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1s ñ =Ỉ 2: similarly, the voltage across 2; 1s llổ =Ï VN The same current goes
through both. Thherefore the total voltage is the sum of the voltages across the
two sections and is equal to Ÿ= ñ + Ññ = (2¡ + 2,)Ï. This means that the
voltage on the complete circuit can be written Ÿ=Í 2.. where the VÀ of the
combined system in series is the sum of the two 2s of the sepDarate pieces:
2, = 2¡+ôa. (25.16)
This is not the only way things may be connected. We may aÌso connect
them in another way, called a parailel connection (Fig. 25-6b). Now we see that a
given voltage across the terminals, if the connecting wires are perfect conductfors,
1s efectively applied to both of the impedances, and will cause currents in each
independently. Therefore the current through Ñ¡ is cqual to ñ = / 2¡. The
current in 2, 1s TP = Ÿ/2¿. Tt is the sœme 0oltage. Now the total current
which is supplied to the terminals is the sưzn of the currents in the two sections:
? =Ÿ/ô\ +Ÿ/2;¿. Thịs can be written as
(1/22)+ (1/22)
1/2„ = 1/2¡ + 1/2. (25.17)
More complicated circuits can sometimes be simplified by taking pieces of
them, working out the succession of Impedances of the pieces, and combining
the circuit together step by step, using the above rules. If we have any kind of
circuit with many impedances connected ín all kinds of ways, and if we include
the voltages in the form of little generators having no impedance (when we pass
charge through it, the generator adds a voltage WV}), then the following principles
apply: (1) At% any junction, the sum oŸ the currents into a junction is zero.
That is, all the current which comes in must come back out. (2) IÝ we carry a
charge around any loop, and back to where it started, the net work done is zero.
These rules are called zchhoff s laas for electrical circuits. Theïr systematic
application to complicated circuits often simplifies the analysis of such circuits.
We mention them here in conjunction with Eqs. (25.16) and (25.17), in case you
have already come across such circuits that you need to analyze in laboratory
work. They will be discussed again in more detail next year.
--- Trang 452 ---
€)pfics: To EPrirtcfpÏlo oŸ Loáist Time©
26-1 Light
This is the fñrst of a number of chapters on the subject of electromagnetic
radiation. Light, with which we see, is only one small part of a vast spectrum of
the same kind of thing, the various parts of this spectrum being distinguished by
diferent values oŸ a certain quantity which varies. 'Phis variable quantity could be
called the “wavelength” As it varies in the visible spectrum, the light apparently
changes color from red to violet. If we explore the spectrum systematically, from
long wavelengths toward shorter ones, we would begin with what are usually called
radiotues. Radiowaves are technically available in a wide range of wavelengths,
some even longer than those used in regular broadcasts; regular broadcasts have
wavelengths corresponding to about 500 meters. 'Phen there are the so-called
“short waves,” i.e., radar waves, millimeter waves, and so on. There are no actual
boundaries between one range of wavelengths and another, because nature did
not present us with sharp edges. The number associated with a given name for
the waves are only approximate and, of course, so are the names we give to the
diferent ranges.
Then, a long way down through the millimeter waves, we come to what
we call the ?mƒrared, and thence to the visible spectrum. Then going in the
other direction, we get into a region which is called the ui#rœoolet. Where the
ultraviolet stops, the x-rays begin, but we cannot defne precisely where this
is; it is roughly at 10” m, or 1072 ø. These are “soft” x-rays; then there are
ordinary x-rays and very hard x-rays; then +-rays, and so on, for smaller and
smaller values of this dimension called the wavelength.
Within this vast range of wavelengths, there are three or more regions of
approximation which are especially interesting. In one of these, a condition exists
in which the wavelengths involved are very small compared with the dimensions
of the equipment available for their study; furthermore, the phobon energies,
--- Trang 453 ---
using the quantum theory, are small compared with the energy sensitivity of the
equipment. nder these conditions we can make a rough frst approximation
by a method called geometrical opfics. TỶ, on the other hand, the wavelengths
are comparable to the dimensions of the equipment, which is difficult to arrange
with visible light but easier with radiowaves, and ïf the photon energies are still
negligibly small, then a very useful approximation can be made by studying the
behavior of the waves, still disregarding the quantum mechanics. This method is
based on the classical theor oƒ electromagnetic radiation, which will be discussed
in a later chapter. Next, If we go to very short wavelengths, where we can
disregard the wave character but the photons have a very Íarge energy compared
with the sensitivity of our equipment, things get simple again. 'This ¡is the simple
photon picture, which we will describe only very roughly. The complete picture,
which unifies the whole thing into one model, will not be available to us for a
long time.
In this chapter our discussion is limited to the geometrical optics region, in
which we forget about the wavelength and the photon character of the lght,
which will all be explained in due time. We do not even bother to say what
the light zs, but just fñnd out ho ?£ behœues on a large scale compared with
the dimensions of interest. All this must be said in order to emphasize the fact
that what we are going to talk about is only a very crude approximation; this
is one of the chapters that we shall have to “unlearn” again. But we shall very
quickly unlearn it, because we shall almost immediately go on to a more accurate
mnethod.
Although geometrical optics is just an approximation, it is of very great
importance technically and of great interest historically. We shall present this
subject more historically than some of the others in order to give some idea. of
the development oŸ a physical theory or physical idea.
tirst, light is, of course, familiar to everybody, and has been familiar since
time mmemorial. NÑow one problem is, by what process do we see light? There
have been many theories, but it finally settled down to one, which is that there
1s something which enters the eye—which bounces of objJects into the eye. We
have heard that idea so long that we accept it, and it is almost impossible for
us to realize that very intelligent men have proposed contrary theories—that
something comes out of the eye and feels for the obJect, for example. Some other
Important observations are that, as light goes from one place to another, it goes
in sứraight lines, 1Ÿ there 1s nothing in the way, and that the rays do not seem
to interfere with one another. hat is, light is crisscrossing in all directions in
--- Trang 454 ---
the room, but the light that is passing across our line of vision does not affect
the light that comes to us from some object. 'This was once a most powerful