The Chemistry Maths Book, Second Edition

11.7 Electric circuits 333

whereLis the inductance. A simple capacitor is made up of two parallel metallic

plates separated by a thin layer of insulating material. The plates become charged

when a current flows. The rate of charging is , the accumulated charge is

, and the potential difference between the plates is

(11.64)

where Cis the capacitance.

EXAMPLE 11.7RL circuit

An RL-circuit containing one resistor and one inductor is

illustrated in Figure 11.5.

By Kirchhoff ’s law,

(11.65)

and this is a first-order linear differential equation. We

havep 1 = 1 R 2 L(constant) andr 1 = 1 E 2 Lin equation (11.47). The

integrating factor is and the general

solution of the equation is

(11.66)

If the e.m.f. is constant,E 1 = 1 E

0

, then

(11.67)

The particular solution for the initial conditionI(0) 1 = 10 is then

(11.68)

When the circuit is closed the current increases from zero to its steady valueE

0

2 Rin

a time that depends on the inductive time constantτ

L

1 = 1 L 2 R.

0 Exercises 43, 44

It

E

R

e

Rt L

()=−

( )

0 −/

1

It e

E

L

edtc

E

R

ce

Rt L Rt L Rt

()=+

















=+

−/ 00 / −/

Z

LL

It e e

E

L

dt c

Rt L Rt L

()=+

















−/ /

Z

Fx e e

RLdt Rt L

()

()

=

∫

=

L

dI

dt

+=RI E

E

Q

C

C

=

QIdt=Z

dQ

dt

=I

• 
  


• 
  

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L

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Figure 11.5