Let us find a particular solution of the differential equation coming from
LCR circuits:
dQ
d2 Q
1
+ Q = E cos ωt
L 2 +R
dt
dt
C
Watch how sweetly it goes if we make it complex-we’ll solve instead:
(∗) L
dQ
d2 Q
1
+R
+ Q = Eeiωt
dt2
dt
C
and the real part of the solution is necessarily a solution to our oringal problem.
Suppose Aei(ωt+θ) is a solution. Then differentiating and plugging in to the
equation we get:
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A{−Lω 2 + iRω + }eiωt = Eeiωt
C
So we get a solution if and only if
A=
E
L(ω02 − ω 2 ) + iRω
where we have used ω0 toqrepresent the natural frequency of the circuit without
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resistance, namely ω0 = LC
.
Now let’s write the complex number A as |A|eiθ :
E
|A| = p
2
2
L (ω0 − ω 2 )2 + R2 ω 2
and the argument is - the argument of the denominator, i.e.
tan θ =
So
Rω
L(ω 2 − ω02 )
Eei(ωt+θ)
p
L2 (ω 2 − ω02 )2 + R2 ω 2
is a solution of (*) and
E
p
L2 (ω 2
− ω02 )2 + R2 ω 2
cos(ωt + θ)
is a solution of our original differential equation.
Differentiating we find that the current I in the circuit oscillates with a
magnitude of
E
q
R2 + (ωL −
1 2
ωC )
Note the interesting fact that this is clearly a maximum when ω = ω0 whereas
the maximum amplitude for the charge oscillations occurs at a different frequency.
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