LECTURE 9CAPACITANCE & ENERGY

Lecture 92

¨ Reading chapter 24-1 to 24-3.¤ Capacitance

n Parallel plate capacitorsn Cylindrical capacitors

¤ The storage of electrical energy¤ Capacitors and batteries in circuits

Capacitors3

¨ A capacitor is a device consisting of two conductors.

¨ One conductor with charge Q, and the other with charge -Q will have a potential difference between them.

¨ The capacitance of the capacitor is a property of the geometry of the objects and defined as

¨ The SI unit of capacitance is the farad (F), which is equivalent to the (C/V).

¨ The symbol for a capacitor in a circuit diagram is

C =QV

Quiz: 1 & 24

Example: 15

¨ A storage capacitor on a random access memory (RAM) chip has a capacitance of C = 55 fF. If the capacitor is charged to V = 5.3V, how many excess electrons are on its negative plate?

Quiz: 36

Parallel plate capacitors7

¨ In a parallel-plate capacitor, two plates of conductors are closely placed parallel to each other.

¨ If a charge +Q is placed on one plate, and –Q on the other, they will be distributed evenly on the inner surfaces of the plates with area, A. The separation of the plates is d.

¨ The E field between the plates is uniform and given by

¨ The potential difference between the plates is given by

¨ The capacitance of a parallel plate capacitor is

¨ The capacitance increases as the plates become larger and decreases as the separation increases. This is a general trend in all capacitors.

E = Qε0A

ΔV =Qdε0A

C =ε0Ad

Demo: 18

¨ Variable capacitor¤ Variable area: As area of overlap between plates increase the capacitance increases.

n 𝐶 = 𝑄𝑉= 𝜀0𝐴

𝑑

n Since charge is constant the potential must decrease.

n 𝑉 ∝ 1𝐴

¤ Variable distance: as distance between plates increase the capacitance decreases.

n 𝐶 = 𝑄𝑉= 𝜀0𝐴

𝑑

n Since charge is constant the potential must increase.

n 𝑉 ∝ 𝑑

Example: 29

¨ Two sheets of aluminum foil have the same area, a separation of d = 1.00 mm, and a capacitance of C = 10. pF, and are charged to V = 12 V.

a) Calculate the area of each sheet, A.

¨ Now, the separation is decreased by 0.10 mm with the charge held constant.

b) What is the new capacitance?c) By how much does the potential difference

change?

Key board10

¨ There is a capacitor under each key.¨ As you press a key, the spacing between the capacitor plates change,

increasing the capacitance.

C =QV=ε0Ad

Condenser microphones11

¨ In a condenser microphone, also called a capacitor microphone or electrostatic microphone, the diaphragm acts as one plate of a parallel-plate capacitor.

¨ The sound (vibrations of air) produces changes in the distance between the plates.

C =QV=ε0Ad

Cylindrical capacitor

¨ Another common capacitor geometry is the cylindrical capacitor

𝐶 =2𝜋𝜀-𝐿

ln   𝑅3 𝑅4⁄

Coaxial cable

Electrostatic field energy13

¨ The energy stored in a capacitor is the total amount of energy required to charge the plates by moving increments of charge, 𝑑𝑄 , from one plate to the other, 𝑈 = 𝑄𝑉78.

¨ Since C = Q/V, the energy stored in a capacitor is given by

¨ The energy stored in a capacitor can be thought as energy stored in the electric filed, electrostatic field energy.

¨ The energy density uE stored in any electric field is

U =12QV =

12Q2

C=12CV 2

uE =energyvolume

=12ε0E

2

Demo: 214

¨ Capacitor energy¤ Three 1.5 mF capacitors connected in parallel are charged to 400 volts. ¤ Capacitors store energy and could be dangerous.

¤𝑈 = 43 𝐶𝑉

3 = 43 4.5  mF 400  V 3 = 360  J

Camera flash and electronics15

¨ A capacitor can store a large amount of charge.¨ A flash unit of a camera uses a large amount of charge in a short

amount of time.¨ In most electronics boards, we use capacitors where we need to use a

lot of charge quickly.

Defibrillator16

¨ A jolt of electric current from a defibrillator can restore normal heartbeat.

¨ A capacitor is used to store a large amount of charge (and energy).

¨ The charged capacitor is discharged quickly to deliver the charge and the energy to a person in distress.

Batteries17

¨ When nothing is connected to the battery, it has the open-circuit terminal voltage across the terminals.

¨ Chemical reaction removes electrons from one terminal and deposits them on the other, maintaining the terminal voltage.

¨ The potential difference between the terminals of a battery is called terminal voltage.

¨ The symbol for a battery in a circuit diagram is

Carbon-zinc cell

Battery and capacitor18

¨ When a capacitor is connected to a battery,¤ the terminal voltage of the battery temporarily drops causing the chemical reactions to take place within

the battery.

¤ the battery acts as a charge pump.

¤ the battery charges the capacitor until the open-circuit terminal voltage is attained.

¨ Since the potential inside a system of connected conductors is constant, when equilibrium is attained,¤ the potentials of the positive terminal, the upper plate and the connecting wire are the same.

¤ the potentials of the negative terminal, the lower plate and the connecting wire are the same.

Example: 3 19

¨ A parallel-plate capacitor has plates of area A = 500 cm2 and is connected across the terminals of a battery. After some time has passed, the capacitor is disconnected from the battery. When the plates are then moved Δd = 0.40 cm farther apart, the charge on each plate remains constant but the potential difference between the plates increases by ΔV = 100 V.a) What is the magnitude of the charge on each plate?

b) Do you expect the energy stored in the capacitor to increase, decrease, or remain constant as the plates are moved this way?

c) Support your answer to Part b) by determining the change in stored energy in the capacitor due to the movement of the plates.