656
14 Electromagnetism II
where we have set R/2Lω = tan φ. This gives
i =
−q m ω e −Rt/2L
cos φ
(sin φ cos ω
t + cos φ sin ω
t)
= −q m ω
e
−Rt/2L sin(ω t + φ)
cos φ
But for low damping φ → 0 as R/2Lω → 0.
∴ cos φ → 1, so that
i = −q m ω
e
−Rt/2L sin(ω
t + φ)
14.17 Equation of the circuit is
L
d 2 q
dt 2 +
1
C
q = ξ
(1)
Multiply (1) by i = dq/dt
Li
di
dt
+
1
C
q
dq
dt
= ξ t
or
d
dt
1
2
Li
2
+
d
dt
q 2
2C
= P input
(2)
Thus, the input power is the sum of the powers delivered to the inductor and
the capacitor.
14.18 The circuit equation is
L
d 2 q
dt 2 + R
dq
dt
+
1
C
q = 0
( 1 )
Multiply (1) by i =
dq
dt
Li
di
dt
+ Ri
2
+
1
C
q
dq
dt
= 0
or
d
dt
1
2
Li
2
+
q 2
2C
= −Ri
2
or
dE
dt
= −i
2 R
(2)
where E =
1
2
Li
2
+
q 2
2C
= total energy