(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 . Show that the maximum emf ε_max induced in the inductor during this time period is (R'/R)V_o. (b) If R' = 45R and V_o = 145 V, determine ε_max. [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.]

Verified step by step guidance

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.
2. 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).
3. 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.
4. 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.
5. 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.

Key Concepts

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.

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.

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.