Capacitors

February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 1

February 3, 2014 Physics for Scientists&Engineers 2 3

Example - Capacitance of a Parallel Plate Capacitor !  We have a parallel plate capacitor

constructed of two parallel plates, each with area 625 cm2 separated by a distance of 1.00 mm.

!  What is the capacitance of this parallel plate capacitor?

C =ε0Ad

A = 625 cm2 = 0.0625 m2

d = 1.00 mm = 1.00 ⋅10−3 m

C =8.85 ⋅10−12 F/m( ) 0.0625 m2( )

1.00 ⋅10−3 m= 5.53 ⋅10-10 F

C = 0.553 nFA parallel plate capacitor constructed out of square conducting plates 25 cm x 25 cm (=625 cm2) separated by 1 mm produces a capacitor with a capacitance of about 0.55 nF

February 3, 2014 Physics for Scientists&Engineers 2 4

Example 2 - Capacitance of a Parallel Plate Capacitor !  We have a parallel plate

capacitor constructed of two parallel plates separated by a distance of 1.00 mm.

!  What area is required to produce a capacitance of 0.55 F?

C =ε0Ad

d = 1.00 mm = 1.00 ⋅10−3 m

A =dCε0

=1.00 ⋅10−3 m( ) 0.55 F( )

8.85 ⋅10−12 F/m( ) = 0.62 ⋅108 m2

A parallel plate capacitor constructed out of square conducting plates 7.9 km x 7.9 km (4.9 miles x 4.9 miles) separated by 1 mm produces a capacitor with a capacitance of 0.55 F

Capacitance !  The potential difference between the two plates is

proportional to the amount of charge on the plates !  The proportionality constant is the the capacitance of the

device or

!  The capacitance of a device depends on the area of the plates and the distance between the plates but not on the charge or potential difference

!  For two parallel plates with area A and separation d:

February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 6

qCV

= q =VC

0ACdε=

February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 8

!  Consider a capacitor constructed of two collinear conducting cylinders of length L

!  The inner cylinder has radius r1 and the outer has radius r2 !  The inner cylinder has charge –q and the outer has charge

+q

Cylindrical Capacitor

February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 10

Cylindrical Capacitor !  We apply Gauss’s Law to get the electric field between the

two cylinders using a Gaussian surface with radius r and length L as illustrated by the black lines

!  Which we can rewrite to get an expression for the electric field between the two cylinders

E•dA∫∫ = −E dA∫∫

= −E 2πrL( ) = −qε0

E = q

ε0 2πrL

February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 11

Cylindrical Capacitor !  As we did for the parallel plate capacitor, we define the

voltage difference across the two cylinders to be

!  Thus, the capacitance of a cylindrical capacitor is

ΔV = −E •ds

i

f

∫ = E drr1

r2∫=

qε0 2πrL

drr1

r2∫

=q

ε0 2πLln r2

r1

⎛⎝⎜

⎞⎠⎟

C = qΔV

= qq

ε0 2πLln

r2

r1

⎛

⎝⎜⎞

⎠⎟

=2πε0 L

ln r2 / r1( )

February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 12

!  Consider a capacitor constructed of two concentric conducting spheres with radii r1 and r2

!  The inner sphere has charge +q and the outer has charge -q

Spherical Capacitor

February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 13

Spherical Capacitor !  We apply Gauss’s Law to get the electric field between the

two spheres using a spherical Gaussian surface with radius r as illustrated by the red line

!  Which we can rewrite to get an expression for the electric field between the two spheres

E•dA∫∫ = EA = E 4πr 2( ) = q

ε0

E = q

4πε0r 2

February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 14

Spherical Capacitor !  As we did for the parallel plate capacitor, we define the

voltage difference across the two spheres to be

!  Thus, the capacitance of a spherical capacitor is

ΔV = −E•ds

i

f

∫ = − Edrr1

r2∫= − q

4πε0r 2 drr1

r2∫

= q4πε0

1r1− 1

r2

⎛⎝⎜

⎞⎠⎟

C = qΔV

= qq

4πε0

1r1

− 1r2

⎛

⎝⎜⎞

⎠⎟

=4πε0

1r1

− 1r2

⎛

⎝⎜⎞

⎠⎟

= 4πε0

r1r2

r2 − r1

February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 15

Capacitance of an Isolated Sphere !  We obtain the capacitance of a single conducting sphere

by taking our result for a spherical capacitor and moving the outer spherical conductor infinitely far away

!  Using our result for a spherical capacitor

!  Looking at the potential of the conducting sphere

!  We get the same result as for a single point charge

C = 4πε0

1r1− 1

r2

⎛⎝⎜

⎞⎠⎟

r2 =∞,r1 = R ⇒ C = 4πε0R

C = qΔV

ΔV =V = qC= q

4πε0R= k q

R

February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 16

Capacitors in Circuits !  A circuit is a set of electrical devices connected with

conducting wires !  Capacitors can be wired together in circuits in parallel or

series •  Capacitors in circuits connected

by wires such that the positively charged plates are connected together and the negatively charged plates are connected together, are connected in parallel

•  Capacitors wired together such that the positively charged plate of one capacitor is connected to the negatively charged plate of the next capacitor are connected in series

February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 17

Capacitors in Parallel !  Consider an electrical circuit with three capacitors wired in

parallel

!  Each of three capacitors has one plate connected to the positive terminal of a battery with voltage V and one plate connected to the negative terminal

!  The potential difference V across each capacitor is the same !  We can write the charge on each capacitor as …

q1 =C1V q2 =C2V q3 =C3V

February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 18

February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 19

Capacitors in Parallel !  We can consider the three capacitors as one equivalent

capacitor Ceq that holds a total charge q given by

!  Thus the equivalent capacitance is

!  A general result for n capacitors in parallel is

!  If we can identify capacitors in a circuit that are wired in parallel, we can replace them with an equivalent capacitance

q = q1 +q2 +q3 =C1ΔV +C2ΔV +C3ΔV = C1 +C2 +C3( )ΔV

Ceq =C1 +C2 +C3

Ceq = Ci

i=1

n

∑

February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 20

Capacitors in Series !  Consider a circuit with three capacitors wired in series

!  The positively charged plate of C3 is connected to the positive terminal of the battery

!  The negatively charge plate of C3 is connected to the positively charged plate of C2

February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 21

Capacitors in Series

!  The negatively charged plate of C2 is connected to the positively charge plate of C1

!  The negatively charge plate of C1 is connected to the negative terminal of the battery

!  The battery produces an equal charge q on each capacitor

February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 22

February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 23

Capacitors in Series !  Knowing that the charge is the same on all three capacitors

we can write

!  We can express an equivalent capacitance as

!  We can generalize to n capacitors in series

!  If we can identify capacitors in a circuit that are wired in series, we can replace them with an equivalent capacitance

ΔV = ΔV1 +ΔV2 +ΔV3 =

qC1

+ qC2

+ qC3

= q 1C1

+ 1C2

+ 1C3

⎛⎝⎜

⎞⎠⎟

1Ceq

= 1Cii=1

n

∑

V = q

Ceq where 1

Ceq= 1

C1+ 1

C2+ 1

C3