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of a d they simply make a “backwards 6,” or Ø. (A Ø should have been used
in the beginning of calculus because we always want to cancel that đ, but we
never want to cancel a Øl) So they write ØỮ/Øz, and furthermore, in moments oŸ
duress, if they want to be øerw careful, they put a line beside it with a little z
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at the bottom (ØU/Øz|y;), which means “Take the derivative of U with respect
to #, keeping and z constant.” Most often we leave out the remark about what
1s kept constant because it is usually evident from the context, so we usually do
not use the line with the and z. However, øas use a Ø instead of a d as a
warning that it is a derivative with some other variables kept constant. 'This is
called a partial derduatiue; ïW 1s a derivative in which we vary only z.
Therefore, we find that the force in the z-direction is minus the partial
derivative of Ư with respect to #:
t„ = —ØU/Ôz. (14.11)
In a similar way, the force in the -direction can be found by diferentiating U
with respect to ø, keeping z and z constant, and the third component, of course,
is the derivative with respect to z, keeping and z constant:
tụ = —8U/0, ty = —8U/Ôz. (14.12)
This 1s the way to get from the potential energy to the force. We get the fcld
from the po#ential in exactly the same way:
Œ„ = —8/Ôz, Œy = —Ø9/Ô, Œ, =—8/Ôz. (14.13)
Incidentally, we shall mention here another notation, which we shall not
actually use for quite a while: Since Πis a vector and has z-, -, and z-components,
the symbolized Ø/Øz, Ø/Øụ, and Ø/9z which produee the ø-, -, and z-components
are something like vectors. 'Phe mathematicians have invented a glorious new
symbol, V, called “grad” or “gradient”, which is not a quantity but an operator
that makes a vector from a scalar. It has the following “components”: 'Phe ø-
component of this “grad” is Ø/Øz the -component is Ø/Øy, and the z-component
is Ø/Øz, and then we have the fun of writing our formulas this way:
t'=_—-YNU, Œ =-VỪ. (14.14)
Using V gives us a quick way of testing whether we have a real vector equation
or not, but actually Eqs. (14.14) mean precisely the same as Eqs. (14.11), (14.12)
and (14.13); it is just another way of writing them, and since we do not want to
write three equations every time, we just write VŨ instead.
One more example of fields and potentials has to do with the electrical case.
In the case of electricity the force on a stationary object is the charge times the
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electric fñeld: ' = g#. (In general, of course, the #-component of foree in an
electrical problem has also a part which depends on the magnetic field. It is easy
to show from Eq. (12.11) that the force on a particle due to magnetic fields is
always at right angles to its velocity, and also at right angles to the ñeld. 5ince
the force due to magnetism on a moving charge is at right angles to the velocity,
no t0ork is done by the magnetism on the moving charge because the motion is at
right angles to the force. Therefore, in calculating theorems of kinetic energy in
electric and magnetic ñelds we can disregard the contribution tom the magnetic
fñeld, since it does not change the kinetic energy.) We suppose that there is only
an electric ñeld. Then we can calculate the energy, or work done, in the same way
as for gravity, and calculate a quantity ó which is minus the integral of # - ds,
from the arbitrary ñxed point to the point where we make the calculation, and
then the potential energy in an electric ñeld is just charge times this quantity ¿:
ðf)== [ Eds
U = qọ.
Let us take, as an example, the case of two parallel metal plates, each with a
surface charge of +øơ per unit area. 'This is called a parallel-plate capacitor. We
found previously that there is zero force outside the plates and that there is a
constant electric field between them, directed from + to — and of magnitude ø/eg
(Fig. 14-5). We would like to know how much work would be done in carrying a
charge from one plate 6o the other. The work would be the (force) - (4s) integral,
which can be written as charge times the potential value at plate 1 minus that at
plate 2:
wr= | đ - ds = q(01 — 92).
W© can actually work out the integral because the force is constant, and 1Ÿ we
Tin non:
1/111) ị
TNnnnnnn"
Fig. 14-5. Field between parallel plates.
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call the separation of the plates đ, then the integral is easy:
l E.ds= =mỊ dự = T”,
1 €0 J1 €0
The diference in potential, Aø = ơd/(cọ, is called the 0olfage đifJerence, and ở is
measured in volts. When we say a pair of plates is charged to a certain voltage,
what we mean is that the diference in electrical potential of the bwo plates is
So-and-so many volts. For a capacitor made of two parallel plates carrying a
surface charge -+ơ, the voltage, or difference in potential, of the pair of plates
is ơd/eọ.
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Tho Spocrerl Thoorgg of lĩocl(fitftgy
15-1 The principle of relativity
For over 200 years the equations of motion enunciated by NÑNewton were believed
to describe nature correctly, and the fñrst time that an error in these laws was
discovered, the way to correct it was also discovered. Both the error and its
correction were discovered by Einstein in 1905.
Newton?s Second Law, which we have expressed by the equation
†' = d(mu)/dt,
was siated with the tacit assumption that ?m is a constant, but we now know
that this is not true, and that the mass of a body increases with velocity. In
Binstein”s corrected formula ?m has the value
m=—————., (15.1)
v1— 12/c2
where the “rest mass” rnọ represents the mass of a body that is not moving and é
is the speed of light, which is about 3 x 105 km -see~1 or about 186,000 mi - sec—1.
For those who want to learn just enough about it so they can solve problems,
that is all there is to the theory of relativity——it just changes Newton's laws by
introducing a correction factor 6o the mass. From the formula itself it is easy
to see that this mass increase is very small in ordinary circumstances. Tf the
velocity 1s even as great as that of a satellite, whiích goes around the earth at
5 mi/sec, then ø/c = 5/186,000: putting this value into the formula shows that
the correction to the mass is only one part in two to three billion, which is nearly
impossible to observe. Actually, the correcbness of the formula has been amply