5 Preflight 3: Two ientical parallel plate capacitors are connecte to a battery, as shown in the figure. C is then isconnecte from the battery, an the separation between the plates of both caps is ouble. 5) What is the relation between the voltages on the two capacitors? a) V > V b) V V c) V < V

6 Lecture3ACT Two ientical parallel plate capacitors are connecte to a battery. C is then isconnecte from the battery an the separation between the plates of both capacitors is ouble. V C V C What is the relation between the, the energy store in C, an the, energy store in C? (a) < (b) (c) >

7 Lecture3ACT Two ientical parallel plate capacitors are connecte to a battery. C is then isconnecte from the battery an the separation between the plates of both capacitors is ouble. V C V C What is the relation between the, the energy store in C, an the, energy store in C? (a) < (b) (c) > What is the ifference between the final states of the two capacitors? The charge on C has not change. The voltage on C has not change. The energy store in C has efinitely increase since work must be one to separate the plates with fixe charge, they attract each other. The energy in C will actually ecrease since charge must leave in orer to reuce the electric fiel so that the potential remains the same. Q Initially: C C Later: CV C

8 Where is the Energy Store? Claim: energy is store in the electric fiel itself. Think of the energy neee to charge the capacitor as being the energy neee to create the fiel. To calculate the energy ensity in the fiel, first consier the constant fiel generate by a parallel plate capacitor, where Q Q The electric fiel is given by: Q Q C ( Aε / ) σ Q E ε ε A ε E A The energy ensity u in the fiel is given by: u εe volume A This is the energy ensity, u, of the electric fiel. nits: J 3 m

9 Energy Density Claim: the expression for the energy ensity of the electrostatic fiel u ε E is general an is not restricte to the special case of the constant fiel in a parallel plate capacitor. Example (Stuent to fill in steps) Consier E- fiel between surfaces of cylinrical capacitor: Calculate the energy in the fiel of the capacitor by integrating the above energy ensity over the volume of the space between cyliners. λ ε E V ε E (π r r l) ε ( ) (π r r l) πε r C ε Lλ π 4π ε πε L b ln( ) a b a r r Lλ πε b ln( ) a L λ πε L b ln( ) a Q C For a cylinrical capacitor

10 Lecture 3, ACT 3 Consier two cylinrical capacitors, each of length L. C has inner raius cm an outer raius.cm. C has inner raius cm an outer raius.cm. If both capacitors are given the same amount of charge, what is the relation between, the energy. store in C, an, the energy store in C?. C C (a) < (b) (c) >

11 Lecture 3, ACT 3 Consier two cylinrical capacitors, each of length L. C has inner raius cm an outer raius.cm. C has inner raius cm an outer raius.cm. If both capacitors are given the same amount of charge, what is the relation between, the energy. store in C, an, the energy store in C?. C C C (a) < (b) (c) > The magnitue of the electric fiel from r to. cm is the same for C an C. But C also has electric energy ensity in the volume. to. cm. In formulas: πε ol C ~ r. C ~. Q / C C. ln( ) outer ln ln ln Q / C C. r inner

DIELECTRICS Consier parallel plate capacitor with vacuum separating plates (left) Suppose we place a material calle a ielectric in between the plates (right) The charge on the plates remain the same,

12 DIELECTRICS Consier parallel plate capacitor with vacuum separating plates (left) Suppose we place a material calle a ielectric in between the plates (right) The charge on the plates remain the same, but a ielectric has a property of having inuce charges on its surface that REDCE the electric fiel in between an the voltage ifference. Since C Q/V, the resulting capacitance will INCREASE. Y&F Figure 4.3

13 DIELECTRICS Suppose the charges on the plate an the ielectric are, s an s i. The electric Fiels before an after are E σ σ σ E i K ; ; ε ε E E σ σ σ We efine the ratio of the original fiel over the new fiel as the ielectric constant, K. Hence, the voltage ifference changes by /K an the capacitance, C o Q/V, changes by CKQ/VK C o i For same Q: C KC o E E o /K V Vo/K But C KC o General

14 DIELECTRICS Materials Glass, mica, plastics are very goo ielectrics

15 DIELECTRICS an permittivity We introuce a convenient reefinition of ε, calle permittivity, as ε Κ ε Consier a parallel plate capacitor with no ielectric C ε A capacitor with a ielectric becomes simply, C A A A KC Kε ε The change in capacitance can be accounte for by changing permittivity.

16 EXAMPLE of parallel plate capacitor problem A parallel plate capacitor is mae by placing polyethylene (K.3) between two sheets of aluminum foil. The area of each sheet is 4 cm, an the thickness of the polyethylene is.3 mm. Fin the capacitance. C K ε o A/ (.3) (8.85 x - C /Nm ) (4 cm )(m / 4 cm ).3 x -3 m.7 nf

17 Lecture3PF 8: Two ientical parallel plate capacitors are connecte to a battery. Remaining connecte, C is fille with a ielectric. 7) Compare the voltages of the two capacitors. a) V > V b) V V c) V < V 8) Compare the charges on the plates of the capacitors. a) Q > Q b) Q Q c) Q < Q Note: nlike constant Q case, here V an E remain the same but C K C o still.

18 EXAMPLE Two parallel plate capacitors, C C μf, are connecte across a V battery in parallel. a.) What energy is store? CV 44μJ T 88μJ b.) A ielectric (K.5) is inserte between the plates of C. Energy? C ' ' KC ' C V.5 μf 36μJ 5μF T 54μJ Note: a ielectric increases amount of energy store in C.

19 Y&F Problems 4.7 an 4.7 A parallel plate capacitor has two ielectrics, sie by sie, show the capacitance is, C ε A K + K A parallel plate capacitor has two ielectrics, stacke, show the capacitance is, C ε A KK K + K

CAPACITANCE: CHAPTER 24. ELECTROSTATIC ENERGY and CAPACITANCE. Capacitance and capacitors Storage of electrical energy. + Example: A charged spherical

CAPACITANCE: CHAPTER 24. ELECTROSTATIC ENERGY and CAPACITANCE. Capacitance and capacitors Storage of electrical energy. + Example: A charged spherical CAPACITANCE: CHAPTER 24 ELECTROSTATIC ENERGY an CAPACITANCE Capacitance an capacitors Storage of electrical energy Energy ensity of an electric fiel Combinations of capacitors In parallel In series Dielectrics