PHYS2326- Week 5
PHYS2326- Week 5 PHYS 2326
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This 28 page Class Notes was uploaded by John Vakidis on Tuesday September 27, 2016. The Class Notes belongs to PHYS 2326 at University of Texas at Dallas taught by Dr. Lindsay King in Fall 2016. Since its upload, it has received 4 views. For similar materials see Electromagnetism and Waves in Physics at University of Texas at Dallas.
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Date Created: 09/27/16
Capacitors - a way to store electric potential energy • How does a camera’s flash unit store energy? • Capacitors are devices that store electric potential energy. • The energy of a capacitor is actually stored in the electric field. • Any two conductors separated by an insulator form a capacitor, as illustrated below. • The definition of capacitance is C = Q/V ab (charge/potential difference) Units Farads (F) (Coulomb/Volt) Note: don’t confuse symbol for charge with symbol for capacitance! When we say a capacitor has charge Q, or that it stores charge Q, we mean that one conductor at higher potential has charge +Q and the other one at lower potential has charge -Q. • A parallel-plate capacitor (see fig a) consists of two parallel conducting plates separated by a distance that is small compared to their dimensions. We can neglect that at the ends the electric field does not exactly point straight between the two plates (see fig b) • The capacitance of a parallel-plate capacitor is C = ε A/d 0 (see notes from class and next slide for proof). • Electric field between plates • Potential difference (substitute in for E from above) • Use definition of capacitance and substitute in for Vab • NB: 1F is an *extremely* large capacitance – usual values of order microfarad (1e-6 F) or picofarad (1e-12F) • Look back at notes/ text on potential. Remember potential outside each shell is as if it was coming from a point charge at centre. Inside a shell there is no E field, and associated potential is a constant (ie contribution from outer shell just gives Va=Vb). This means Va -Vb is all due to difference at a and b from inner shell. • Look back at your notes for potential associated with cylinders. • Capacitors are in series if they are connected one after the other, as illustrated below. • The equivalent capacitance of a series combination is given by 1/C eq = 1/C +11/C + 12C + … (s3e next slide) • Write down potential diff across each capacitor in terms of capacitance and charge. • You know that these values add to give the total Vab. • Rearrange to show how to • NB for capacitors in series combine we used the fact that Q is the capacitors in same across each capacitor! series! • Capacitors are connected in parallel between a and b if the potential difference V ab is the same for all the capacitors. • The equivalent capacitance of a parallel combination is the sum of the individual capacitances: C eq = C 1 C + 2 + ..3 Always draw a diagram to see which capacitors are in series or parallel. Then start replacing groups of them until things are as simple as possible! See notes from class for example! • The potential energy stored in a capacitor is See class notes or book for proofs U = Q /2C = 1/2 CV = 1/2 QV • The capacitor energy is stored in the electric field between the plates. The energy density (energy/volume) - see later in notes for proof - u = 1/2 ε E 0 2 • The "Z machine" shown below can produce up to 2.9 × 10 14 W using capacitors in parallel! Energy density stored in E field • Energy density u = energy/volume • e.g. for parallel plate capactior, • generally true - - not just for parallel plate capacitor NB: a vacuum with an electric field in it has energy! • A dielectric is a nonconducting material. Most capacitors have dielectric between their plates. • The dielectric constant of the material is K = C/C > 1. (See next slide for 0 some examples) • Dielectric increases the capacitance. • A dielectric affects the electric field between the plates, as we discussed in class. You don't have to remember these values - but be roughly aware of a typical possible value - ie K can be of order ~1, ~10, ~100 • The figure below shows polarization of the dielectric and how the induced charges reduce the magnitude of the resultant electric field. • If the electric field is strong enough, dielectric breakdown occurs and the dielectric becomes a conductor. • The dielectric strength is the maximum electric field the material can withstand before breakdown occurs. • The table shows the dielectric strength of some insulators. Don't try to remember exact values but be aware of a typical value! Electric field without (Eo) and with (E) dielectric, when the capacitor is charged up, isolated (removed from a battery), and dielectric inserted. So like we said, electric field is reduced. Going back to the energy and energy density , now with dielectric; again take as an example parallel plates: With constant Q (battery disconnected), put in dielectric: Stored energy U=(Q^2)/2C = (Q^2)/2KCo; down by factor K from vacuum. This also makes sense using equation in lower box above, since E^2 is down by a factor K^2, but permittivity is up by factor K! In a more unusual case, consider that the battery is still connected across the capacitor - we have constant V . If we insert a dielectric, electric field E remains constant. More charge comes from battery to build up on the plates - compensates for the dielectric's opposing induced field. So.... Energy U=C(V^2)/2; KCo(V^2)/2; up by factor K. This also makes sense if we look at the equation in the second box on the previous page - now E is constant but permittivity is up by a factor of K!
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