Capacitors. February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 1

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Geoffrey Pearson
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1 Capacitors February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 1

2 Example - Capacitance of a Parallel Plate Capacitor! We have a parallel plate capacitor constructed of two parallel plates, each with area 625 cm 2 separated by a distance of 1.00 mm.! What is the capacitance of this parallel plate capacitor? C = ε 0A d A = 625 cm 2 = m 2 d = 1.00 mm = m ( )( m 2 ) C = F/m m = F C = nf A parallel plate capacitor constructed out of suare conducting plates 25 cm x 25 cm (=625 cm 2 ) separated by 1 mm produces a capacitor with a capacitance of about 0.55 nf February 3, 2014 Physics for Scientists&Engineers 2 3

3 Example 2 - Capacitance of a Parallel Plate Capacitor! We have a parallel plate capacitor constructed of two parallel plates separated by a distance of 1.00 mm.! What area is reuired to produce a capacitance of 0.55 F? C = ε 0 A d d = 1.00 mm = m A = dc ε 0 = ( m) 0.55 F ( ) ( F/m) = m 2 A parallel plate capacitor constructed out of suare conducting plates 7.9 km x 7.9 km (4.9 miles x 4.9 miles) separated by 1 mm produces a capacitor with a capacitance of 0.55 F February 3, 2014 Physics for Scientists&Engineers 2 4

4 Capacitance! The potential difference between the two plates is proportional to the amount of charge on the plates! The proportionality constant is the the capacitance of the device or C = = VC V! The capacitance of a device depends on the area of the plates and the distance between the plates but not on the charge or potential difference! For two parallel plates with area A and separation d: C ε = d 0 A February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 6

5 Cylindrical Capacitor! Consider a capacitor constructed of two collinear conducting cylinders of length L! The inner cylinder has radius r 1 and the outer has radius r 2! The inner cylinder has charge and the outer has charge + February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 8

6 Cylindrical Capacitor! We apply Gauss s Law to get the electric field between the two cylinders using a Gaussian surface with radius r and length L as illustrated by the black lines E da = E da = E( 2πrL) = ε 0! Which we can rewrite to get an expression for the electric field between the two cylinders E = ε 0 2πrL February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 10

7 Cylindrical Capacitor! As we did for the parallel plate capacitor, we define the voltage difference across the two cylinders to be f ΔV = E ds r = 2 E dr = = r 2 r 1 i ε 0 2πrL dr ε 0 2π L ln r 2 r 1 r 1! Thus, the capacitance of a cylindrical capacitor is C = ΔV = ε 0 2π L ln r 2 r 1 = 2πε L 0 ln( r 2 / r ) 1 February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 11

8 Spherical Capacitor! Consider a capacitor constructed of two concentric conducting spheres with radii r1 and r2! The inner sphere has charge + and the outer has charge - February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 12

9 Spherical Capacitor! We apply Gauss s Law to get the electric field between the two spheres using a spherical Gaussian surface with radius r as illustrated by the red line 2 E da = EA = E ( 4π r ) = ε 0! Which we can rewrite to get an expression for the electric field between the two spheres E= 4πε 0r 2 February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 13

10 Spherical Capacitor! As we did for the parallel plate capacitor, we define the voltage diﬀerence across the two spheres to be ΔV = i = r2 r1 f r2 E ds = E dr r1 dr 2 4πε 0r 1 1 = 4πε 0 r1 r2! Thus, the capacitance of a spherical capacitor is C= = ΔV February 3, πε 0 r1r2 = = 4πε 0 r2 r πε 0 r1 r2 r1 r2 Physics for Scientists & Engineers 2, Chapter 24 14

11 Capacitance of an Isolated Sphere! We obtain the capacitance of a single conducting sphere by taking our result for a spherical capacitor and moving the outer spherical conductor infinitely far away! Using our result for a spherical capacitor C = 4πε r 1 r 2! Looking at the potential of the conducting sphere C = ΔV ΔV = V = C = r 2 =,r 1 = R C = 4πε 0 R 4πε 0 R = k R! We get the same result as for a single point charge February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 15

12 Capacitors in Circuits! A circuit is a set of electrical devices connected with conducting wires! Capacitors can be wired together in circuits in parallel or series Capacitors in circuits connected by wires such that the positively charged plates are connected together and the negatively charged plates are connected together, are connected in parallel Capacitors wired together such that the positively charged plate of one capacitor is connected to the negatively charged plate of the next capacitor are connected in series February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 16

13 Capacitors in Parallel! Consider an electrical circuit with three capacitors wired in parallel! Each of three capacitors has one plate connected to the positive terminal of a battery with voltage V and one plate connected to the negative terminal! The potential difference V across each capacitor is the same! We can write the charge on each capacitor as 1 = C 1 V 2 = C 2 V 3 = C 3 V February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 17

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15 Capacitors in Parallel! We can consider the three capacitors as one euivalent capacitor C e that holds a total charge given by = = C 1 ΔV + C 2 ΔV + C 3 ΔV = ( C 1 + C 2 + C 3 )ΔV! Thus the euivalent capacitance is C e = C 1 + C 2 + C 3! A general result for n capacitors in parallel is C e = n C i i=1! If we can identify capacitors in a circuit that are wired in parallel, we can replace them with an euivalent capacitance February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 19

16 Capacitors in Series! Consider a circuit with three capacitors wired in series! The positively charged plate of C 3 is connected to the positive terminal of the battery! The negatively charge plate of C 3 is connected to the positively charged plate of C 2 February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 20

17 Capacitors in Series! The negatively charged plate of C 2 is connected to the positively charge plate of C 1! The negatively charge plate of C 1 is connected to the negative terminal of the battery! The battery produces an eual charge on each capacitor February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 21

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19 Capacitors in Series! Knowing that the charge is the same on all three capacitors we can write ΔV = ΔV 1 + ΔV 2 + ΔV 3 = + + = C 1 C 2 C 3 C 1 C 2 C 3! We can express an euivalent capacitance as V = C e! We can generalize to n capacitors in series 1 C e = n i=1 where 1 C i 1 C e = 1 C C C 3! If we can identify capacitors in a circuit that are wired in series, we can replace them with an euivalent capacitance February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 23

CHAPTER 26 ELECTROSTATIC ENERGY AND CAPACITORS

CHAPTER 26 ELECTROSTATIC ENERGY AND CAPACITORS CHAPTER 6 ELECTROSTATIC ENERGY AND CAPACITORS. Three point charges, each of +q, are moved from infinity to the vertices of an equilateral triangle of side l. How much work is required? The sentence preceding