ch24 capacitance

8/2/2019 Ch24 Capacitance

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Capacitors in series and parallel (sec. 24.2)

3.1.5

C 2009 J. F. Becker


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3.1.5


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.

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,

a b,

(-Q) (+Q).


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A charged parallel platecapacitor.

Q = C V where C = eo A / dfor a parallel plate capacitor,

where eo is the permittivity ofthe insulating material

(dielectric) between plates.

Recall that we used Gauss'sLaw to calculate the electricfield (E) between the plates

of a charged capacitor:

E = s / eo where there is avacuum between the plates.Vab = E d, so E = Vab /d

The unit of capacitance is called the Farad (F).


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(a) Two capacitors in series,(b) the equivalent capacitor.

1 / Ceq= 1 / C1 + 1 / C2


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(a) Two capacitors in parallel,(b) the equivalent circuit.

Ceq = C1 + C2


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Capacitors can store charge and ENERGYdU = 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, wehave to integrate over all the little chargesdq being added to a plate: dU = q dV gives

U = V dq = q/c dq = 1/C q dq = Q2 / 2C.And using Q = C V, we getU = Q2 / 2C = C V2 / 2 = Q V / 2

So the energy stored in a capacitor can bethought of as the potential energy stored inthe system of positive charges that are

separated from the negative charges, muchlike a stretched s rin has otential ener .


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ELECTRIC FIELD ENERGY

Here's another way to think of the energy

stored in a charged capacitor: If we considerthe space between the plates to contain theenergy (equal to 1/2 C V2) we can calculate an

energy DENSITY (Joules per volume). Thevolume between the plates is area x plateseparation, or A d. Then the energy density uis

u = 1/2 C V2 / A d =eoE2 / 2Recall C =eoA / d and V =E d.

C 2009 J. F. Becker


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Energy density: u = eoE2 / 2This is an important result because it tells usthat empty space contains energy if there isan electric field (E) in the "empty" space.

If we can get an electric field to travel (orpropagate) we can send or transmit energyand information through empty space!!!

C 2009 J. F. Becker


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Effect of a dielectric between the plates of aparallel plate capacitor. Note the charge isconstant !

DIELECTRICCONSTANT:

K= C / Co= ratio of thecapacitances

V = Vo / K


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A dielectric is added between the plates of acharged 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


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


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Polarization of a

dielectric in an electricfield E gives rise tothin layers of bound

charges on thedielectrics surfaces,

creating surfacecharge densities

+si and si.


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Polarization of a dielectric inan electric field E.


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A neutral sphere

B in the electricfield of a charged

sphere A is

attracted to thecharged spherebecause ofpolarization.


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DISCHARGING: CHARGING:An RC circuit

that can be usedto charge anddischarge a

capacitor(through aresistor).


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DISCHARGING A CAPACITOR:current vs time

DISCHARGING A CAPACITOR:

charge vs time


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See www.physics.edu/becker/physics51

ReviewC 2009 J F Becker

ch24 capacitance

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