Page 25 - Schaum's Outline of Theory and Problems of Applied Physics

10 USEFUL MATH [CHAP. 1

of capacitance C is V, the charge on the capacitor is Q = CV. Find the charge on a 200-pF capacitor
when it is connected to a 3-kV source.

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Here C = 200 pF = 200 picofarads = 200 × 10 −12 F = 2 × 10 −10 F and V = 3kV = 3 kilovolts = 3 × 10 V.
Hence
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Q = CV = (2 × 10 −10 F)(3 × 10 V) = 6 × 10 −7 C
The answer can be expressed as Q = 0.6 µC = 0.6 microcoulomb since 1 µC = 10 −6 C.

SIGNIFICANT FIGURES

An advantage of powers-of-10 notation is that it provides us with a clear way to express the accuracy with which
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a quantity is known. For instance, the speed of light c in empty space is often given as 2.998 × 10 m/s. If this
value were written out as 299,800,000 m/s, we might think that this speed is precisely equal to this many meters
per second, right down to the last zero. Actually, the speed of light is 299,792,458 m/s. If we do not need this
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much detail, we write c = 2.998 × 10 m/s to indicate both how large the number is (the 10 tells us this) and
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how precise the quoted figure is (the 2.998 tells us that c is closer to 2.998 × 10 m/s than it is to either 2.997 or
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2.999 × 10 m/s).
Thus, giving the speed of light as
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c = 2.998 × 10 m/s
means that
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c = (2.998 ± 0.0005) × 10 m/s
The accurately known digits, plus one uncertain digit, are called significant figures. Here c has four significant
figures; 2, 9, 9, and 8. The first three digits are accurately known, whereas the last (the 8) is uncertain; it could
bea7ora9.
To be sure, sometimes one or more final zeros in a number are meaningful in their own right. In the case of
the speed of light, we can legitimately say that, to three-digit accuracy,

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c = 3.00 × 10 m/s
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because c is closer to this figure than to either 2.99 or 3.01 × 10 m/s.
When two or more quantities are combined arithmetically, the result is no more accurate than the quantity
with the largest uncertainty. For instance, a 72 kg person might pick up a 0.33-kg apple. The total mass of person
+ apple is still considered to be 72 kg because we have only two significant figures. All we know of the person’s
mass is that it is somewhere between 71.5 and 72.5 kg, which means an uncertainty of more than the apple’s
mass. If the person’s mass is given instead as 72.0 kg, the mass of person + apple is 72.3 kg; if it is given as
72.00 kg, the mass of person + apple is 72.33 kg. Thus

72 kg + 0.33 kg = 72 kg (2 significant figures)
72.0kg + 0.33 kg = 72.3 kg (3 significant figures)
72.00 kg + 0.33 kg = 72.33 kg (4 significant figures)
Significant figures must be taken into account in other arithmetical operations also. If we divide 7.9 by 3.24,
we are not justified in writing
7.9
= 2.43827 ...
3.24
The 7.9 has only two significant figures, and the answer cannot have more than this. Hence the correct answer
is 7.9/3.24 = 2.4.