Unformatted text preview: Capacitance
1. Electrostatics 2. Circuits Capacitors and Dielectrics: Storing Charge Current and Resistance Direct Current Circuits (Steady State and RC)
Symbols: Capacitor and Capacitance
Capacitor: A device with two distinct metal surfaces (electrodes) separated by an insulator. Each surface holds an equal but opposite amount of charge. Electrolitic Capacitor 3. Magnetics 4. Optics The ratio of the Stored Charge to the Potential Difference between two between electrodes is the Capacitance: Capacitance: C = Q /V The unit for capacitance is the farad: farad: 1 F = 1 C/V Capacitance is, by convention, always positive. Therefore, always use always the absolute value of the charge on one electrode, and the magnitude of magnitude the potential difference.
1 Monday, 5 February 2007 2 Monday, 5 February 2007 Question: Capacitance depends on ... A. The charge being stored B. The potential difference between the electrodes C. Both A & B D. None of the above
3 C = Q /V
+Q Parallel Plate Capacitor
V = - E d E d
E= E= 2 o d o =- d o =- d o
E= 2 o E= E= 2 o 2 o o =- Monday, 5 February 2007 Monday, 5 February 2007 Q o A C= = V d Qd A o Capacitance depends on "geometry" & geometry" "material" material" 4 1 Capacitance of Concentric Spheres
1. C = Q / V 2. Assume a charge
-Q Capacitance of Concentric Spheres
E dA =
+Q Qencl o a
r b Since E dA and E is constant everywhere on the Gaussian surface 3. Use Gauss's Law to Gauss' find the E-field E4. Use V = - E dl to find the potential 5. Solve for capacitance Qencl
1. C = Q / V 2. Assume a charge 3. Use Gauss's Law to find the Gauss' E-field 4. Use V = - E dl to find the potential 5. Solve for capacitance E dA = E dA = E dA = E 4r 2 Q Q E= = +Q E 4r 2 = 4 o r 2 o b b Q dr V = - E dl = - 4 o r 2 a a ( ) ( ) Monday, 5 February 2007 5 Monday, 5 February 2007 Q 1 1 - 4 o b a a 4 o Q Q C = = = V 1 1 1 1 Q - - 4 o b a b a
V = 4 o r Q b = 6 Energy Stored in Capacitors
q dU = Vdq = dq C
Q Question: In a capacitor, where is the potential energy stored? A. In the electric field B. In the positive plate C. In the negative plate D. Both B & C E. None of the above
7 Monday, 5 February 2007 8 U = dU = 1 Q2 q dq = C 2 C 0 1 Q2 1 1 = QV = CV 2 2 C 2 2
V = - E d Consider the Parallel-Plate Capacitor: Parallel- 1 o A 1 U = CV 2 = 2 d 2 (Ed )2 = 1 o E 2 ( Ad )
Energy Density: True for any situation! uE = energy 1 = E2 volume 2 o Monday, 5 February 2007 2 Constant Charge vs. Constant Voltage
+Q -Q Capacitors in Parallel
Two (or more) capacitors are said to be attached in parallel if the plates of each capacitor are connected to each other such that the the potential difference between the plates of each capacitor is the same.
i Battery emf Constant Voltage Source
V constant Charges cannot move (isolated) Can separated the plates, deforms them, etc ... C1 C2 C3 Qi = V (C1 + C2 + C3 ) Qi = V
i i Qi = C1V1 + C2V2 + C3V 1 1 U = QV = CV 2 2 2 C1 = Q Q Q1 , C2 = 2 , C3 = 3 V3 V2 V1 i Ci = VCeq Ceq, parallel = Ci
i 1 Q2 1 U= = QV 2 C 2
Monday, 5 February 2007 9 Conductors are equipotential surfaces everywhere @ one voltage! Why parallel? V1 = V2 = V3 Q1 Q2 Q3 = = C1 C2 C3 1 U = CV 2 2
When C increases, U also increases (for a given V)!!
10 Monday, 5 February 2007 Capacitors in Series
Two (or more) capacitors are said to be attached in series if one of their plates share a common potential, but the other plate does does not. Capacitors in series each store the same charge.
C1 C2 C3 Arbitrary Combinations of Capacitors
C1 C2 C3 C4 C7 C8 Start @ inner-most spot & work inneryour way out!
C6 C5 C1 = Q1 V1 V = - E dl C56 = C5 + C6 1 1 C3 series C4 C34 = + C3 C 4 C5 C6 C7 series C8 C78 = 1 1 + C7 C8 1 1 + C1 C2 -1 -1 Q C2 = 2 V2 C3 = Q3 V3 Vtotal =
i Vi = Q1 Q2 Q3 + + C1 C2 C3
1 Ci C1 C2
-1 Q's in series are the same!! Vtotal = Q
Vtotal 1 1 1 + + =Q C1 C2 C3
1 Ceq, series =
i C34 C56 C78 C1 series C2 C56 C78 C12 = i Monday, 5 February 2007 Q = Ceq 1 Ci C5678 = C56 + C78
12 11 Monday, 5 February 2007 3 Arbitrary Combinations of Capacitors
-1 What if an insulator (dielectric) other than air is present between between the plates? The insulator becomes polarized! C34 C5678 C12 series C5678 C125678 = 1 1 + C12 C5678 C125678 C34 C34 C125678
C1 C2 C12345678 = C34 + C125678 = Ceq Edielectric
C3 C4 C5 C6 C7 C8 Ceq So what's the bottom line? what'
13 Monday, 5 February 2007 14 Monday, 5 February 2007 The Bottom Line on Dielectrics in Capacitors - dielectric Eo = E = Eo - Edielectric = o K Question:
A capacitor is charged up, and then carefully removed from the power supply. A dielectric is then slowly inserted into this now charged, isolated capacitor. As the dielectric is inserted, the energy stored in the capacitor ... A. B. C. D. Increases Decreases Remains the same Changes in an indeterminate way When the charge remains constant, the electric field is reduced by a factor of K due to the polarization of the dielectric material. V V = - E d So potential is reduced: V = o reduced: K C= Q Q KQ So capacitance is increased: C = increased: = = KCo V (Vo K ) Vo Everything we have already learned is still valid in the presence presence of dielectrics. Just make one substitution: o = K o
Permittivity of material Dielectric constant of material 15 Monday, 5 February 2007 Monday, 5 February 2007 16 4 Question:
A. Where did that energy go? B. C. Thin air Polarizing the dielectric No way of knowing More than one Dielectric +Q A capacitor with two different dielectrics next to each other is identical to two separate capacitors in parallel! -Q
Dielectric gets sucked in! E-field doing work! Frictionless: Oscillates!
Monday, 5 February 2007 Friction: Heats up!
17 A capacitor with two different dielectrics stacked on top of each other is identical to two separate capacitors in series!
Monday, 5 February 2007 18 5 ...
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- Fall '07
- Energy, Dielectric, Vtotal