capacitors - michigan state university
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Capacitors
February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 1
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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
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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
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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=
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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
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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
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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
ε0 2πLln
r2
r1
⎛
⎝⎜⎞
⎠⎟
=2πε0 L
ln r2 / r1( )
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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
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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
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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
4πε0
1r1
− 1r2
⎛
⎝⎜⎞
⎠⎟
=4πε0
1r1
− 1r2
⎛
⎝⎜⎞
⎠⎟
= 4πε0
r1r2
r2 − r1
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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
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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
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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
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February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 18
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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
∑
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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
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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
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February 3, 2014 Physics for Scientists & Engineers 2, Chapter 24 22
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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