Capacitance and Dielectrics

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PowerPoint® Lectures for University Physics, Twelfth Edition – Hugh D. Young and Roger A. Freedman

Lectures by James Pazun Modified by P. Lam 7_15_2008

Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Topics for Chapter 24 • I. Capacitors and capacitance • II. Energy stored in a charged capacitor • III. Capacitors with dielectrics Intermission • IV. Dielectric breakdown • V. Capacitors in series an parallel circuits

• If we pull a positive charge and negative charge apart, it requires energy; this energy is stored as electrostatic potential energy in this charge arrangement. If the charges are allowed to move, they will move toward each other and gain kinetic energy (one can use this kinetic energy to push something to do work or to light up a light bulb) • A capacitor is device which can hold these separated charges.

Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley How do we build a capacitor? • The basic components of a capacitor are two conductors separated by a distance. • The “parallel-plate capacitor is the easiest to analyze. • Connect the plates to the two terminals of a battery (see figure below). Electrons will move from the top plate to the battery and then enter the bottom plate; leaving a net positive charge on the top plate and net negative charge on the bottom plate. The electrons will stop moving when the voltage difference between

the two plates (Vab) equal to the voltage of the battery (). •Disconnect the battery and the capacitor holds the charge.

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How much charge (Q) a capacitor can hold for a given voltage

difference (Vab) between the plates depends on the size and shape of the capacitor and it is called the capacitance (C). Q Definition : C = Vab For parallel- plate capacitor with area (A) and separation (d), A C= (if d << A) o d Q Qd Q A Derivation : d << A E = = ; Vab = Ed = C= = o o o A o A Vab d

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Q Coulomb Definition : C = unit = Farad Vab volt Example : The separation of a parallel plate capacitor is 1 mm, what is the area of the plate in order for C =1 Farad? A Answer : C = ; = 8.85x1012 Farad /m o d o 3 Cd (1 Farad)(10 m) 8 2 A = = 12 1x10 m !!! o 8.85x10 Farad /m

Common capacitances are pF =10-12 Farad or μF=10-6 Farad. Note : Even if C =1μF, the area is still 100m2!! (1)Large area can be accommodated by rolling the two plates into a cylinder. (2) Capacitance can also be increased by inserting an insulating material(called dielectrics) between the plates (will be discussed later). Electrolytic dielectrics capacitor can have a capacitance of 1 Farad and yet of very small size. Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Cylindrical capacitance - coaxial cable

A cylindrical capacitor is formed by two concentric cylinders (eg. a coaxial cable); the inner conductor is typically a solid cylinder, the outer conductor is a thin cylindrical shell. Suppose we connect this capacitor to a battery; when it is fully charged, the charge is Q.

If ra, rb << L, then we can use the infinite line cylinder approximation. Q Using Gauss's Law E= for ra < r < rb 2 rLo b Q rb Q 2L Vab =| Edr |= ln C = o a 2Lo ra Vab ln(rb /ra )

Note : Let d rb ra , d d ln(rb /ra ) = ln(1+ ) if d << ra ra ra

2L 2ra L A C = o o = o (same formula as for parallel- plate) ln(rb /ra ) d d

Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Commercial capacitors • Commercial capacitors for home electronics have sizes range from a grain of rice to that of a large cigar. • Capacitors like those mentioned above and pictured at right are microfarad capacitors. • Note: Even though the exterior shape is a cylinder, it doesn’t mean that it is cylindrical capacitor; it could be a rolled-up parallel-plate capacitor.

Q Altough the definition C involves Q and Vab : C= Vab The capacitance depends on the size and geometry of the capacitor, Example : A C= for parallel- plate capacitor o d 2L C = o for cylindrical capacitor ln(rb /ra )

If we plot the charge (Q) vs the voltage across the capacitor (Vab), then C is the slope of the linear graph. Q slope=C

Vab

Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Example Example 24.2. The dimensions of a parallel-plate capacitor are d=5mm and A=2 m2. A charge (Q) of 35.4μC is each plate (a) What is the capacitance (C) ?

(b) What is the voltage (Vab) across the capacitor? (c) What is the magnitude of the electric field between the plates? (d) Now pull the plates apart to d=10mm, what are the values for

C, Q, Vab, and E? Do you have to apply energy to pull the plates apart?

Q slope=C The energy stored in a charged capacitor is 1 1 Q2 1 U= QV = = CV 2 2 ab 2 C 2 ab Vab Refer to the example in the previous slide, find the energy stored in the capacitor when d=5mm and the energy stored after you pull it apart to d=10mm. How much energy did you have to apply to pull the plates apart? If the capacitor is connected to a battery (10 kV) while you are pulling the plates apart, what is the stored energy after you have pulled the plates to d=10mm?

• The potential difference between the parallel plates of a capacitor decreases when a dielectric material is inserted between the plates because the dielectric weakens the electric field (explanation in next slide).

Q Since C , V decreasing V while Q remains the same increaseing C.

E A E = o ; K = dielectric constant of the material C=K K o d

Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley IV. Dielectric breakdown • A very strong electrical field can exceed the strength of the dielectric to contain it. Table 24.2 at the bottom of the page lists some limits. •For a given capacitor separation (d), the dielectric strength sets a limit on the maximum voltage (max V=Emd) that can be sustained by the capacitor.

Series :

Q1 = Q2

V1 + V2 = V

Parallel :

Q1 + Q2 = Q

V1 = V2 = V

The energy stored in a charged capacitor is 1 Q2 U = 2 C A Consider a parallel- plate capacitor, C = o d 1 Q2 1 Q2d 1 Q2 U = = = 2 2 d * A *o 2 C 2 o A 2 o A

Q 1 2 E = = U = o E *Volume o o A 2 1 E 2 = energy density (energy/volume) 2 o That is, one can think of energy being stored as separated positive and negative charges OR being stored in the E - field. The second interpretation is useful when we discuss energy carried by electromagnetic waves.