27. The assumption stated at the end of the problem is equivalent to setting φ = 0 in Eq. 33-25. Since the
maximum energy in the capacitor (each cycle) is given by q
2
max
/2C, where q
max
is the maximum charge
(during a given cycle), then we seek the time for which
q
2
max
2C
=
1
2
Q
2
2C
=
⇒ q
max
=
Q
√
2
.
Now q
max
(referred to as the exponentially decaying amplitude in
§33-5) is related to Q (and the other
parameters of the circuit) by
q
max
= Qe
−Rt/2L
=
⇒ ln
q
max
Q
=
−
Rt
2L
.
Setting q
max
= Q/
√
2, we solve for t:
t =
−
2L
R
ln
q
max
Q
=
−
2L
R
ln
1
√
2
=
L
R
ln 2 .
The identities ln(1/
√
2) =
− ln
√
2 =
−
1
2
ln 2 were used to obtain the final form of the result.
Document Outline
- Main Menu
- Chapter 1 Measurement
- Chapter 2 Motion Along a Straight Line
- Chapter 3 Vectors
- Chapter 4 Motion in Two and Three Dimensions
- Chapter 5 Force and Motion I
- Chapter 6 Force and Motion II
- Chapter 7 Kinetic Energy and Work
- Chapter 8 Potential Energy and Conservation of Energy
- Chapter 9 Systems of Particles
- Chapter 10 Collisions
- Chapter 11 Rotation
- Chapter 12 Rolling, Torque, and Angular Momentum
- Chapter 13 Equilibrium and Elasticity
- Chapter 14 Gravitation
- Chapter 15 Fluids
- Chapter 16 Oscillations
- Chapter 17 Waves—I
- Chapter 18 Waves—II
- Chapter 19 Temperature, Heat, and the First Law of Thermodynamics
- Chapter 20 The Kinetic Theory of Gases
- Chapter 21 Entropy and the Second Law of Thermodynamics
- Chapter 22 Electric Charge
- Chapter 23 Electric Fields
- Chapter 24 Gauss’ Law
- Chapter 25 Electric Potential
- Chapter 26 Capacitance
- Chapter 27 Current and Resistance
- Chapter 28 Circuits
- Chapter 29 Magnetic Fields
- Chapter 30 Magnetic Fields Due to Currents
- Chapter 31 Induction and Inductance
- Chapter 32 Magnetism of Matter: Maxwell’s Equation
- Chapter 33 Electromagnetic Oscillations and Alternating Current
- 33.1 - 33.10
- 33.1
- 33.2
- 33.3
- 33.4
- 33.5
- 33.6
- 33.7
- 33.8
- 33.9
- 33.10
- 33.11 - 33.20
- 33.11
- 33.12
- 33.13
- 33.14
- 33.15
- 33.16
- 33.17
- 33.18
- 33.19
- 33.20
- 33.21 - 33.30
- 33.21
- 33.22
- 33.23
- 33.24
- 33.25
- 33.26
- 33.27
- 33.28
- 33.29
- 33.30
- 33.31 - 33.41
- 33.31
- 33.32
- 33.33
- 33.34
- 33.35
- 33.36
- 33.37
- 33.38
- 33.39
- 33.40
- 33.41 - 33.50
- 33.41
- 33.42
- 33.43
- 33.44
- 33.45
- 33.46
- 33.47
- 33.48
- 33.49
- 33.50
- 33.51 - 33.60
- 33.51
- 33.52
- 33.53
- 33.54
- 33.55
- 33.56
- 33.57
- 33.58
- 33.59
- 33.60
- 33.61 - 33.70
- 33.61
- 33.62
- 33.63
- 33.64
- 33.65
- 33.66
- 33.67
- 33.68
- 33.69
- 33.70
- 33.71 - 33.80
- 33.71
- 33.72
- 33.73
- 33.74
- 33.75
- 33.76
- 33.77
- 33.78
- 33.79
- 33.80
- 33.81 - 33.90
- 33.81
- 33.82
- 33.83
- 33.84
- 33.85
- 33.86
- 33.87
- 33.88
- 33.89
- 33.90
- 33.91 - 33.92
- 33.91
- 33.92
- 33.1 - 33.10
- Chapter 34 Electromagnetic Waves
- Chapter 35 Images
- Chapter 36 Interference
- Chapter 37 Diffraction
- Chapter 38 Special Theory of Relativity
- Chapter 39 Photons and Matter Waves
- Chapter 40 More About Matter Waves
- Chapter 41 All About Atoms
- Chapter 42 Conduction of Electricity in Solids
- Chapter 43 Nuclear Physics
- Chapter 44 Energy from the Nucleus
- Chapter 45 Quarks, Leptons, and the Big Bang
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