chapter 24 capacitance, dielectrics, electric energy storage

Copyright © 2009 Pearson Education, Inc.

Chapter 24Capacitance, Dielectrics, Electric Energy Storage


A charged capacitor stores electric energy; the energy stored is equal to the work done to charge the capacitor:

24-4 Electric Energy Storage



Conceptual Example 24-9: Capacitor plate separation increased.

A parallel-plate capacitor carries charge Q and is then disconnected from a battery. The two plates are initially separated by a distance d. Suppose the plates are pulled apart until the separation is 2d. How has the energy stored in this capacitor changed?


The energy density, defined as the energy per unit volume, is the same no matter the origin of the electric field:

The sudden discharge of electric energy can be harmful or fatal. Capacitors can retain their charge indefinitely even when disconnected from a voltage source – be careful!


22 20 0

20

1 1 12 2 2

12

AU CV Ed AdE

dEnergy U

u EVolume Ad


Heart defibrillators use electric discharge to “jump-start” the heart, and can save lives.




National Ignition Facility (NIF)

Lawrence Livermore National Laboratory


NIFLaser system driven by 4000 300 μF capacitorswhich store a total of 422 MJ. They take 60 s tocharge and are discharged in 400 μs.1)What is the potential difference across each capacitor?2)What is the power delivered during the discharge?


NIFLaser system driven by 4000 300 μF capacitorswhich store a total of 422 MJ. They take 60 s tocharge and are discharged in 400 μs.1)What is the potential difference across each capacitor?2)What is the power delivered during the discharge?

Solution:1)U = CV2/2 →V = (2U/C)1/2 →V = [2(422x106)/4000/300x10-6] ½ = 26.5 kV2)P = W/t = U/t = 422x106 /400x10-6 ~ 1012 W = 1000 GW! cf. 1.0-1.5 GW for power plant


The molecules in a dielectric, when in an external electric field, tend to become oriented in a way that opposes the external field.

A dielectric is an insulator, and is characterized by a dielectric constant K.

24-6 Molecular Description of Dielectrics


This means that the electric field within the dielectric is less than it would be in air, allowing more charge to be stored for the same potential. This reorientation of the molecules results in an induced charge – there is no net charge on the dielectric, but the charge is asymmetrically distributed.

The magnitude of the induced charge depends on the dielectric constant:

24-6 Molecular Description of Dielectrics

00 0 00 0

1 11 1ind indeff

QQ Q Q Q Q Q

K KQ Q

K


Capacitance of a parallel-plate capacitor filled with dielectric:

24-5 Dielectrics

Using the dielectric constant, we define the permittivity:


Dielectric strength is the maximum field a dielectric can experience without breaking down.

24-5 Dielectrics


24-5 DielectricsHere are two experiments where we insert and remove a dielectric from a capacitor. In the first, the capacitor is connected to a battery, so the voltage remains constant. The capacitance increases, and therefore the charge on the plates increases as well.

20 0 0

12

U C V 2 20 0 0 0 0

1 12 2K KC V K C V KU U U


24-5 Dielectrics

In this second experiment, we charge a capacitor, disconnect it, and then insert the dielectric. In this case, the charge remains constant. Since the dielectric increases the capacitance, the potential across the capacitor drops.

2 2 22 0 0 0

0 0 00 0

1 1 12 2 2

C V QU C V

C C

2 22 22 0 0

000

1 1 1 1 12 2 2 2

K KK K

KK

K

Q QC VC V U

C C KC KU U


24-5 DielectricsExample 24-11: Dielectric removal.

A parallel-plate capacitor, filled with a dielectric with K = 3.4, is connected to a 100-V battery. After the capacitor is fully charged, the battery is disconnected. The plates have area A = 4.0 m2 and are separated by d = 4.0 mm. (a) Find the capacitance, the charge on the capacitor, the electric field strength, and the energy stored in the capacitor. (b) The dielectric is carefully removed, without changing the plate separation nor does any charge leave the capacitor. Find the new values of capacitance, electric field strength, voltage between the plates, and the energy stored in the capacitor.


• Capacitor: nontouching conductors carrying equal and opposite charge.

• Capacitance:

• Capacitance of a parallel-plate capacitor:

Summary of Chapter 24



• Capacitors in parallel:

• Capacitors in series:


• Energy density in electric field:

• A dielectric is an insulator.

• Dielectric constant gives ratio of total field to external field.

• For a parallel-plate capacitor:



Chapter 25Electric Currents and

Resistance


Volta discovered that electricity could be created if dissimilar metals were connected by a conductive solution called an electrolyte.

This is a simple electric cell.

25-1 The Electric Battery


Several cells connected together make a battery, although now we refer to a single cell as a battery as well.

25-1 The Electric Battery


Electric current is the rate of flow of charge through a conductor:

Unit of electric current: the ampere, A:

1 A = 1 C/s.

25-2 Electric Current

The instantaneous current is given by:


A complete circuit is one where current can flow all the way around. Note that the schematic drawing doesn’t look much like the physical circuit!




Example 25-1: Current is flow of charge.

A steady current of 2.5 A exists in a wire for 4.0 min. (a) How much total charge passed by a given point in the circuit during those 4.0 min? (b) How many electrons would this be?

ConcepTest 25.1ConcepTest 25.1 Connect the Battery Connect the Battery

Which is the correct way to Which is the correct way to

light the lightbulb with the light the lightbulb with the

battery?battery?

4) all are correct

5) none are correct

1) 3)2)

Current can flow only if there is a continuous connectioncontinuous connection from

the negative terminal through the bulb to the positive terminal.

This is the case for only Fig. (3).

ConcepTest 25.1ConcepTest 25.1 Connect the Battery Connect the Battery

Which is the correct way to Which is the correct way to

light the lightbulb with the light the lightbulb with the

battery?battery?

4) all are correct

5) none are correct

1) 3)2)


By convention, current is defined as flowing from + to -. Electrons actually flow in the opposite direction, but not all currents consist of electrons.



Experimentally, it is found that the current in a wire is proportional to the potential difference between its ends:

25-3 Ohm’s Law: Resistance and Resistors


The ratio of voltage to current is called the resistance:



In many conductors, the resistance is independent of the voltage; this relationship is called Ohm’s law. Materials that do not follow Ohm’s law are called nonohmic.

Unit of resistance: the ohm, Ω:

1 Ω = 1 V/A.




Conceptual Example 25-3: Current and potential.

Current I enters a resistor R as shown. (a) Is the potential higher at point A or at point B? (b) Is the current greater at point A or at point B?



Example 25-4: Flashlight bulb resistance.

A small flashlight bulb draws 300 mA from its 1.5-V battery. (a) What is the resistance of the bulb? (b) If the battery becomes weak and the voltage drops to 1.2 V, how would the current change?


Some clarifications:

• Batteries maintain a (nearly) constant potential difference; the current varies.

• Resistance is a property of a material or device.

• Current is not a vector but it does have a direction.

• Current and charge do not get used up. Whatever charge goes in one end of a circuit comes out the other end.


1) Ohm’s law is obeyed since the Ohm’s law is obeyed since the current still increases when current still increases when VV increasesincreases

2) Ohm’s law is not obeyedOhm’s law is not obeyed

3) this has nothing to do with Ohm’s this has nothing to do with Ohm’s lawlaw

ConcepTest 25.2ConcepTest 25.2 Ohm’s Law Ohm’s Law

You double the You double the voltagevoltage across across

a certain conductor and you a certain conductor and you

observe the observe the currentcurrent increases increases

three times. What can you three times. What can you

conclude?conclude?

1) Ohm’s law is obeyed since the Ohm’s law is obeyed since the current still increases when current still increases when VV increasesincreases

2) Ohm’s law is not obeyedOhm’s law is not obeyed

3) this has nothing to do with Ohm’s this has nothing to do with Ohm’s lawlaw

Ohm’s law, V = IRV = IR, states that the

relationship between voltage and

current is linearlinear. Thus, for a conductor

that obeys Ohm’s law, the current must

double when you double the voltage.

ConcepTest 25.2ConcepTest 25.2 Ohm’s Law Ohm’s Law

You double the You double the voltagevoltage across across

a certain conductor and you a certain conductor and you

observe the observe the currentcurrent increases increases

three times. What can you three times. What can you

conclude?conclude?

Follow-up:Follow-up: Where could this situation occur? Where could this situation occur?


The resistance of a wire is directly proportional to its length and inversely proportional to its cross-sectional area:

The constant ρ, the resistivity, is characteristic of the material.

25-4 Resistivity


25-4 ResistivityThis table gives the resistivity and temperature coefficients of typical conductors, semiconductors, and insulators.


25-4 ResistivityExample 25-5: Speaker wires.

Suppose you want to connect your stereo to remote speakers. (a) If each wire must be 20 m long, what diameter copper wire should you use to keep the resistance less than 0.10 Ω per wire? (b) If the current to each speaker is 4.0 A, what is the potential difference, or voltage drop, across each wire?


For any given material, the resistivity increases with temperature:

Semiconductors are complex materials, and may have resistivities that decrease with temperature.

25-4 Resistivity


25-4 ResistivityExample 25-7: Resistance thermometer.

The variation in electrical resistance with temperature can be used to make precise temperature measurements. Platinum is commonly used since it is relatively free from corrosive effects and has a high melting point. Suppose at 20.0°C the resistance of a platinum resistance thermometer is 164.2 Ω. When placed in a particular solution, the resistance is 187.4 Ω. What is the temperature of this solution?

ConcepTest 25.3aConcepTest 25.3a Wires IWires I

Two wires, Two wires, AA and and BB, are made of the , are made of the

same metalsame metal and have and have equal lengthequal length, ,

but the resistance of wire but the resistance of wire AA is is four four

timestimes the resistance of wire the resistance of wire BB. How . How

do their diameters compare?do their diameters compare?

1) ddAA = 4 = 4ddBB

2) ddAA = 2 = 2ddBB

3) ddAA = = ddBB

4) 4) ddAA = 1/2 = 1/2ddBB

5) 5) ddAA = 1/4 = 1/4ddBB

The resistance of wire A is greater because its area is lessarea is less than

wire B. Since areaarea is related to radiusradius (or diameter) squaredsquared, the

diameter of A must be two times less than the diameter of Bdiameter of A must be two times less than the diameter of B.

ConcepTest 25.3aConcepTest 25.3a Wires IWires I

Two wires, Two wires, AA and and BB, are made of the , are made of the

same metalsame metal and have and have equal lengthequal length, ,

but the resistance of wire but the resistance of wire AA is is four four

timestimes the resistance of wire the resistance of wire BB. How . How

do their diameters compare?do their diameters compare?

1) ddAA = 4 = 4ddBB

2) ddAA = 2 = 2ddBB

3) ddAA = = ddBB

4) 4) ddAA = 1/2 = 1/2ddBB

5) 5) ddAA = 1/4 = 1/4ddBB

RA


Power, as in kinematics, is the energy transformed by a device per unit time:

25-5 Electric Power

or


The unit of power is the watt, W.

For ohmic devices, we can make the substitutions:

25-5 Electric Power


25-5 Electric Power

Example 25-8: Headlights.

Calculate the resistance of a 40-W automobile headlight designed for 12 V.


What you pay for on your electric bill is not power, but energy – the power consumption multiplied by the time.

We have been measuring energy in joules, but the electric company measures it in kilowatt-hours, kWh:

1 kWh = (1000 W)(3600 s) = 3.60 x 106 J.

25-5 Electric Power


25-5 Electric Power

Example 25-9: Electric heater.

An electric heater draws a steady 15.0 A on a 120-V line. How much power does it require and how much does it cost per month (30 days) if it operates 3.0 h per day and the electric company charges 9.2 cents per kWh?


25-6 Power in Household Circuits

Conceptual Example 25-12: A dangerous extension cord.

Your 1800-W portable electric heater is too far from your desk to warm your feet. Its cord is too short, so you plug it into an extension cord rated at 11 A. Why is this dangerous?


Current from a battery flows steadily in one direction (direct current, DC). Current from a power plant varies sinusoidally (alternating current, AC).

25-7 Alternating Current


The voltage varies sinusoidally with time:

as does the current:


,,


Multiplying the current and the voltage gives the power:



Usually we are interested in the average power:


.


The current and voltage both have average values of zero, so we square them, take the average, then take the square root, yielding the root-mean-square (rms) value:



25-7 Alternating CurrentExample 25-13: Hair dryer.

(a) Calculate the resistance and the peak current in a 1000-W hair dryer connected to a 120-V line. (b) What happens if it is connected to a 240-V line in Britain?


Electrons in a conductor have large, random speeds just due to their temperature. When a potential difference is applied, the electrons also acquire an average drift velocity, which is generally considerably smaller than the thermal velocity.

25-8 Microscopic View of Electric Current: Current Density and Drift

Velocity



Velocity

We define the current density (current per unit area) – this is a convenient concept for relating the microscopic motions of electrons to the macroscopic current:

If the current is not uniform:

.


This drift speed is related to the current in the wire, and also to the number of electrons per unit volume:


Velocity

and



Velocity

Example 25-14: Electron speeds in a wire.

A copper wire 3.2 mm in diameter carries a 5.0-A current. Determine (a) the current density in the wire, and (b) the drift velocity of the free electrons. (c) Estimate the rms speed of electrons assuming they behave like an ideal gas at 20°C. Assume that one electron per Cu atom is free to move (the others remain bound to the atom).



Velocity

The electric field inside a current-carrying wire can be found from the relationship between the current, voltage, and resistance. Writing R = ρ l/A, I = jA, and V = El , and substituting in Ohm’s law gives:



VelocityExample 25-15: Electric field inside a wire.

What is the electric field inside the wire of Example 25–14? (The current density was found to be 6.2 x 105 A/m2.)


In general, resistivity decreases as temperature decreases. Some materials, however, have resistivity that falls abruptly to zero at a very low temperature, called the critical temperature, TC.

25-9 Superconductivity


Experiments have shown that currents, once started, can flow through these materials for years without decreasing even without a potential difference.

Critical temperatures are low; for many years no material was found to be superconducting above 23 K.

Since 1987, new materials have been found that are superconducting below 90 K, and work on higher temperature superconductors is continuing.

25-9 Superconductivity


• A battery is a source of constant potential difference.

• Electric current is the rate of flow of electric charge.

• Conventional current is in the direction that positive charge would flow.

• Resistance is the ratio of voltage to current:



• Ohmic materials have constant resistance, independent of voltage.

• Resistance is determined by shape and material:

• ρ is the resistivity.



• Power in an electric circuit:

• Direct current is constant.

• Alternating current varies sinusoidally:



• The average (rms) current and voltage:

• Relation between drift speed and current:



Homework Assignment # 5

Chapter 24 – 60, 82

Chapter 25 – 10, 20, 40, 54, 58

Tentative HW # 6:

Chapter 26 –

chapter 24 capacitance, dielectrics, electric energy storage

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