CAPACITORS February 2008 Capacitors Part I A simple
CAPACITORS February, 2008
Capacitors Part I
A simple Capacitor TWO PLATES WIRES Battery WIRES o Remove the battery o Charge Remains on the plates. o The battery did WORK to charge the plates o That work can be recovered in the form of electrical energy – Potential Difference
INSIDE THE DEVICE
Two Charged Plates (Neglect Fringing Fields) d Air or Vacuum E -Q Area A V=Potential Difference +Q Symbol ADDED CHARGE
Where is the charge? - Q- d Air or Vacuum E Area A V=Potential Difference + + + +Q AREA=A s=Q/A
One Way to Charge: o Start with two isolated uncharged plates. o Take electrons and move them from the + to the – plate through the region between. o As the charge builds up, an electric field forms between the plates. o You therefore have to do work against the field as you continue to move charge from one plate to another.
Capacitor
More on Capacitors d Air or Vacuum -Q E +Q Area A Gaussian Surface V=Potential Difference Same result from other plate!
DEFINITION - Capacity o The Potential Difference is APPLIED by a battery or a circuit. o The charge q on the capacitor is found to be proportional to the applied voltage. o The proportionality constant is C and is referred to as the CAPACITANCE of the device.
UNITS o A capacitor which acquires a charge of 1 coulomb on each plate with the application of one volt is defined to have a capacitance of 1 FARAD o One Farad is one Coulomb/Volt
The two metal objects in the figure have net charges of +79 p. C and -79 p. C, which result in a 10 V potential difference between them. (a) What is the capacitance of the system? [7. 9] p. F (b) If the charges are changed to +222 p. C and -222 p. C, what does the capacitance become? [7. 9] p. F (c) What does the potential difference become? [28. 1] V
NOTE o Work to move a charge from one side of a capacitor to the other is q. Ed. o Work to move a charge from one side of a capacitor to the other is q. V o Thus q. V=q. Ed o E=V/d As before ( we omitted the pesky negative sign but you know it is there, right? )
Continuing… o The capacitance of a parallel plate capacitor depends only on the Area and separation between the plates. o C is dependent only on the geometry of the device!
Units of e 0 pico
Simple Capacitor Circuits o Batteries n Apply potential differences o Capacitors o Wires n Wires are METALS. n Continuous strands of wire all at the same potential. n Separate strands of wire connected to circuit elements may be at DIFFERENT potentials.
Size Matters! o A Random Access Memory stores information on small capacitors which are either charged (bit=1) or uncharged (bit=0). o Voltage across one of these capacitors ie either zero or the power source voltage (5. 3 volts in this example). o Typical capacitance is 55 f. F (femto=10 -15) o Question: How many electrons are stored on one of these capacitors in the +1 state?
Small is better in the IC world!
TWO Types of Connections SERIES PARALLEL
Parallel Connection V CEquivalent=CE
Series Connection q V -q C 1 q -q C 2 The charge on each capacitor is the same !
Series Connection Continued q V C 1 -q q -q C 2
More General
Example C 1 C 2 (12+5. 3)pf V series C 3 C 1=12. 0 uf C 2= 5. 3 uf C 3= 4. 5 ud
More on the Big C E=e 0 A/d +dq +q -q o We move a charge dq from the (-) plate to the (+) one. o The (-) plate becomes more (-) o The (+) plate becomes more (+). o d. W=Fd=dq x E x d
Not All Capacitors are Created Equal o Parallel Plate o Cylindric al o Spherical
Spherical Capacitor
Calculate Potential Difference V (-) sign because E and ds are in OPPOSITE directions.
Continuing… Lost (-) sign due to switch of limits.
Capacitor Circuits
A Thunker If a drop of liquid has capacitance 1. 00 p. F, what is its radius? STEPS Assume a charge on the drop. Calculate the potential See what happens
Anudder Thunker Find the equivalent capacitance between points a and b in the combination of capacitors shown in the figure. V(a b) same across each
Thunk some more … C 1 V C 2 (12+5. 3)pf C 3 C 1=12. 0 uf C 2= 5. 3 uf C 3= 4. 5 ud
More on the Big C E=e 0 A/d +dq +q -q o We move a charge dq from the (-) plate to the (+) one. o The (-) plate becomes more (-) o The (+) plate becomes more (+). o d. W=Fd=dq x E x d
So…. Sorta like (1/2)mv 2
DIELECTRIC
Polar Materials (Water)
Apply an Electric Field Some LOCAL ordering Larger Scale Ordering
Adding things up. . - + Net effect REDUCES the field
Non-Polar Material
Non-Polar Material Effective Charge is REDUCED
We can measure the C of a capacitor (later) C 0 = Vacuum or air Value C = With dielectric in place C=k. C 0 (we show this later)
How to Check This C 0 V V 0 Charge to V 0 and then disconnect from The battery. Connect the two together C 0 will lose some charge to the capacitor with the dielectric. We can measure V with a voltmeter (later).
Checking the idea. . V Note: When two Capacitors are the same (No dielectric), then V=V 0/2.
Messing with Capacitors + The battery means that the potential difference across the capacitor remains constant. + V - - For this case, we insert the dielectric but hold the voltage constant, + + q=CV V Remember – We hold V constant with the battery. - since C k. C 0 qk k. C 0 V THE EXTRA CHARGE COMES FROM THE BATTERY!
Another Case o We charge the capacitor to a voltage V 0. o We disconnect the battery. o We slip a dielectric in between the two plates. o We look at the voltage across the capacitor to see what happens.
No Battery + q 0 V 0 - q 0 =C 0 Vo When the dielectric is inserted, no charge is added so the charge must be the same. + - qk V
Another Way to Think About This o There is an original charge q on the capacitor. o If you slide the dielectric into the capacitor, you are adding no additional STORED charge. Just moving some charge around in the dielectric material. o If you short the capacitors with your fingers, only the original charge on the capacitor can burn your fingers to a crisp! o The charge in q=CV must therefore be the free charge on the metal plates of the capacitor.
A Closer Look at this stuff. . q Consider this virgin capacitor. No dielectric experience. Applied Voltage via a battery. ++++++ V 0 C 0 -q ---------
Remove the Battery q ++++++ V 0 The Voltage across the capacitor remains V 0 q remains the same as well. -q --------- The capacitor is fat (charged), dumb and happy.
Slip in a Dielectric Almost, but not quite, filling the space Gaussian Surface q V 0 -q ++++++ - - - - -q’ + + + +q’ --------- E 0 E E’ from induced charges
A little-q’sheet from the past. . +q’ - -q - + q+ + 0 2 x. Esheet 0
Some more sheet…
A Few slides back No Battery + V 0 - + - q=C 0 Vo q 0 When the dielectric is inserted, no charge is added so the charge must be the same. qk V
From this last equation
Another look Vo + -
Add Dielectric to Capacitor Vo o Original Structure + - V 0 + o Disconnect Battery + - o Slip in Dielectric Note: Charge on plate does not change!
What happens? so + so - si + si Potential Difference is REDUCED by insertion of dielectric. Charge on plate is Unchanged! Capacitance increases by a factor of k as we showed previously
SUMMARY OF RESULTS
APPLICATION OF GAUSS’ LAW
New Gauss for Dielectrics
- Slides: 63