غريبة أيامي
18-03-2012, 15:39
السلام عليكم
ممكن ترجمة هذه الجزئية من كتاب ضروري...
15.1 Introduction
One of Newton's great achievements was to show that all of the phenomena of classical mechanics
can be deduced as consequences of three basic, fundamental laws, namely Newton's laws of
motion. It was likewise one of Maxwell's great achievements to show that all of the phenomena of
classical electricity and magnetism – all of the phenomena discovered by Oersted, Ampère, Henry,
Faraday and others whose names are commemorated in several electrical units – can be deduced as
consequences of four basic, fundamental equations. We describe these four equations in this
chapter, and, in passing, we also mention Poisson's and Laplace's equations. We also show how
Maxwell's equations predict the existence of electromagnetic waves that travel at a speed of 3 % 108
m s−1. This is the speed at which light is measured to move, and one of the most important bases of
our belief that light is an electromagnetic wave
Before embarking upon this, we may need a reminder of two mathematical theorems, as well as a
reminder of the differential equation that describes wave motion.
The two mathematical theorems that we need to remind ourselves of are:
The surface integral of a vector field over a closed surface is equal to the volume integral of its
divergence.
The line integral of a vector field around a closed plane curve is equal to the surface integral of its
curl.
A function f (x − vt)represents a function that is moving with speed v in the positive x-direction,
and a function g(x + vt) represents a function that is moving with speed v in the negative xdirection.
It is easy to verify by substitution that y = Af +Bg is a solution of the differential
equation
.
2
2
2
2
2
dx
d y
dt
d y = v 15.1.1
Indeed it is the most general solution, since f and g are quite general functions, and the function y
already contains the only two arbitrary integration constants to be expected from a second order
differential equation. Equation 15.1.1 is, then, the differential equation for a wave in one
dimension. For a function ψ(x, y, z) in three dimensions, the corresponding wave equation is
ψ&& = v 2∇2ψ. 15.1.2
It is easy to remember which side of the equation v2 is on from dimensional considerations.
One last small point before proceeding – I may be running out of symbols! I may need to refer to
surface charge density, a scalar quantity for which the usual symbol is σ. I shall also need to refer
to magnetic vector potential, for which the usual symbol is A. And I shall need to refer to area, for
which either of the symbols A or σ are commonly used – or, if the vector nature of area is to be
emphasized, A or σ. What I shall try to do, then, to avoid this difficulty, is to use A for magnetic
vector potential, and σ for area, and I shall try to avoid using surface charge density in any
equation. However, the reader is warned to be on the lookout and to be sure what each symbol
means in a particular context.
ممكن ترجمة هذه الجزئية من كتاب ضروري...
15.1 Introduction
One of Newton's great achievements was to show that all of the phenomena of classical mechanics
can be deduced as consequences of three basic, fundamental laws, namely Newton's laws of
motion. It was likewise one of Maxwell's great achievements to show that all of the phenomena of
classical electricity and magnetism – all of the phenomena discovered by Oersted, Ampère, Henry,
Faraday and others whose names are commemorated in several electrical units – can be deduced as
consequences of four basic, fundamental equations. We describe these four equations in this
chapter, and, in passing, we also mention Poisson's and Laplace's equations. We also show how
Maxwell's equations predict the existence of electromagnetic waves that travel at a speed of 3 % 108
m s−1. This is the speed at which light is measured to move, and one of the most important bases of
our belief that light is an electromagnetic wave
Before embarking upon this, we may need a reminder of two mathematical theorems, as well as a
reminder of the differential equation that describes wave motion.
The two mathematical theorems that we need to remind ourselves of are:
The surface integral of a vector field over a closed surface is equal to the volume integral of its
divergence.
The line integral of a vector field around a closed plane curve is equal to the surface integral of its
curl.
A function f (x − vt)represents a function that is moving with speed v in the positive x-direction,
and a function g(x + vt) represents a function that is moving with speed v in the negative xdirection.
It is easy to verify by substitution that y = Af +Bg is a solution of the differential
equation
.
2
2
2
2
2
dx
d y
dt
d y = v 15.1.1
Indeed it is the most general solution, since f and g are quite general functions, and the function y
already contains the only two arbitrary integration constants to be expected from a second order
differential equation. Equation 15.1.1 is, then, the differential equation for a wave in one
dimension. For a function ψ(x, y, z) in three dimensions, the corresponding wave equation is
ψ&& = v 2∇2ψ. 15.1.2
It is easy to remember which side of the equation v2 is on from dimensional considerations.
One last small point before proceeding – I may be running out of symbols! I may need to refer to
surface charge density, a scalar quantity for which the usual symbol is σ. I shall also need to refer
to magnetic vector potential, for which the usual symbol is A. And I shall need to refer to area, for
which either of the symbols A or σ are commonly used – or, if the vector nature of area is to be
emphasized, A or σ. What I shall try to do, then, to avoid this difficulty, is to use A for magnetic
vector potential, and σ for area, and I shall try to avoid using surface charge density in any
equation. However, the reader is warned to be on the lookout and to be sure what each symbol
means in a particular context.