1. Chapter 24 Capacitance
A capacitor consists of two spatially separated conductors which can be charged to +Q and -Q.
Capacitor is a device that stores electrostatic potential energy.
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2. Capacitance
The capacitance is defined as the ratio of the charge on one conductor of the capacitor to the potential difference between the conductors.
Capacitance is measured in Faradays represented by the letter F which is equal to (coulomb/volt)
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3. Capacitance between Parallel Plates
Calculate the capacitance. We assume +, - charge densities on each plate with potential difference V: A d +Q -Q d E x y
First determine E field produced by charged conductors:
What is s ?
A = area of plate
Choose integration path,dy, from negative plate to positive plate
Dy & E field point in opposite directions
The dot porduct of E*dy will be negative
V=- integral of – E dy = integral E dy = positive quantity
Second, integrate E to find the potential difference V
As promised, V is proportional to Q !
C determined by geometry !
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4. Example
What is the capacitance of a parallel plate capacitor made out of two square plates 10 x10 cm2 separated by 1 mm wide gap?
d=1 mm
A=0.01 m2
If we connect 1.5V battery to its plates, how much charge could be stored in this capacitor?
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5. Demos
Demo 5A-28: Capacitance versus d
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6. Question
Does the capacitance C of a capacitor increase, decrease or remain the same when the charge q on it is doubled?
Decreases
Remains the Same
Increases
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7. Question Which line goes with which parallel plate capacitor? qd 1 2 2A 3 2d a b c q V
1:b, 2:a, 3:c
1:b, 2:c, 3:a
1:a, 2:b, 3:c
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8. A Cylindrical Capacitor: Step 1
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9. Calculating Capacitance
Put + Q , – Q on the plates.
Find between the plates i.e., given the charges, using Gauss’ Law
Find V across the plates
Use the definition
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10. A Cylindrical Capacitor : Step 2
The electric field in between the plates is radial. Use a cylindrical shaped Gaussian surface of radius R (R1<R<R2) of length l<<L
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11. A Cylindrical Capacitor : Step 3-Q
In general, the path of integration is taken from the – to the + plate. The vectors E and ds are opposite, so the dot product of E * ds is negative yielding: r R2 +Q ds R1 E
For the case of parallel plate capacitors, ds = -dr because the integration path is radially inwards giving:
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12. A Cylindrical Capacitor : Step 4
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13. QUIZ (A) A > B > C (B) B > A = C (C) B = C < A
An isolated solid metal sphere A of radius R and two spherical capacitors B and C with inner and out radii R and 2R are shown. The inner “spherical” plate of capacitor B is a spherical shell; that of capacitor C is a solid sphere. Rank the objects A, B, and C according to their capacitance, greatest first.
(A) A > B > C
(B) B > A = C
(C) B = C < A
(D) C < A < B
(E) A < B = C A B C
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14. Energy Stored in a Capacitor
If a small amount of additional positive charge dq is transferred from the negative conductor to the positive conductor through a potential increase of V, the potential energy of the charge and thus the capacitor is increased by
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15. Energy stored in a capacitor
Charging capacitor requires work: V
must move charges from one plate to another through rising V
charging a capacitor involves work: W=q V
this work is equal to the energy stored in a capacitor
Energy stored in a capacitor:
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16. Energy stored in a capacitor
Alternative equations for energy stored in a capacitor
Using
Using
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17. Demo
Demo 5A-29: Energy stored in a capacitor
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18. QUIZ
A parallel plate capacitor is given a charge q. The plates are then pulled a small distance farther apart. Which of the following apply to the situation after the plates have been moved?
The energy stored in the capacitor decreases.
(A) True
(B) False +q -q + - d
pull
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19. QUIZ:
A parallel plate capacitor is given a charge q. The plates are then pushed a small distance closer together. Which of the following apply to the situation after the plates have been moved?
The energy stored in the capacitor decreases.
(A) True
(B) False +q -q + - d
push
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20. QUIZ
A capacitor has a capacitance of 25 µF and is initially uncharged. The battery provides a potential difference of 120 Volts. After the switch, S, is closed, how much charge will pass through it? S
3 mC
14 mC
16 mC
30 mC
140 mC
3 milli Coulomb + C -
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21. Demonstration: Energy Stored in a Capacitor
P.S.
Power supply (P.S.) supplies voltage V.
can produce a big jolt.
Application: defibrillator A fibrillating heart is one
in which the cardiac muscles go into uncontrolled
twitching and quivering. A defibrillator is used to
stop this.
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22. Demonstration: Energy Stored in a Capacitor
P.S.
Power supply (P.S.) supplies voltage V.
Defibrillation is the definitive treatment for the life-threatening cardiac arrhythmias, ventricular fibrillation and pulseless ventricular tachycardia. Defibrillation consists of delivering a therapeutic dose of electrical energy to the affected heart with a device called a defibrillator. This depolarizes a critical mass of the heart muscle, terminates the arrhythmia, and allows normal sinus rhythm to be reestablished by the body's natural pacemaker, in the sinoatrial node of the heart.
In 1962 Bernard Lown introduced the external DC defibrillator. This device applied a direct current from a discharging capacitor through the chest wall into the heart to stop heart fibrillation.[5]
can produce a big jolt.
Application: defibrillator A fibrillating heart is one
in which the cardiac muscles go into uncontrolled
twitching and quivering. A defibrillator is used to
stop this.
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23. Numbers
About 200 J of this is sent through the heart attack victim in 2 milliseconds.
short time
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24. Capacitance
Capacitance is defined for any pair of spatially separated conductors.
How do we understand this definition ? +Q -Q d
Consider two conductors, one with excess charge = +Q
and the other with excess charge = -Q V
We can integrate the electric field between them to find the potential difference between the conductor
These charges create an electric field in the space between them E
This potential difference should be proportional to Q !
The ratio of Q to the potential difference is the capacitance
and only depends on the geometry of the conductors