bei48482_FM

It is worth emphasizing the difference between a conservedquantity, such as total

energy, and an invariantquantity, such as proper mass. Conservation of Emeans that,

in a given reference frame, the total energy of some isolated system remains the same

regardless of what events occur in the system. However, the total energy may be dif-

ferent as measured from another frame. On the other hand, the invariance of mmeans

that mhas the same value in all inertial frames.

The conversion factor between the unit of mass (the kilogram, kg) and the unit of

energy (the joule, J) is c^2 , so 1 kg of matter—the mass of this book is about that—has

an energy content of mc^2 (1 kg)(3  108 m/s)^2  9  1016 J. This is enough to

send a payload of a million tons to the moon. How is it possible for so much energy

to be bottled up in even a modest amount of matter without anybody having been

aware of it until Einstein’s work?

In fact, processes in which rest energy is liberated are very familiar. It is simply that

we do not usually think of them in such terms. In every chemical reaction that evolves

energy, a certain amount of matter disappears, but the lost mass is so small a fraction

of the total mass of the reacting substances that it is imperceptible. Hence the “law” of

conservation of mass in chemistry. For instance, only about 6  10 ^11 kg of matter

vanishes when 1 kg of dynamite explodes, which is impossible to measure directly, but

the more than 5 million joules of energy that is released is hard to avoid noticing.

Example 1.7

Solar energy reaches the earth at the rate of about 1.4 kW per square meter of surface perpen-

dicular to the direction of the sun (Fig. 1.15). By how much does the mass of the sun decrease

per second owing to this energy loss? The mean radius of the earth’s orbit is 1.5  1011 m.

Solution

The surface area of a sphere of radius ris A 4 r^2. The total power radiated by the sun, which

is equal to the power received by a sphere whose radius is that of the earth’s orbit, is therefore

P A (4r^2 ) (1.4 103 W/m^2 )(4)(1.5 1011 m)^2 4.0 1026 W

Thus the sun loses E 0 4.0  1026 J of rest energy per second, which means that the sun’s rest

mass decreases by

m4.4 109 kg

per second. Since the sun’s mass is 2.0  1030 kg, it is in no immediate danger of running out

of matter. The chief energy-producing process in the sun and most other stars is the conversion

of hydrogen to helium in its interior. The formation of each helium nucleus is accompanied by

the release of 4.0  10 ^11 J of energy, so 10^37 helium nuclei are produced in the sun per second.

4.0 1026 J



(3.0 (^108) m/s)^2
E 0

c^2
P

A
P

A
28 Chapter One
Figure 1.15
Solar
radiation
1.4 kW/m^2
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