332 Chapter 11First-order differential equations
amount of intermediate Bremains small throughout the reaction because Bis rapidly
converted into C. Figure (c), withk
12 k
21 = 110 , shows the behaviour whenk
11 >> 1 k
2; the
rate-determining step isB 1 → 1 C, and the amount of Bbecomes large because the
processB 1 → 1 Cis slow, and Bdecreases only when the supply of Ahas been exhausted.
0 Exercises 41, 42
11.7 Electric circuits
The modelling of electric circuits is an important application
of linear differential equations. A simple circuit consists of a
source of electric energy, such as a generator or a battery, that
generates an electromotive force (e.m.f.) E, and one or more
elements (Figure 11.4) connected in series; resistors (R),
inductors (L), and capacitors (C). The e.m.f. of the source is
equal to the difference of electric potential between the two
points at which it is attached to the circuit. When the circuit is
closed, a currentIflows and each element of the circuit causes
a drop in voltage (electric potential) in the circuit. The total
drop in voltage is equal to the e.m.f. (Kirchhoff ’s voltage law).
A resistor is a circuit element that offers resistance to the flow of charge in the
circuit. The voltage drop E
Racross a resistor is proportional to the current,
E
R1 = 1 RI (Ohm’s law) (11.62)
and the constant of proportionality Ris called the resistanceof the resistor (see
Tables 1.1 and 1.2 for the appropriate SI units). An inductor is a coil which builds
up a magnetic field within itself when a current flows. A changing magnetic field
develops an induced voltage across the coil that is proportional to the rate of change
of current and opposes the current flow:
(11.63)
ELdI
dt
L=
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Figure 11.3
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Q -Q
Figure 11.4