Capacitors:Part 1

Phys 122 Lecture 11G. Rybka

• Long-­answer REGRADEs DUE today by 4 pm;; turn in directly to Alexis Olsho;; no special form required

• Dielectrics and more Capacitor stuff on Monday• no new Smart Physics due

• Go over tough significant figure problem on homework

C1 C2

a

b

C3C

a bº

C QV

≡

Energy in Capacitors

• The work done (usually by a battery) to separate the charges on a capacitor is stored (and can later be retrieved) as Electric Potential Energy

• We charge a capacitor by moving dq through an existing V(q), and integrate to get energy stored:

U q VΔ = Δ

Equivalent Expressions: Energy in Capacitors

• Energy Density:– The energy stored (per volume) in ANY electric field

» Not just our parallel plate case» Units: J / m3

C = Q/V

V = Q/C

2 21 1 20 202 2 1 1

02 2 2/CV AV d VUu E

Vol Ad Ad d

ε εε≡ = = = =

Recall: (V/d = E)

• Calculate the capacitance:• Assume +Q, -­Q on surface of cylinders with potential difference V.

Example:Cylindrical Capacitor

a

b L

r

+ + + +

Recall: Cylindrical Symmetry

ÞLr

QE02πε

=

• Gaussian surface is cylinder of radius r and length L

• Cylinder has charge Q

• Apply Gauss' Law:

0

2ε

πQrLESdE ==•∫

!!

Er

L

Er

+ + +Q

This is not new

Example: Cylindrical Capacitor

• Calculate the capacitance:• Assume +Q, -­Q on surface of cylinders with potential difference V.

• From Gauss’ Law:

E QrL

=2 0πε

V E dl Edr QrLdr Q

Lbab

a

b

a

a

b= − •∫ = −∫ = ∫ = ⎛

⎝⎜ ⎞⎠⎟

! !

2 20 0πε πεln

If inner cylinder has +Q, then the potential V is positive if we take the zero of potential to be defined at r = b:

Þ C QV

Lba

≡ =⎛⎝⎜ ⎞⎠⎟

2 0πε

ln

a

b L

r

+Q

- Q

Clicker• In each case below, a charge of +Q is placed on a solid spherical conductor and a charge of -­Q is placed on a concentric conducting spherical shell. – Let V1 be the potential difference between the spheres with (a1, b). – Let V2 be the potential difference between the spheres with (a2, b). – What is the relationship between the absolute value of V1 and V2?

(a) V1 < V2 (b) V1 = V2 (c) V1 > V2

a2

b

+Q-­Q

a1

b

+Q

-­Q

• Two spherical capacitors.• Intuition: for parallel plate capacitors: V = (Q/C) = (Qd)/(Aε0).

Therefore you might expect that V1 > V2 since (b-­a1) > (b-­a2).• In fact this is the case as we can show directly from the definition of V!

Follow upCompute V1 from E (let Vb = 0)

∫∫∫ +=−=b

a

a

a

a

b

drrEdrrEdrrEV2

2

1

1

)()()(1

V E r dr Va

a1 2

1

2= +∫ ( )

Since E(r) has the same form for r > a2,

(c) V1 > V2

The actual functional form for the spherical capacitor:

⎟⎠

⎞⎜⎝

⎛ −=⎥⎦

⎤⎢⎣

⎡=−=−= ∫∫ baQ

rQ

rdrQdrrEV

a

b

a

b

a

b

114

144

)(00

20 πεπεπε

C QV

a b

abb a

≡ =−⎛

⎝⎜ ⎞

⎠⎟=

−41 1

400

πεπε

(Abs. value)

Capacitors in Parallel

• Find “equivalent” capacitance C

VC1 C2

a

b

Q2Q1 º VC

a

b

Q

Parallel Combination:

Equivalent Capacitor:

2

2

1

1

CQ

CQV ==

Þ 21 CCC +=

VVCVC

VQQ

VQC 2121 +

=+

=≡Þ

V

Q1 = C1V Q2 = C2V

C2C1

Key point: V is the same for both capacitors

Key Point: Qtotal = Q1 + Q2 = VC1 + VC2 = V(C1 + C2)

Ctotal = C1 + C2

Qtotal

Qtotal

Parallel Capacitor Circuit

C1 + C2= Ceq

Capacitors in Series

• Find “equivalent” capacitance C

• Charge on C1 same as charge on C2– since applying a potential difference across ab cannot produce a net

charge on the inner plates of C1 and C2 .

C a bº +Q -­Q

C1 C2a b

+Q -­Q

V QCab =

V V V QC

QCab = + = +1 2

1 2

RHS:

LHS:Þ 1 1 1

1 2C C C= +

Examples:Combinations of Capacitors

C1 C2

a

b

C3C

a bº

Note: C3 is in series with the parallel combination on C1 and C2. i.e.

1 1 13 1 2C C C C

= ++ Þ C C C C

C C C=

++ +3 1 2

1 2 3

( )

PreLecture Trouble … (click)Suppose you have two identical capacitors, each having capacitance C. Cmax is the biggest possible equivalent capacitance that can be made by combining these two, and Cmin is the smallest.

How does Cmax compare to Cmin?

A. Cmax = 4CminB. Cmax = 3CminC. Cmax = 2CminD. Cmax = (3/2)CminE. Cmax = Cmin

2/111

min

CCCC

⇒+= CCCC 221max =+=

4/ minmax =CC

A circuit consists of three unequal capacitors C1, C2, and C3 which are connected to a battery of voltage V0. The capacitance of C2 is twice that of C1. The capacitance of C3 is three times that of C1. The capacitors obtain charges Q1, Q2, and Q3.

Compare Q1, Q2, and Q3.

Q1 > Q3 > Q2Q1 > Q2 > Q3Q1 > Q2 = Q3Q1 = Q2 = Q3Q1 < Q2 = Q3

Checkpoint difficulty

Great responseQ2 and Q3 will be the same since C2 and C3 are in series. The capacitance of the series C2 and C3 is 1.2 times the capacitance of C1. V0 voltage will be running through the two parallel branches, so C1 will have less charge than C2 and C3 because C2 and C3 have a combined capacitance of 1.2C, where C is the capacitance of C1.

40%

A circuit consists of three unequal capacitors C1, C2, and C3 which are connected to a battery of voltage V0. The capacitance of C2 is twice that of C1. The capacitance of C3 is three times that of C1. The capacitors obtain charges Q1, Q2, and Q3.

Compare Q1, Q2, and Q3.

Q1 > Q3 > Q2Q1 > Q2 > Q3Q1 > Q2 = Q3Q1 = Q2 = Q3Q1 < Q2 = Q3

Checkpoint difficulty

• V0 = VC1 = VC23• Which is larger? C1 or C23 ?

• 𝑪𝟐𝟑 =𝟏𝟐𝑪+ 𝟏𝟑𝑪

'𝟏= 𝟔

𝟓𝑪 SO, C23 > C1

• Recall Q2 = Q3 for series capacitors = Q23• Q = CV• V’s the same, so Q23 > Q1 since C23 > C1

C232C

3CC1 =C

ClickerWhat is the equivalent capacitance, Ceq, of the combination shown?

(a) Ceq= (3/2)C (b) Ceq= (2/3)C (c) Ceq= 3C

o

o

C CCCeq

CC C ≡ C C1

1 1 11C C C= + C C

1 2= C C C Ceq = + =

232

Which of the following is NOT necessarily true:A) V0 = V1B) Ctotal > C1C) V2 = V3D) Q2 = Q3E) V1 = V2 + V3

Similar to CheckPoint

Ctotal

C1

C2

C3

V0

V1 Q1

V2

Q2

V3 Q3