PH -A Spring 15 Capacitance Lecture 7 Chapter 5 (Halliday/Resnick/Walker, Fundamentals of Physics 9 th edition) 1
Chapter 5 Capacitance In this chapter we will cover the following topics: -Capacitance C of a system of two isolated conductors. -Calculation of the capacitance for some simple geometries. -Methods of connecting capacitors (in series, in parallel). -Euivalent capacitance. -Energy stored in a capacitor. -Behavior of an insulator (a.k.a. dielectric) when placed in the electric field created in the space between the plates of a capacitor. -Gauss law in the presence of dielectrics.
C V V + V - Capacitance A system of two isolated conductors, one with a charge and the other -, separated by an insulator (this can be vacuum or air) is known as a "capacitor." The symbol used to indicate a capacitor is two parallel lines. We refer to the conductors as "plates." We refer to the = CV V O "charge" of the capacitor as the absolute value of the charge on either plate. As shown in the figure, the charges on the capacitor plates create an electric field in the surrounding space. The electric potential of the positive and negative plate are V and V, respectively. We use the symbol V for the potential difference V V between the plates ( V would be more appropriate). If we plot the charge as a function of V we get the straight line shown in the figure. The capacitance C is defined as the ratio / V. SI Unit : Farad (symbol F) We define a capacitor of C 1 F as one that acuires a charge = 1 C if we apply a voltage difference V 1 V between its plates. 3
Parallel Plate Capacitor A parallel plate capacitor is defined as a capacitor made up from two parallel plane plates of area A separated by a distance d. The electric field between the plates and away from the plate edges is uniform. Close to the plates' edges the electric field (known as "fringing field") becomes nonuniform. V + - _ Batteries A battery is a device that maintains a constant potential difference V between its two terminals. These are indicated in the battery symbol using two parallel lines uneual in length. The longer line indicates the terminal at higher potential while the shorter line denotes the lower-potential terminal. 4
+ - Charging a Capacitor One method to charge a capacitor is shown in the figure. When the switch S is closed, the electric field of the battery drives electrons from the battery negative terminal to the capacitor plate connected to it (labeled " " for low). The battery positive terminal removes an eual number of electrons from + - the plate connected to it (labeled " h" for high). Initially the potential difference V between the capacitor plates is zero. The charge on the plates as well as the potential difference between the plates increase, and the charge movement from the battery terminals to and from the plates decreases. All charge movement stops when the potential difference between the plates becomes eual to the potential difference between the battery terminals. 5
V + V - Calculating the Capacitance The capacitance depends on the geometry of the plates (shape, size, and relative position of one with respect to the other). Below we give a procedure for calculating C. Recipe : 1. Assume that the plates have charges and -.. Use Gauss' law to determine the electric field E between the plates ε EdA. enc 3. Determine the potential difference V between the plates using the euation V Eds along any path that connects the negative with the positive plate. 4. The capacitance C is given by the euation C. V 6
ˆn P S Capacitance of a Parallel Plate Capacitor The plates in the figure have area A and are separated by a distance d. The upper plate has a charge and the lower plate a charge -. N We apply Gauss' law using the Gaussian surface S shown in the figure. The electric flux EAcos EA. From Gauss' law we have: EA E. A The potential difference V between the positive and the negative plate is d given by: V Edscos Eds Ed. A A C V d / A d The capacitance. C A d 7
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S ˆn Cylindrical Capacitor It consists of two cylinders of radii a and b with a common axis. The two cylinders have a height L. We choose a Gaussian surface S that is also a cylinder with radius r and height L. The flux of the electric field through S is rle cos rle. Using Gauss' law we have:. If we combine these euations we have: E. rl The potential difference V between the positive and the negative plate is dr a b given by: V Edrcos18 ln r ln. L r L L a The capacitance b b L C V L b a / ln / ln b/ a a. 11
Spherical Capacitor It consists of two concentric spheres of radii a and b. S ˆn We choose a Gaussian surface S that is also a sphere with radius r. The flux of the electric field through S is: 4 cos 4 re re Using Gauss' law we have:. If we combine these euations we have: E 4 r The potential difference V between the positive and the negative plate is dr 1 1 1 given by: V Edrcos18. 4 r 4 r 4 a b The capacitance C a. b b 4 V 1 1 1 1 4 a b a b a. ab 4. b a 1
Euivalent Capacitor Consider the combination of capacitors shown in the figure to the left and to the right (upper part). We will substitute these combinations of capacitor with a single capacitor C e that is "electrically euivalent" to the capacitor group it substitutes. This means that if we apply the same voltage V across the capacitors in fig. a and fig. b (either right or left) by connecting to a battery, the same charge is provided by the battery. Alternatively, if we place the same charge on plates of the capacitors in fig. a and fig. b (either right or left), the voltage V across them is identical. This can be stated in the following manner: If we place the capacitor combination and the euivalent capacitor in separate black boxes, by doing electrical measurements we cannot distinguish between the two. 13
Ce C1 C C3 Capacitors in Parallel In fig. a we show three capacitors connected in parallel. This means that the plate of each capacitor is connected to the terminals of a battery of voltage V. We will substitute the parallel combination of fig. a with a single euivalent capacitor shown in fig. b, which is also connected to an identical battery. The three capacitors have the same potential difference V across their plates. The charge on C is CV. The charge on C is CV. 1 1 1 The charge on C is CV. The net charge C C C e 1 3 3 3 1 3 1 3 C1C C3 V The euivalent capacitance Ce C1C C3. V V For a parallel combination of n capacitors it is given by the expression: C C C... C n n j1 C j V. 14
Capacitors in Series In fig. a we show three capacitors connected in series. This means that one capacitor is connected after the other. The combination is connected to the terminals of a battery of voltage V. We will substitute the series combination of fig. a with a single euivalent capacitor shown in fig. b, which is also connected to an identical battery. The three capacitors have the same charge on their plates. The voltage across C is V / C. 1 1 1 The voltage across C is V / C. The voltage across C is V / C. 3 3 3 The net voltage across the combination V V V V. Thus we have: V 1 1 C 1 C The euivalent capacitance 1 1 1 1 C C C C C 3 e 1 3 1. C V 1 1 1 C C C 1 3 15 e 1 3
More Complex Capacitor Systems In general, a capacitor system may consist of smaller capacitor groups that can be identified as connected "in parallel" or "in series." For example, C and C in fig. a are connected in parallel. 1 They can be substituted by the euivalent capacitor C C C as 1 3 1 1 shown in fig. b. Capacitors C and C in fig. b are connected in series. They can be substituted by a single capacitor C as shown in fig. c. 13 C 13 is given by the euation 1 1 1. C C C 13 1 3 16
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' -' - - - - - Charge O d' V' A V' V ' + + + + + B Voltage Energy Stored in an Electric Field Consider a capacitor C that has a charge. We can calculate the work W reuired to charge the capacitor by assuming that we transfer a charge d from the negative plate to the positive plate. We assume that the capacitor charge is and the corresponding voltage V. The work dw reuired for the charge transfer is given by: dw V d d. C We continue this process till the capacitor charge is 1 eual to. The total work W Vd d. C 1 CV V W If we substitute CV we get: or. C W W C Work W can also be calculated by determining the area A of triangle OAB, which is V eual to Vd : Area W. 1
- - - - - - U A E d + + A + + + CV C E u Potential Energy Stored in a Capacitor The work W spent to charge a capacitor is stored in the form of potential energy U W that can be retrieved when the capacitor is CV V discharged. Thus U. C Energy Density We can ask the uestion: Where is the potential energy of a charged capacitor stored? The answer is counterintuitive. The energy is stored in the space between the capacitor plates where a uniform electric field E V / d is generated by the capacitor charges. In other words, the electric field can store energy in empty space! U We define energy density (symbol u ) as the potential energy per unit volume: u. V The volume V between the plates is V Ad, where A is the plate area. U CV A V V E Thus the energy density u Ad Ad d Ad d This result, derived for the parallel plate capacitor, holds in general.
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- C C air V Capacitor with a Dielectric In 1837 Michael Faraday investigated what happens to the capacitance C of a capacitor when the gap between the plates is completely filled with an insulator (a.k.a. dielectric). ' -' - V V Faraday discovered that the new capacitance is given by C C air. Here C is the capacitance before the insertion air of the dielectric between the plates. The factor is known as the dielectric constant of the material. Faraday's experiment can be carried out in two ways: 1. With the voltage V across the plates remaining constant. In this case a battery remains connected to the plates. This is shown in fig. a. - V'. With the charge of the plates remaining constant. In this case the plates are isolated from the battery. This is shown in fig. b. 4
- ' -' C C air V V Fig. a: Capacitor voltage V remains constant This is because the battery remains connected to the plates. After the dielectric is inserted between the capacitor plates the plate charge changes from to. κ The new capacitan ce C κ κcair. V V V - - V V' Fig. b: Capacitor charge remains constant This is because the plates are isolated. After the dielectric is inserted between the capacitor plates V the plate voltage changes from V to V. The new capacitance C Cair. V V / V 5
In a region completely filled with an insulator of dielectric constant, all electrostatic euations containing the constant are to be modified by replacing with. Example 1: Electric field of a point charge inside a dielectric is: E 1. 4 r conductor dielectric Example : The electric field outside an isolated conductor immersed in a dielectric becomes: E. 6
U pecos Dielectrics : An Atomic View Dielectrics are classified as "polar" and "nonpolar." Polar dielectrics consist of molecules that have a nonzero electric dipole moment even at zero electric field due to the asymmetric distribution of charge within the molecule (e.g., H O ). At zero electric field (see fig. a) the electric dipole moments are randomly oriented. When an external electric field E is applied (see fig. b) the electric dipole moments tend to align preferentially along the direction of E because this configuration corresponds to a minimum of the potential energy and thus is a position of stable euilibrium. Thermal random motion opposes the alignment and thus ordering is incomplete. Even so, the partial alignment produced by the external electric field generates an internal electric field that opposes E. Thus the net electric field E is weaker than E. 7
A nonpolar dielectric, on the other hand, consists of molecules that in the absence of an electric field have zero electric dipole moment (see fig. a). If we place the dielectric between the plates of a capacitor the external electric field E induces an electric dipole moment p that becomes aligned with E (see fig. b). The aligned molecules do not create any net charge inside the dielectric. A net charge appears at the left and right surfaces of the dielectric opposite the capacitor plates. These charges come from negative and positive ends of the electric dipoles. These induced surface charges have sign opposite that of the opposing plate charges. Thus the induced charges create an electric field E that opposes the applied field E (see fig. c). As a result, the net electric field E between the capacitor plates is weaker. E E 8
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S S ˆn ˆn Gauss' Law and Dielectrics In Chapter 3 we formulated Gauss' law assuming that the charges existed in vacuum: EdA or. In this section we will write Gauss' law in a form that is suitable for cases in which dielectrics are present. Consider first the parallel plate capacitor shown in fig. a. We will use the Gaussian surface S. The flux E A E. Now we fill the space between the plates A with an insulator of dielectric constant (see fig. b). We will apply Gauss' law for the same surface S. Inside S in addition to the plate charge we also have the induced charge on the surface of the dielectric: EA E E (e. 1). From Faraday's experiments we have: E (e. ). A A If we compare e. 1 with e. we have: E da. 3
Gauss' Law in the Presence of Dielectrics E da Even though the euation above was derived for the parallel plate capacitor, it is true in general. Note 1 : The flux integral now involves E. Note : The charge that is used is the plate charge, also known as "free charge." Using the euation above we can ignore the induced charge. Note 3 : The dielectric constant is kept inside the integral to describe the most general case in which is not constant over the Gaussian surface. 31