Capacitors and DielectricsCapacitorsConductors are commonly used as places to store chargeYou cant just create some positive charge somewhere, you have to have corresponding negative charge somewhere elseDefinition of a capacitor:Two conductors, one of which stores charge +Q, and the other of which stores charge Q.Vab+QQCan we relate the charge Q that develops to the voltage difference V?Gausss Law tells us the electric field between the conductors:Integration tells us the potential difference
The relationship between voltage difference and charge is normally linearThis allows us to define capacitanceCapacitance has units of Coulomb/VoltAlso known as a Farad, abbreviated FA Farad is a very large amount of capacitanceLets work it out for concentric conducting spheres:CapacitanceWhats the capacitance of the Earth, if we put the other part of the charge at infinity?
Parallel Plate CapacitorsA more typical geometry is two large, closely spaced, parallel conducting platesArea A, separation d.Lets find the capacitance:Charge will all accumulate on the inner surfaceLet + and be the charges on each surfaceAs we already showed using Gausss law, this means there will be an electric field given by:If you integrate the electric field over the distance d, you get the potential differenceAdTo get a large capacitance, make the area large and the spacing smallCircuit symbol for a capacitor:
Capacitors in Parallel VC1C2When capacitors are joined at both ends like this, they are said to be in parallelThey have the same voltage across themThey can be treated like a single capacitor:When capacitors are joined at one end, with nothing else, they are said to be in seriesThey have the same voltage across themThey can be treated like a single capacitor:Capacitors in Series
Series and ParallelWhen two circuit elements are connected at one end, and nothing else is connected there, they are said to be in seriesWhen two circuit elements are connected at both ends, they are said to be in parallelThese formulas work for more than two circuit elements as well.
Complicated Capacitor Circuits4The capacitance of the capacitors in pF at right are marked. What is the effective capacitance of all the capacitors shown?For complex combinations of capacitors, you can replace small structures by equivalent capacitors, eventually simplifying everything10 V3215Capacitors 1 and 5 are connected at both ends- therefore they are parallelCapacitors 3 and 6 are connected at just one end therefore they are seriesAll three capacitors are now connected at both ends they are all in parallel
Energy in a CapacitorSuppose you have a capacitor with charge q already on it, and you try to add a small additional charge dq to it, where dq is small. How much energy would this take?The side with +q has a higher potentialMoving the charge there takes energyThe small change in energy is:+q q dq Now, imagine we start with zero charge and build it up gradually to q = QIt makes sense to say an uncharged capacitor has U = 0
Energy density in a capacitorSuppose you have a parallel plate capacitor with area A, separation d, and charged to voltage V. (1) Whats the energy divided by the volume between the plates?(2) Write this in terms of the electric field magnitudeAdEnergy density is energy over volumeWe can associate the energy with the electric field itselfThis formula can be shown to be completely generalizableIt has nothing in particular to do with capacitors
Dielectrics in CapacitorsWhat should I put between the metal plates of a capacitor?Goal make the capacitance largeThe closer you put the plates together, the bigger the capacitanceIts hard to put things close together unless you put something between themWhen they get charged, they are also very attracted to each otherPlacing an insulating material a dielectric allows you to place them very close togetherThe charges in the dielectric will also shiftThis partly cancels the electric fieldSmall field means smaller potential differenceC = Q/V, so C gets bigger too++++++++++++
Choosing a dielectricWhat makes a good dielectric?Have a high dielectric constant The combination 0 is also called , the permittivityMust be a good insulatorOtherwise charge will slowly bleed awayHave a high dielectric strengthThe maximum electric field at which the insulator suddenly (catastrophically) becomes a conductorThere is a corresponding breakdown voltage where the capacitor fails
What are capacitors good for?They store energyThe energy stored is not extremely large, and it tends to leak away over timeGasoline or fuel cells are better for this purposeThey can release their energy very quicklyCamera flashes, defibrillators, research usesThey resist changes in voltagePower supplies for electronic devices, etc.They can be used for timing, frequency filtering, etc.In conjunction with other parts
DipolesWeve done a lot with charges in electric fieldsHowever, in nature, neutral combinations are much more common than charged objectsThis doesnt mean there are no electric effects!A dipole is any collection of 2+ charges that have no total charge, but the charge is lopsided on one side or the otherMany molecules are dipolesThe dipole moment for a pair of charges, is just a vector equal to the charge q times the separation vector rFor more complicated objects, it is harder+q -q r
Dipoles in Uniform Electric FieldsElectric fields are often uniform, or nearly uniform, as seen by a molecule After all, molecules are pretty small!A dipole in a uniform electric field feels no total forceHowever, there is a torque, or twisting force, on a dipole+q -q r F F There is also an energy associated with a dipole in an electric field
Dipoles as DielectricsIn the absence of an electric field, dipoles will orient randomly in different directions due to random thermal motionWhen you turn the electric field on, the random motions still continue, but a fraction of the molecules will reorient to match the electric fieldNow there is an excess of positive charge on the right and negative charge on the leftThis creates a weaker counter-balancing electric field that partly cancels the imposed field