Tài liệu Chapter IX Conductors, Capacitors ppt

Charges and electric field on conductors: 1.1 The balance of charges on conductors: In conductors there are charged particles which can be freely move under any small force.. Therefore

GENERAL PHYSICS II

Electromagnetism

7

Thermal Physics

Chapter IX

Conductors, Capacitors

§1 Charges and electric field on conductors

§2 Capacitance of conductors and capacitors

§3 Energy storage in capacitors and electric field energy

§4 Electric current, resistance and electromotive force

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§1 Charges and electric field on conductors:

1.1 The balance of charges on conductors:

In conductors there are charged particles which can be freely move under any small force Therefore the balance of charges on conductors can be observed under these circumstances:

@ Ihe electric field equals zero everywhere inside the conductor

= The surface of conductors is equipotential

Inside conductors there is no charge This conclusion can be proved

by applying the Gauss’s law for any arbitrary closed surface inside conductor All the charge is distributed on the surface of conductors

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Since the distribution of charge on conductors does not depend on the distribution of the

matter = the distribution of charges is the same for hollow and solid conductors

The fact that the distribution of charges only

on the surface of conductors can be understood as follows:

Suppose that we provide the conductor with

an amount of charges ™ charges repulse

mutually and tend to leave as far as possible each from other

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1.2 The electric field at the surface of

a conductor:

Consider the red small cylindrical surface

with the base dS Applying the Gauss’s law

for this closed surface we have

The electric field in vacuum

near the surface

(O is the surface charge density

at the considered point on the surface)

5lectric field in dielectric iment near the surface

+ Near convexes of the surface:

the equipotential surfaces are dense = E ¡s large

= the charge density is large

+ Near deepenings of the surface:

the equipotential surfaces are rare = E is smaller

=) the charge density is small

§2 Capacitance of conductors and capacitors:

charged Metal sphere rod

A charge body (rod) can give another body a charge of

opposite sign, without losing Tsulating

any of its own charge stand ~~

on sphere rearrange themselves

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2.2 Capacitors and capacitance:

A capacitor is a device whose purpose is to store electrical energy which can then be released in a controlled manner during a short period of time

A capacitor consists of 2 spatially separated conductors which can be

charged to +Q and -Q respectively

Definition: The capacitance of the capacitor is the ratio of the charge on one conductor of the capacitor to the potential difference between the

$ Example 1: Parallel Plate Capacitor

¢« Calculate the capacitance We assume +0, -o Charge densities on each plate with potential difference V:

Need V: from definition:

@ Use Gauss Law to find E (in next slide)

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Recall the formula for the electric field of two infinite sheets:

¢ Field outside the sheets is zero

Gaussian surface encloses zero net

¢ Field inside sheets is not zero:

¢ Gaussian surface encloses non-zero

qE x ds q TƯ: =

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¢ The capacitance of this capacitor depends only on its shape and size

¢ This formula is true for parallel-plate capacitor (shape), and C depends

on A, d (size) (for another shape one has other formula)

¢ When the space between the metal plates is filled with a dielectric material, the capacitance increases by a factor k (see the previous chapter)

(Recall: k - dielectric constant; € - permitivity of the dielectric)

In order to increase C =) om (limitatively), and one must increase A, €

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$ Example 2: Cylindrical Capacitor

¢ Calculate the capacitance:

¢« Assume +Q, -Q on surface of cylinders with potential difference V

¢ Gaussian surface is cylinder of radius; (a

2.4 Capacitors in Series:

ee

iain TIỆ

ome me eC

Find “equivalent” capacitance C in the sense that no measurement at

a, b could distinguish the above two situations

The charge on C, must be the same as the charge on C, since applying a potential difference across ab cannot produce a net

charge on the inner plates of C, and C,

—> assume there is no net charge on node between C, and C,

$ Examples: Combinations of Capacitors

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§3.Energy of a Capacitor and electric field energy:

3.1 Energy of a charged capacitor:

¢ How much energy is stored in a charged capacitor?

— Calculate the work provided (usually by a battery) to charge a Capacitor to +/- Q:

Calculate incremental work dW needed to add charge dq to capacitor at

voltage V (there is a trick here!): ee

W=— |adq =——>r~_ Look at this!

oF 2C là ways to write W

3.2 Energy storage in capacitors:

Where is the Energy Stored?

Energy is stored in the electric field itself Think of the energy needed

to charge the capacitor as being the energy needed to create the field

To calculate the energy density in the field, first consider the constant field generated by a parallel plate capacitor, where

¬ j Q

ears Donan Ọ ¬_ 2 DEN

This is the energy

%# Note 1: The expression for the energy density of the electrostatic

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$4 Electric current, resistance and electromotive force

4.1 The definition of current and current density:

_ Charges, e.g free electrons, exists in conductors with a density, n„ (nạ approx 1023 m3)

“Somehow” put that charge in motion:

effective picture all charge moves with a velocity, v,

real picture a lot of “random motion” of charges with a small average

equal to v,

We need a quantity which can characterize flows of moving charges

=) Definition of current: The rate at which charge flows

Phe Definition of Current

In metal wires, the electrons are the carriers

of charge We have the following equations:

cit ei) s{) |

trons which pass through the cross-section A during f)

eure section 21091: Đa

nis free electron density

The formula gives relation between

J a current density, electron density and

velocity of electrons 20

| H

Consider a simple circuit (picture)

We can calculate the acceleration

* The units of resistance are

* From the formula of the resistance we see that the resistance depends

On the length /, the cross-section A of the wire, and the resistivity Ð

* The resistivity is dependent on the miroscopic properties of conducting

material It is dificult to determine T, n in order to calculate p

= The resistivity is usually measured by experiment, it’s values are given in tables

% The resistivity of conductors depends on temperature For almost of metals the temperature dependence of resistivity is as follows

P = Po (1 + af’),

where temperature t is in the Celsius scale, Og — the resistivity at 0° C,

O = 1/273

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4.3 Energy transfer in electric circuits:

From the formula for the current we can calculate the amount of charge

dQ passes through the conductor in a time interval at: dQ = [.dt

The change of potential energy for this amount of charge (dQ) is

V.d@ = V.I.dt The rate at which energy is transferred into (or out) the conductor (the amount of energy which is transferred in a time unit) is power

4.3 Electromotive force:

lf between two ends of a conductor there is initially a difference of electric potential = the electric field E# 0 = (positively) charged particles move from the higher potential end to the lower (and negative charges move in the opposite direction) After a very short time the

electric field disappears = the current will stop shortly

In order to keep up the current during a long time one must continuously take (pos.) charges from the lower potential end and provide charges to the higher potential end ==every circuit with a Steady current must include some devices performing this task (battery, generator, )

F,: the electrostatic force 0;>0›

Fn: must be non-electrostatic ¥; #2

force For example: cf

¢ force of chemical nature

in batteries

¢ force gven rise by rotating

2/2p/2oosagnetic field in generators

Every battery or generator is specified by the quantity called electromotive force (emf ) &

Definition of emf & : the work done by non-electrostatic forces on the charge +1 unit over all the closed circuit:

From the definition ™ emf 6 has the same unit as the electric potential

Denote by f,, the non-electrostatic force acting on q:

=> Ent is the circular (closed path integral) of the non-electrostatic electric

field

Batteries or generators are called sources of emf

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One can define the electromotive force acting on a section †1—›2 of circuIt:

The work done by both electrostatic and non-electrostatic forces on this section:

This is the Ohm’s law for a section of circuit:

- In the absent of non-electrostatic forces: F.Í = (01 - 02)

¢ For a closed circuit (the point 1 coincides with 2)

2/20/2008 It is the Ohm’s law for a closed circuit a

mm Denote by r the inner resistance of the

source we have the Ohm’s law for closed circuits in the form:

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