麻省理工的电磁场讲义MIT8_02SC_notes1.pdf

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Module 1: Fields

Table of Contents

1.1

Action at a Distance versus Field Theory ................................................................ 1-2

1.2 Scalar Fields ......................................................................................................... 1-3

Example 1.1: Half-Frozen /Half-Baked Planet ........................................................ 1-5

1.2.1 Representations of a Scalar Field................................................................ 1-5

1.3 Vector Fields........................................................................................................ 1-7

1.4 Fluid Flow............................................................................................................ 1-8

1.4.1 Sources and Sinks (link) ............................................................................. 1-8

1.4.2 Circulations (link) ..................................................................................... 1-11

1.4.3 Relationship between Fluid Fields and Electromagnetic Fields ............... 1-13

1.5 Gravitational Field ............................................................................................. 1-13

1.6 Electric Fields .................................................................................................... 1-14

1.7 Magnetic Field ................................................................................................... 1-15

1.8 Representations of a Vector Field...................................................................... 1-16

1.8.1 Vector Field Representation ..................................................................... 1-17

1.8.2 Field Line Representation ......................................................................... 1-18

1.8.3 Grass Seeds and Iron Filings Representations.......................................... 1-18

1.8.4 Motion of Electric and Magnetic Field Lines ........................................... 1-19

1.9 Summary ............................................................................................................ 1-20

1.10 Solved Problems ................................................................................................ 1-20

1.10.1 Vector Fields............................................................................................. 1-20

1.10.2 Scalar Fields .............................................................................................. 1-22

These notes are excerpted “Introduction to Electricity and Magnetism” by Sen-Ben Liao, Peter

1-1

Fields

1.1 Action at a Distance versus Field Theory

“… In order therefore to appreciate the requirements of the science [of

electromagnetism], the student must make himself familiar with a

considerable body of most intricate mathematics, the mere retention of

which in the memory materially interferes with further progress …”

James Clerk Maxwell [1855]

Classical electromagnetic field theory emerged in more or less complete form in 1873 in

James Clerk Maxwell’s A Treatise on Electricity and Magnetism. Maxwell based his

theory in large part on the intuitive insights of Michael Faraday. The wide acceptance of

Maxwell’s theory has caused a fundamental shift in our understanding of physical reality.

In this theory, electromagnetic fields are the mediators of the interaction between

material objects. This view differs radically from the older “action at a distance” view

that preceded field theory.

What is “action at a distance?” It is a worldview in which the interaction of two material

objects requires no mechanism other than the objects themselves and the empty space

between them. That is, two objects exert a force on each other simply because they are

present. Any mutual force between them (for example, gravitational attraction or electric

repulsion) is instantaneously transmitted from one object to the other through empty

space. There is no need to take into account any method or agent of transmission of that

force, or any finite speed for the propagation of that agent of transmission. This is known

as “action at a distance” because objects exert forces on one another (“action”) with

nothing but empty space (“distance”) between them. No other agent or mechanism is

needed.

Many natural philosophers objected to the “action at a distance” model because in our

everyday experience, forces are exerted by one object on another only when the objects

are in direct contact. In the field theory view, this is always true in some sense. That is,

objects that are not in direct contact (objects separated by apparently empty space) must

exert a force on one another through the presence of an intervening medium or

mechanism existing in the space between the objects.

The force between the two objects is transmitted by direct “contact” from the first object

to an intervening mechanism immediately surrounding that object, and then from one

element of space to a neighboring element, in a continuous manner, until the force is

transmitted to the region of space contiguous to the second object, and thus ultimately to

the second object itself.

1-2

Although the two objects are not in direct contact with one another, they are in direct

contact with a medium or mechanism that exists between them. The force between the

objects is transmitted (at a finite speed) by stresses induced in the intervening space by

the presence of the objects. The “field theory” view thus avoids the concept of “action at

a distance” and replaces it by the concept of “action by continuous contact.” The

“contact” is provided by a stress, or “field,” induced in the space between the objects by

their presence.

This is the essence of field theory, and is the foundation of all modern approaches to

understanding the world around us. Classical electromagnetism was the first field theory.

It involves many concepts that are mathematically complex. As a result, even now it is

difficult to appreciate. In this first chapter of your introduction to field theory, we discuss

what a field is, and how we represent fields. We begin with scalar fields.

1.2 Scalar Fields

A scalar field is a function that gives us a single value of some variable for every point in

space. As an example, the image in Figure 1.2.1 shows the nighttime temperatures

measured by the Thermal Emission Spectrometer instrument on the Mars Global

Surveyor (MGS). The data were acquired during the first 500 orbits of the MGS mapping

mission. The coldest temperatures, shown in purple, are −120° C while the warmest,

shown in white, are

−65°

The view is centered on Isidis Planitia (15N, 270W), which is covered with warm

material, indicating a sandy and rocky surface. The small, cold (blue) circular region to

the right is the area of the Elysium volcanoes, which are covered in dust that cools off

rapidly at night. At this season the north polar region is in full sunlight and is relatively

warm at night. It is winter in the southern hemisphere and the temperatures are extremely

low.

Figure 1.2.1 Nighttime temperature map for Mars

The various colors on the map represent the surface temperature. This map, however, is

limited to representing only the temperature on a two-dimensional surface and thus, it

does not show how temperature varies as a function of altitude. In principal, a scalar

1-3

Example 1.1: Half-Frozen /Half-Baked Planet

As an example of a scalar field, consider a planet with an atmosphere that rotates with the

same angular frequency about its axis as the planet orbits about a nearby star, i.e., one

hemisphere always faces the star. Let

denote the radius of the planet. Use spherical

coordinates

(

) with the origin at the center of the planet, and choose

2 for the

center of the hemisphere facing the star. A simplistic model for the temperature variation

at any point is given by

θφ

⎡

− )

(1.2.1)

(, , )

⎣

sin

)

⎤

⎦

−

(rR

where

, T

, and

are constants. The dependence on the variable r in the term

−

(rR− )

indicates that the temperature decreases exponentially as we move radially away

from the surface of the planet. The dependence on the variable

in the term

sin

implies that the temperature decreases as we move toward the poles. Finally, the

dependence in the term

(1 + sin

)

indicates that the temperature decreases as we move

away from the center of the hemisphere facing the star.

A scalar field can also be used to describe other physical quantities such as the

atmospheric pressure. However, a single number (magnitude) at every point in space is

not sufficient to characterize quantities such as the wind velocity since a direction at

every point in space is needed as well.

1.2.1 Representations of a Scalar Field

A field, as stated earlier, is a function that has a different value at every point in space. A

scalar field is a field for which there is a single number associated with every point in

space. We have seen that the temperature of the Earth’s atmosphere at the surface is an

example of a scalar field. Another example is

(,xyz , )

−

1/ 3

(1.2.2)

(

)

(

)

−

This expression defines the value of the scalar function

at every point

yz, )

in space.

How do visually represent a scalar field defined by an equation such as Eq. (1.2.2)?

Below we discuss three possible representations.

1. Contour Maps

One way is to fix one of our independent variables (

z, for example) and then show a

contour map for the two remaining dimensions, in which the curves represent lines of

constant values of the function

. A series of these maps for various (fixed) values of z

1-5

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