0

0

00

,

.

,

.

.

EA

QCV

V V Eds Ed

V V V EdQE dA EA

Q EAEA ACEd d

εεε ε

−− + +

+ −

=

− = − = −

= − =

Φ = ⋅ = =

=

= =

∫

∫rr

Ñ

Q

Q−

++++++

d )(uniformE0ε

A

plate) conducting (

.ACdε=

Parallel-plate Capacitor

If there is dielectric ( )Between the plates：

ε

0C C=

d 0ε 0 rε ε ε= b

?C =

A

0 0 1 2Let Q on the plate, in , and in .E E Eε ε,QC V Eds

V+−= = −∫

0

1

2

0 00 0

1 1

2 20 0

, ,

, ,

, .

A

A

A

Q QE dA E A EA

Q QE dA E A EA

Q QE dA E A EA

ε ε

ε ε

ε ε

⋅ = = =

⋅ = = =

−⋅ = − = =

∫

∫

∫

rr

rr

rr

Ñ

Ñ

Ñ

0 1

00 0.

0 1

( ) ,

1( ) 1( ) (1 )r

V Eds E d b E b

AQC A CE d b E b dd b

εε

ε

+−⇒ = − = − +

= = > =− + − −

∫

Ex：

0E

2EQ−

Gaussian

surface

0AQ

2A

1A

1EQ−

Q

o E dA qε κ ⋅ =∫rr

Ñ

(25 - 21)

Gauss’ law in the presence of dielectricsEven though the equation above was derived for the parallel plate capacitor is true in general.Note 1：The flux integral now involves Eκ

r

Note 2：The charge q that is used is the plate charge, also known as “free charge”. Using the equation above we can ignore the induced charge Note 3：The dielectric constant K is kept inside the integral to describe the most general case in which K is not constant over the Gaussian surface.

q′

222

3

11 ( ) 122 ( )2

ee

A EdCVU d Ju Evolume Ad Ad m

εε= = = =

Energy density of capacitor

Consider a capacitor made of two parallelmetallic plates separated by a distance d. The top plate has a surface charge density +σ,the bottom plate – σ.A slab of metal of thicknessl<d is inserted between the plates, not con-nected to either one. Upon insertion of the metal slab, the potential difference between the plates1.increases.2.decreases.3.remains the same.

Consider two capacitors, each having plateseparation d. In each case, a slab of metal ofthickness d/3 is inserted between the plates.In case (a), the slab is not connected to ei-ther plate. In case (b), it is connected to the upper plate. The capacitance is higher for

1.case(a).2.case(b).3. The two capacitances are equal.

Consider a simple parallel-plate capacitorwhose plates are given equal and oppositecharges and are separated by a distance d.Suppose the plates are pulled apart untilthey are separated by a distance D>d. Theelectrostatic energy stored in the capacitor is1.greater than2.the same as3.smaller than

before the plates were pulled apart.

Chapter 26

Capacitance and Dielectrics

Quick Quiz 26.1

A capacitor stores charge Q at a potential difference ΔV. If the voltage applied by a battery to the capacitor is doubled to 2ΔV:(a) the capacitance falls to half its initial value and the charge remains the same (b) the capacitance and the charge both fall to half their initial values (c) the capacitance and the charge both double (d) the capacitance remains the same and the charge doubles

Answer: (d). The capacitance is a property of the physical system and does not vary with applied voltage. According to Equation 26.1, if the voltage is doubled, the charge is doubled.

Quick Quiz 26.1

Quick Quiz 26.2

Many computer keyboard buttons are constructed of capacitors, as shown in the figure below. When a key is pushed down, the soft insulator between the movable plate and the fixed plate is compressed. When the key is pressed, the capacitance (a) increases(b) decreases(c) changes in a way that we cannot determine because the complicated electric circuit connected to the keyboard button may cause a change in ΔV.

Answer: (a). When the key is pressed, the plate separation is decreased and the capacitance increases. Capacitance depends only on how a capacitor is constructed and not on the external circuit.

Quick Quiz 26.2

Quick Quiz 26.3

Two capacitors are identical. They can be connected in series or in parallel. If you want the smallest equivalent capacitance for the combination, you should connect them in (a) series(b) parallel(c) Either combination has the same capacitance.

Answer: (a). When connecting capacitors in series, the inverses of the capacitances add, resulting in a smaller overall equivalent capacitance.

Quick Quiz 26.3

Quick Quiz 26.4

Consider the two capacitors in question 3 again. Each capacitor is charged to a voltage of 10 V. If you want the largest combined potential difference across the combination, you should connect them in (a) series(b) parallel(c) Either combination has the same potential difference.

Answer: (a). When capacitors are connected in series, the voltages add, for a total of 20 V in this case. If they are combined in parallel, the voltage across the combination is still 10 V.

Quick Quiz 26.4

Quick Quiz 26.5

You have three capacitors and a battery. In which of the following combinations of the three capacitors will the maximum possible energy be stored when the combination is attached to the battery? (a) series (b) parallel (c) Both combinations will store the same amount of energy.

Answer: (b). For a given voltage, the energy stored in a capacitor is proportional to C: U = C(ΔV)2/2. Thus, you want to maximize the equivalent capacitance. You do this by connecting the three capacitors in parallel, so that the capacitances add.

Quick Quiz 26.5

Quick Quiz 26.6

You charge a parallel-plate capacitor, remove it from the battery, and prevent the wires connected to the plates from touching each other. When you pull the plates apart to a larger separation, do the following quantities increase, decrease, or stay the same? (a) C; (b) Q; (c) E between the plates; (d) ΔV ; (e) energy stored in the capacitor.

Answer: (a) C decreases (Eq. 26.3). (b) Q stays the same because there is no place for the charge to flow. (c) E remains constant (see Eq. 24.8 and the paragraph following it). (d) ΔV increases because ΔV = Q/C, Q is constant (part b), and C decreases (part a). (e) The energy stored in the capacitor is proportional to both Q and ΔV (Eq. 26.11) and thus increases. The additional energy comes from the work you do in pulling the two plates apart.

Quick Quiz 26.6