Capacitance and dielectrics(sec. 24.1) Capacitors in series and parallel (sec. 24.2) Energy storage in capacitors and electric field energy(sec. 24.3) Dielectrics(sec. 24.4) Molecular model / polarization(sec. 24.5) R-C circuits(sec. 26.4) Chapter 24 Capacitance and Dielectrics C 2009 J. F. Becker

Any two conductors insulated from one another form a CAPACITOR. A "charged" capacitor can store charge. When a capacitor is being charged, negative charge is removed from one side of the capacitor and placed onto the other, leaving one side with a negative charge (-q) and the other side with a positive charge (+q).

A charged parallel plate capacitor. Q = C V where C = e o A / d for a parallel plate capacitor, where e o is the permittivity of the insulating material (dielectric) between plates. Recall that we used Gauss's Law to calculate the electric field (E) between the plates of a charged capacitor: E = s / e o where there is a vacuum between the plates. V ab = E d, so E = V ab /d The unit of capacitance is called the Farad (F).

(a) Two capacitors in series, (b) the equivalent capacitor. 1 / Ceq = 1 / C / C 2

(a) Two capacitors in parallel, (b) the equivalent circuit. Ceq = C 1 + C 2

Capacitors can store charge and ENERGY dU = q dV, and the potential V increases as the charge is placed on the plates (V = Q / C). Since the V changes as the Q is increased, we have to integrate over all the little charges “dq” being added to a plate: dU = q dV gives U =  V dq =  q/c dq = 1/C  q dq = Q 2 / 2C. And using Q = C V, we get U = Q 2 / 2C = C V 2 / 2 = Q V / 2 So the energy stored in a capacitor can be thought of as the potential energy stored in the system of positive charges that are separated from the negative charges, much like a stretched spring has potential energy.

ELECTRIC FIELD ENERGY Here's another way to think of the energy stored in a charged capacitor: If we consider the space between the plates to contain the energy (equal to 1/2 C V 2 ) we can calculate an energy DENSITY (Joules per volume). The volume between the plates is area x plate separation, or A d. Then the energy density u is u = 1/2 C V 2 / A d =  o E 2 / 2 Recall C =  o A / d and V =E d. C 2009 J. F. Becker

Energy density: u =  o E 2 / 2 This is an important result because it tells us that empty space contains energy if there is an electric field (E) in the "empty" space. If we can get an electric field to travel (or propagate) we can send or transmit energy and information through empty space!!! C 2009 J. F. Becker

Effect of a dielectric between the plates of a parallel plate capacitor. Note – the charge is constant ! DIELECTRIC CONSTANT: K = C / Co = ratio of the capacitances V = Vo / K

A dielectric is added between the plates of a charged capacitor (battery not connected): Q = Q, therefore Q = C V and Q = Co Vo Co Vo = C V, and if Vo decreases to V, Co must increase to C to keep equation balanced, and V = Vo Co/C Definition of DIELECTRIC CONSTANT: K = C / Co = ratio of the capacitances V = Vo / K C 2009 J. F. Becker

The charges induced on the surface of the dielectric reduce the electric field.

“Polarization” of a dielectric in an electric field E gives rise to thin layers of bound charges on the dielectric’s surfaces, creating surface charge densities +  i and –  i.

“Polarization” of a dielectric in an electric field E.

A neutral sphere B in the electric field of a charged sphere A is attracted to the charged sphere because of polarization.

DISCHARGING: CHARGING: An RC circuit that can be used to charge and discharge a capacitor (through a resistor).

CHARGING A CAPACITOR: current vs time CHARGING A CAPACITOR: charge vs time

DISCHARGING A CAPACITOR: current vs time DISCHARGING A CAPACITOR: charge vs time

See Review C 2009 J. F. Becker