Understanding Capacitance in Physics

1. Physics 121 - Electricity and MagnetismLecture 06 - Capacitance Y&F Chapter 24 Sec. 1 - 6 • Overview • Definition of Capacitance • Calculating the Capacitance • Parallel Plate Capacitor • Spherical and Cylindrical Capacitors • Capacitors in Parallel and Series • Energy Stored in an Electric Field • Electric Dipole in an Electric Field • Atomic Physics View of Dielectrics • Capacitors with a Dielectric • Dielectrics and Gauss Law

2. DV = 1.5 V Definition: + charges • Example: A primitive capacitor • + ball at same potential as + side of battery. • Similarly for the – side. • Charge will flow to produce potential difference of 1.5 V across the balls. • The conductors’ geometry (i.e., capacitance) • determines how much charge it takes. _ + • Example: Capacitance of an isolated sphere • Charged, single conducting sphere, radius R • Apply point charge potential formula (symmetry), V( infinity ) = 0 + 1.5 V battery _ + charges Capacitance measures charge storage per volt • Apply a potential difference to a conductor arrangement. • The conductor GEOMETRY determines how much charge is stored. • Equations below are for measuring capacitance C given Q and DV

3. Varying geometry varies capacitance: • Move the balls closer together while still connected to the battery. • Potential difference DV cannot change. • But recall: • As the distance Ds between the balls decreases • the E field has to increase. • Charge flows from the battery to the balls to increase E. • After moving closer together the two balls hold more charge for the same potential difference: i.e. the capacitance increases. + charges _ + + 1.5 V battery constant _ + charges increases increases Capacitance Definition: • measures charge stored on plates per volt of DV • depends on plate geometry and dielectric material if any. Always positive. • does not depend on applied DV or charge Q. • units: 1 FARAD = 1 Coulomb / Volt. Farads are very large • 1 mF = 10-6 F. 1 pF = 1 pico-Farad = 10-12 F = 10-6mF = 1 mmF

4. +Q -Q + - d DV Capacitors are charge storage devices • Consist of two conductors, insulated • Electrically neutral before & after being charged • Qenc= Qnet=0 • Current can flow from + plate to – plate if • there is a conducting path connecting plates - a • “complete circuit” – external to the plates. • Capacitors store charge and potential energy • Capacitors are everywhere in circuits • power supplies • radio circuits • memory bits • Common type: “parallel plate”, often tubular

5. d EXAMPLE: Parallel Plate Capacitor S Find E between plates using Gauss’ Law Method for calculating capacitance from geometry: +Q - Q • Find potential difference DV: • Choose Vi = 0 on negative plate (grounded) • Choose path from – plate to + plate, opposite to E field • Assume two conducting plates (equipotentials) with • equal and opposite charges +Q and –Q • Find E between the plates: possibly using Gauss’ Law • Calculate DV between plates using a convenient path • Capacitance C = Q/DV • Insulating materials (“dielectrics”) can reduce the E field • between plates by “polarizing”, increasing capacitance. • C DEPENDS ONLY ON GEOMETRY • C  infinity as plate separation d  0 • C directly proportional to plate area A • Dielectrics increase C

6. Charge on a capacitor - no battery attached 6-1: Suppose that we charge an ideal parallel plate capacitor to 20 V, then disconnect the battery. What then happens to the charge and voltage on it? A. The charge stays on the plates indefinitely, and the voltage stays constant at 20 V. • The charge leaks out the bottom quickly, and the voltage goes to 0 V. • The charge jumps quickly across the air gap, and the voltage goes to 0 V. D. The charge stays on the plates, but the voltage drops to 0 V. • The charge instantly disappears, but the voltage stays constant at 20 V.

7. A giant, natural capacitor Where are the plates? Why the sparks?

8. To find potential difference use outward radial • integration path from r = a to r = b. Negative For b > a Vb < Va Let outer radius b  infinity. Then a/b  0 and result becomes the formula for the isolated sphere: Ex 24.3: Capacitance of a spherical capacitor • 2 concentric spherical, conducting shells, radii a & b • Equal charges +q (inner sphere), -q (outer sphere) • All charge on the outer sphere is on its inner • surface (by Gauss’s Law, GS within outer shell) • Choose another spherical Gaussian surface S as • shown with (a<r<b). Find field using Gauss’s Law: E between shells is point-charge-like As before:

9. CROSS SECTION VIEW • To find potential difference use outward radial • integration path from r = a to r = b. Vb < Va For b > a C depends only on geometrical parameters C  0 as b/a  inf C  inf as b/a  1 Ex 24.4: Capacitance of a cylindrical capacitor • 2 concentric, long cylindrical conductors • Radii a & b and length L >> b => neglect end effects • Charges are +q (inner) and -q (outer), l is uniform • All charge on the outer conductor is on its inner • surface (by Gauss’s Law, GS within outer cylinder) • Choose cylindrical Gaussian surface S, radius r • between plates and find field at radius r. • E is perpendicular to endcaps => zero flux contribution • So: E between cylinders is as for straight line

10. All these capacitance formulas depend only on geometrical factors • Capacitance for isolated Sphere • Parallel Plate Capacitor • Concentric Cylinders Capacitor • Concentric Spheres Capacitor • Units: F (Farad)= C2/Nm = C/ Volt = e0length - named after Michael Faraday. [note: e0 = 8.85 pF/m]

11. + - Example: Current flow through a circuit i • Close switch S, completing circuit. Current starts flowing. • Battery (EMF) maintains DV (= EMF E) & supplies energy. It moves • free + charges from – to + terminal inside battery, raising PE. + + C • Convention: current i flows from + to – outside of battery • After switch closes, current (charge) flows until DV across • capacitor equals battery voltage E. • Then current stops as E field in wire  0 E - - S Capacitors in circuits CIRCUIT SYMBOLS: CIRCUIT DEFINITIONS: Open Circuit: NO closed path. No current. Conductors are equipotentials Closed Circuit: Current can flow through closed completed paths. Loop Rule: Potential field is conservative  Potential CHANGE around ANY closed path = 0 • DEFINITION: EQUIVALENT CAPACITANCE • Capacitors can be connected in series, parallel, or hybrid combinations • The “equivalent capacitance” to a combination is the capacitance of a SINGLE • fictional capacitor that would have the same effect on the rest of the circuit. • The equivalent capacitance can replace the original combination in analysis.

12. .... Q1 Q2 Q3 .... DV E DV C1 C2 C3 Qtot The parallel capacitors are just like a single capacitor with larger plates so.... E Ceq Charges on parallel capacitors add Parallel capacitances add directly Question: Why is DV the same for all elements in parallel? Answer: Potential is conservative field, for ANY closed loop around circuit: Parallel capacitors - Equivalent capacitance An actual parallel circuit... ...and the equivalent circuit: DV is the same for each branch

13. DV2 DV3 DV1 Qtot C2 C3 C1 DVtot DVtot Ceq Gaussian surface Qenc = 0 neutral..so Q1 = - Q2 Charges on series capacitors are all equal - here’s why..... Q2 Q1 Reciprocals of series capacitances add For two capacitors in series: Series capacitors - equivalent capacitance An actual series circuit... The equivalent circuit... DVtot DVi can be different for capacitors in series

14. A 33mF and a 47 mF capacitor are connected in parallel Find the equivalent capacitance Example 1: Example 2: Same two capacitors as above, but now in series connection Solution: Example 3: • Two capacitors are connected as in the sketch • C1 = 10 mF, charged initially to Vi = 100V • C2 = 20 mF, uncharged initially • Close switches. Find final potentials across C1 & C2. • Solution : • C1 & C2 in parallel  come to same potential Vf • Total initial charge: • Qtot redistributes on both C1 & C2 • Final charge on each: Solution:

15. Three Capacitors in Series 6-2: The equivalent capacitance for two capacitors in series is: Which of the following is the equivalent capacitance formula for three capacitors in series? Apply formula for Ceq twice or Look for a clever way around

16. After Step 1 After Step 2 C1 C2 C12 parallel V V C123 V series C3 C3 Values: C1 = 12.0 mF, C2 = 5.3 mF, C3 = 4.5 mF C123 = (12 + 5.3) 4.5 / (12 + 5.3 + 4.5) mF = 3.57 mF Example: Reduce circuit to find Ceq=C123 for mixed series-parallel capacitors

17. C2 II I C1 E E E E C3 IV III C1 C2 C1 C2 C3 C3 C2 C3 C1 Series or Parallel? 6-3: In the circuits below, which ones show capacitors 1 and 2 connected in series? • I, II, III • I, III • II, IV • III, IV • None

18. V=1.5 V _ + + charges + 1.5 V battery • The energy stored by charging a capacitor from charge 0 to charge Q is the integral: _ + charges Energy Stored in a Capacitor When current begins to flow in the circuit, battery energy is drawn off. Where does it go? • Charge separation in a neutral body stores potential energy in the field. • Plate potential difference starts at zero and grows to battery potential V as current flows. Current falls off to zero when C is fully charged • Battery applies constant V while a small element dq of charge is moved from plate to plate, storing additional potential energy: dU = V dq = qdq/C • Can current really flow across vacuum between the plates?

19. Demonstration Source: Pearson Study Area - VTD Discussion: • How do series and parallel capacitors add together to make equivalent capacitances? • When the capacitors are charged, which arrangement stores more energy? • What determines how long the bulbs stay lit? https://mediaplayer.pearsoncmg.com/assets/secs-vtd45_seriesparallel

20. Energy Density: Capacitors Store Energy in the Electrostatic Field • The total energy in a parallel plate capacitor is • The volume of space filled by the electric field in the capacitor is = Ad, so the energy density u is • But for a parallel plate capacitor, • so Energy density is the energy stored in the electric field per unit volume. • Same Formula applies generally

21. What Changes? 6-4: A parallel plate capacitor is connected to a battery of constant voltage V. If the plate separation is decreased, which of the following increase? • II, III and IV. • I, IV, V and VI. • I, II and III. • All except II. • All increase.

22. Polarization: An external field E0 aligns dipole molecules in an insulator - the polarization field Epolreduces the net field Ediel + - + - + - + - + - Polarization in DielectricsA dipole molecule in a uniform external field.. ..feels torque, stores electrostatic potential energy • |torque| = 0 at q = 0 or q = p • |torque| = pE at q = +/- p/2 • RESTORING TORQUE: t(-q) = t(+q) See Lecture 3

23. MOLECULAR VIEW E0= Evac is due to free charge sfree on the plates Epol is response to Evac Weakened net field inside is Ediel =Enet = E0 – Epol NO EXTERNAL FIELD WITH EXTERNAL FIELD Eo Polar Material permanent dipole Non-polar Material induced dipole Polarization surface charge density reduces free surface charge density snet=sfree – spol (sfree = sext = svac) Dielectrics increase capacitance For a given DV = Enetd, more movable charge sfreeis needed Dielectric constant Inside conductors, polarization reduces Enet to zero Dielectric materials/Insulators • Insulators become POLARIZED in an external electric field • The NET field inside the material is lowered below vacuum E0.

24. The capacitance increases when the space between plates is filled with a dielectric. For a parallel plate capacitor: • A dielectric weakens the field, compared to what it would be for vacuum • The induced polarization charge density on the dielectric surface is proportional to the “free” surface charge density on the plates due to the external circuit: Dielectric Constant K inCapacitance Calculations • All materials (water, paper, plastic, air) polarize to some extent and have different permittivities e = ke0 • e0 is the “free space permittivity”.kis called the dielectric constant - a dimensionless number. • Wherever you see e0 for a vacuum, substitute ke0 to account for a dielectric.

27. Detach battery, then insert dielectric • Q remains constant, Enet is reduced • Voltage (fixed Q) drops to Vf. • Dielectric reduces Enet and V. • Keep batteryattached, insert dielectric • Battery maintains voltage Vf = V0 but Enet and V try to decrease • More charge flows to the capacitor as dielectric is inserted until V and Enet are back to original values. E E What happens as you insert a dielectric? Two scenarios.Initially: charge capacitor C0 to voltage V0, charge Q0, field E0. OR

28. { free charge on plates field not counting polarization = e0Evac • K could vary over Gaussian surface S. Usually it is constant and factors • Flux can still be measured by field without dielectric: • Count only the free charge qfree (excluding polarization) in qenc • above. Ignore polarization charges inside the Gaussian surface Gauss’ Law with a dielectric OPTIONAL TOPIC Alternatively: The “Electric Displacement”D measures field that would be present due to the “free” charge only, i.e. without polarization field from dielectric