Skip to main content
Ch. 30 - Inductance, Electromagnetic Oscillations, and AC Circuits
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Giancoli Douglas 5th editionCh. 30 - Inductance, Electromagnetic Oscillations, and AC Circuits문제 25
29장, 문제 25

(II) (a) In Fig. 30–28, assume that the switch has been in position A for sufficient time so that a steady current I₀ = V₀/R flows through the resistor R. At time t = 0, the switch is quickly switched to position B and the current decays through resistor R' (which is much greater than R) according to I=I0e−t/τ′I = I_0 e^{-t/\(\tau\)'}I=I0e−t/τ′I = I_0 e^{-t/\(\tau\)'}. Show that the maximum emf εmax induced in the inductor during this time period is (R'/R)Vo. (b) If R' = 45R and Vo = 145 V, determine εmax. [When a mechanical switch is opened, a high-resistance air gap is created, which is modeled as R' here. This Problem illustrates why high-voltage sparking can occur if a current-carrying inductor is suddenly cut off from its power source. The very high voltage can produce an electric field great enough to ionize atoms of air, which emit light when electrons recombine with the ions.]
Circuit diagram showing a switch S, resistors R and R', an inductor L, and a voltage source V₀, illustrating current decay.

검증된 단계별 안내
1
Step 1: Understand the problem setup. Initially, the switch is in position A, and a steady current I₀ = V₀/R flows through the resistor R. At time t = 0, the switch is moved to position B, and the current begins to decay through a much larger resistance R'. The goal is to find the maximum emf (εₘₐₓ) induced in the inductor during this transition and show that it equals (R'/R)V₀.
Step 2: Recall Faraday's Law of Induction. The emf induced in the inductor is given by ε = -L(dI/dt), where L is the inductance of the inductor and dI/dt is the rate of change of current. The negative sign indicates the direction of the induced emf opposes the change in current (Lenz's Law).
Step 3: Write the expression for the decaying current. After the switch is moved to position B, the current decays according to I(t) = I₀ e⁻ᵗ/ʳ, where I₀ = V₀/R and ʳ = L/R'. Differentiate this expression with respect to time to find dI/dt. Using the chain rule, dI/dt = -I₀/ʳ * e⁻ᵗ/ʳ.
Step 4: Substitute dI/dt into Faraday's Law. The induced emf becomes ε = -L(-I₀/ʳ * e⁻ᵗ/ʳ). Simplify this to ε = (L * I₀/ʳ) * e⁻ᵗ/ʳ. The maximum emf occurs at t = 0 because the exponential term e⁻ᵗ/ʳ decreases with time. At t = 0, εₘₐₓ = L * I₀/ʳ.
Step 5: Express εₘₐₓ in terms of the given quantities. Substitute I₀ = V₀/R and ʳ = L/R' into the expression for εₘₐₓ. This gives εₘₐₓ = (L * (V₀/R)) / (L/R') = (R'/R)V₀. This proves the first part of the problem. For part (b), substitute R' = 45R and V₀ = 145 V into εₘₐₓ = (R'/R)V₀ to calculate the numerical value of εₘₐₓ.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
5m
도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Induced EMF

Induced electromotive force (emf) occurs when a change in magnetic flux through a circuit induces a voltage. According to Faraday's law of electromagnetic induction, the induced emf is proportional to the rate of change of magnetic flux. In the context of inductors, when the current through an inductor changes, it generates an emf that opposes the change, described by Lenz's law.
추천 영상:
가이드 코스
09:57
Motional EMF

Exponential Decay of Current

The exponential decay of current in an RL circuit occurs when the switch is opened, causing the current to decrease over time. The current can be modeled by the equation I(t) = I₀ e⁻ᵗ/ʳ, where I₀ is the initial current, t is time, and r is the time constant. This behavior illustrates how the inductor releases stored energy gradually, leading to a transient response in the circuit.
추천 영상:
가이드 코스
04:24
Amplitude Decay in an LRC Circuit

Voltage Division in Resistors

Voltage division is a principle that describes how the total voltage across a series of resistors is distributed among them. The voltage across a resistor in a series circuit is proportional to its resistance. In this problem, the relationship between resistors R and R' is crucial for determining the maximum induced emf, as it affects how the initial voltage V₀ is divided when the switch is moved to position B.
추천 영상:
가이드 코스
07:14
RMS Current and Voltage
관련 실천
교과서 질문

Two tightly wound solenoids have the same length and circular cross-sectional area. But solenoid 1 uses wire that is 1.5 times as thick as solenoid 2. What is the ratio of their inductive time constants? (Assume the only resistance in the circuits is that of the wire itself.)

97
views
교과서 질문

(II) For the toroid of Fig. 30–26, determine the energy density in the magnetic field as a function of r(r₁ < r < r₂) and integrate this over the volume to obtain the total energy stored in the toroid, which carries a current I in each of its N loops.


<IMAGE>

124
views
교과서 질문

Two tightly wound solenoids have the same length and circular cross-sectional area. But solenoid 1 uses wire that is 1.5 times as thick as solenoid 2.

(a) What is the ratio of their inductances?

(b) What is the ratio of their inductive time constants? (Assume the only resistance in the circuits is that of the wire itself.)

1362
views
교과서 질문

(III) Determine the emf induced in the square loop in Fig. 29–52 if the loop stays at rest and the current in the straight wire is given by I(t) = (15.0 A) sin (2200 t) where t is in seconds. The distance a is 12.0 cm, and b is 15.0 cm.

175
views
교과서 질문

(II) Suppose that the U-shaped conductor and connecting rod in Fig. 29–12a are oriented vertically (but still in contact) so that the rod is falling due to the gravitational force. Find the terminal speed of the rod if it has mass m = 3.6 grams, length 𝓁 = 18 cm, and resistance R = 0.0013 Ω. It is falling in a uniform horizontal field B = 0.080 T. Neglect the resistance of the U-shaped conductor, and friction.

1460
views
교과서 질문

(II) (a) Determine the energy stored in the inductor L as a function of time for the LR circuit of Fig. 30–6a. (b) After how many time constants does the stored energy reach 99.9% of its maximum value?

1542
views