Main Points of Chapter 25 Definition of capacitance Calculation of capacitance Energy in capacitors and in electric fields Equivalent capacitance for series and parallel connections Dielectrics Microscopic description of dielectrics
25-1 Capacitance Simplest capacitor – two equal and oppositely charged conductors Parallel-plate capacitor:
25-1 Capacitance Conductors of arbitrary shape and size: Isolated conductor: Q: Where’s the other plate? A: At infinity, effectively.
25-1 Capacitance Potential difference between the conductors depends on the charge on them V α Q Definition of capacitance: C = Q/V Depends only on geometry and materials
25-1 Capacitance Capacitance of a parallel-plate capacitor Field is uniform in middle of capacitor Capacitance (ignoring edge effects): (25-4)
25-2 Energy in Capacitors Each bit of charge added increases the electric field; subsequent charges take more work Total work to charge capacitor to Q: (25-7)
25-2 Energy in Capacitors Potential energy of a charged capacitor: (25-8) (25-9) (25-10) All three expressions are equivalent!
25-3 Energy in Electric Fields Using the known electric field inside a parallel-plate capacitor: (25-11) Then dividing by the capacitor’s volume Ad, we find the energy density: (25-12)
This is a general expression for the local energy density in free space, for a constant or variable electric field—not just for a parallel-plate capacitor. 25-3 Energy in Electric Fields (25-12)
25-4 Capacitors in Parallel and in Series Finding the equivalent capacitance: Parallel connection (25-14)
25-4 Capacitors in Parallel and in Series Finding the equivalent capacitance: Series connection (25-14) Just remember—the equivalent capacitance is C eq, not 1/C eq !
25-5 Dielectrics Dielectric = insulator Molecules act as dipoles, permanent or induced This effectively reduces the electric field
25-5 Dielectrics Dielectric also increases capacitance: (25-18) Some dielectric constants:
25-5 Dielectrics If the charge is held constant, insertion of a dielectric causes the voltage to decrease:
25-5 Dielectrics If the voltage is held constant, insertion of a dielectric causes the charge to increase:
25-5 Dielectrics Addition of dielectric to capacitor modifies capacitance equation – use ε instead of ε 0 : (25-23) Dielectric strength – maximum electric field that a material can sustain before breaking down (becoming conductive as electrons are ripped off atoms by intense field)
25-6 The Microscopic Description of Dielectrics For polar molecules (having a permanent dipole moment), an external field tends to rotate them
25-6 The Microscopic Description of Dielectrics For nonpolar molecules, an external field tends to polarize them
25-6 The Microscopic Description of Dielectrics In either case, the induced electric field reduces the overall field:
25-6 The Microscopic Description of Dielectrics Gauss’ Law If the dielectric is uniform (same throughout): If not: (25-28) (25-29)
Summary of Chapter 25 Capacitor is two equal and oppositely charged conductors Capacitance depends only on geometry and dielectrics Electric fields carry energy Capacitors in parallel add; capacitors in series add reciprocals Dielectrics reduce electric field, increasing capacitance