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# Lesson 2 Capacitors

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### Lesson 2 Capacitors

1. 1. Capacitors <ul><li>A capacitor is a device that stores electric charge. </li></ul><ul><li>A capacitor consists of two conductors separated by an insulator. </li></ul><ul><li>Capacitors have many applications: </li></ul><ul><ul><li>Computer RAM memory and keyboards. </li></ul></ul><ul><ul><li>Electronic flashes for cameras. </li></ul></ul><ul><ul><li>Electric power surge protectors. </li></ul></ul><ul><ul><li>Radios and electronic circuits. </li></ul></ul>
2. 2. Types of Capacitors Parallel-Plate Capacitor Cylindrical Capacitor A cylindrical capacitor is a parallel-plate capacitor that has been rolled up with an insulating layer between the plates.
3. 3. Capacitors and Capacitance Charge Q stored: The stored charge Q is proportional to the potential difference V between the plates. The capacitance C is the constant of proportionality, measured in Farads. Farad = Coulomb / Volt A capacitor in a simple electric circuit.
4. 4. Parallel-Plate Capacitor <ul><li>A simple parallel-plate capacitor consists of two conducting plates of area A separated by a distance d . </li></ul><ul><li>Charge +Q is placed on one plate and –Q on the other plate. </li></ul><ul><li>An electric field E is created between the plates. </li></ul>+Q -Q +Q -Q
5. 5. Energy Storage in Capacitors <ul><li>Since capacitors store electric charge, they store electric potential energy. </li></ul><ul><li>Consider a capacitor with capacitance C , potential difference V and charge q . </li></ul><ul><li>The work dW required to transfer an elemental charge dq to the capacitor: </li></ul>The work required to charge the capacitor from q=0 to q=Q : Energy Stored by a Capacitor = ½CV 2 = ½QV
6. 6. Example: Electronic Flash for a Camera <ul><li>A digital camera charges a 100 μF capacitor to 250 V. </li></ul><ul><ul><li>How much electrical energy is stored in the capacitor? </li></ul></ul><ul><ul><li>If the stored charge is delivered to a krypton flash bulb in 10 milliseconds, what is the power output of the flash bulb? </li></ul></ul>
7. 7. Dielectrics <ul><li>A dielectric is an insulating material (e.g. paper, plastic, glass) . </li></ul><ul><li>A dielectric placed between the conductors of a capacitor increases its capacitance by a factor κ , called the dielectric constant . </li></ul><ul><li> C= κC o ( C o =capacitance without dielectric) </li></ul><ul><li>For a parallel-plate capacitor: </li></ul>ε = κε o = permittivity of the material.
8. 8. Properties of Dielectric Materials <ul><li>Dielectric strength is the maximum electric field that a dielectric can withstand without becoming a conductor. </li></ul><ul><li>Dielectric materials </li></ul><ul><ul><li>increase capacitance. </li></ul></ul><ul><ul><li>increase electric breakdown potential of capacitors. </li></ul></ul><ul><ul><li>provide mechanical support. </li></ul></ul>Dielectric Strength (V/m) Dielectric Constant κ Material 3 x 10 6 1.0006 air 8 x 10 6 300 strontium titanate 150 x 10 6 7 mica 15 x 10 6 3.7 paper
9. 9. Practice Quiz <ul><li>A charge Q is initially placed on a parallel-plate capacitor with an air gap between the electrodes, then the capacitor is electrically isolated. </li></ul><ul><li>A sheet of paper is then inserted between the capacitor plates. </li></ul><ul><li>What happens to: </li></ul><ul><ul><li>the capacitance? </li></ul></ul><ul><ul><li>the charge on the capacitor? </li></ul></ul><ul><ul><li>the potential difference between the plates? </li></ul></ul><ul><ul><li>the energy stored in the capacitor? </li></ul></ul>
10. 10. Discharging a Capacitor <ul><li>Initially, the rate of discharge is high because the potential difference across the plates is large. </li></ul><ul><li>As the potential difference falls, so too does the current flowing </li></ul><ul><li>Think pressure </li></ul>As water level falls, rate of flow decreases
11. 11. <ul><li>At some time t, with charge Q on the capacitor, the current that flows in an interval  t is: </li></ul><ul><li>I =  Q/  t </li></ul><ul><li>And I = V/R </li></ul><ul><li>But since V=Q/C, we can say that </li></ul><ul><li>I = Q/RC </li></ul><ul><li>So the discharge current is proportional to the charge still on the plates. </li></ul>
12. 12. <ul><li>For a changing current, the drop in charge,  Q is given by: </li></ul><ul><li> Q = -I  t (minus because charge Iarge at t = 0 and falls as t increases) </li></ul><ul><li>So  Q = -Q  t/RC (because I = Q/RC) </li></ul><ul><li>Or -  Q/Q =  t/RC </li></ul>
13. 13. <ul><li>If we make the change infinitesimally small we get… </li></ul>
14. 14. <ul><li>Applying the laws of logs </li></ul>
15. 15. <ul><li>Of course, we can’t easily measure Q but since Q = VC, and C is just constant, we can measure V and plot a graph of ln V against t. </li></ul>ln V t Gradient = 1/RC ln V 0
16. 16. <ul><li>Alternatively, if you want to get rid of the ln, just multiply both sides by e. </li></ul><ul><li>ln Q = ln Q 0 –t/RC </li></ul><ul><li>Becomes </li></ul><ul><li>Q = Q 0 e -t/RC </li></ul><ul><li>And gives a graph that looks like this… </li></ul>
17. 17. Incidentally, graphs can be V or Q against t, they all have the same basic shape
18. 18. The time constant. <ul><li>The time constant  =RC . </li></ul><ul><li>The units are seconds ( t/RC is dimensionless). </li></ul><ul><li>The time taken for the charge to fall to (1/e) of the initial value in the circuit. </li></ul><ul><li>The time taken for the voltage to fall by ( 1/e) of its initial value. </li></ul>
19. 19. Charging a capacitor <ul><li>At time t=0 the switch is closed, with the capacitor initially uncharged. </li></ul><ul><li>A current will flow  =V c +V R =I 0 R , as initially V c =0. Thus the initial current is I 0 =  /R . </li></ul><ul><li>Now a charge begins to build on the capacitor, introducing a reverse voltage. The current falls, and stops when the P.D. across C is  . </li></ul><ul><li>Final charge is given by &quot; Q=CV &quot; => Q 0 =C  . </li></ul>
20. 20. Charging a capacitor (quantitative). <ul><li>Apply Kirchoff's loop rule. </li></ul>
21. 21. Charging a capacitor (cont) <ul><li>Where Q 0 = C  = the final charge on the capacitor. </li></ul>
22. 22. Charging a capacitor (cont). <ul><li>To find the current, differentiate since I=dQ/dt . </li></ul><ul><li>By considering time zero, when the current is I 0 , </li></ul>
23. 24. Energy Considerations. <ul><li>During charging, a total charge Q=C  flows through the battery. </li></ul><ul><li>The battery does work W=Q 0  =C  2 . </li></ul><ul><li>The energy stored in the capacitor is ½ QV= ½ Q 0  = ½ C  2 . </li></ul><ul><li>Where's the other half? </li></ul>
24. 25. Finally… <ul><li>Electromagnetism depends a lot on integrals, vectors etc. shows how useful they are. </li></ul><ul><li>It is one of the foundations of physics but: </li></ul><ul><ul><li>it can be rather formal, encouraging the precise thinking that we expect of any academic training; </li></ul></ul><ul><ul><li>it is rather far removed from the everyday, but that develops the imagination we expect from a physicist. </li></ul></ul>