30. Induction and Inductance

Consider the LR circuit shown below. Initially, both switches are open. Switch 1 is closed. a) What is the maximum current in the circuit after a long time? Then, S₁ is opened and S₂ is closed. b) What is the current in the circuit after 0.05s?

An LR circuit with L = 0.1 H and R = 10 Ω are connected to a battery with the circuit initially broken. When the circuit is closed, how much time passes until the current reaches half of its maximum value?

(II) It takes 2.85 ms for the current in an LR circuit to increase from zero to 0.75 its maximum value. Determine (a) the time constant of the circuit, (b) the resistance of the circuit if L = 31.0 mH.

(III) 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.)

(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?

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(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 = I₀ e⁻ᵗ/ʳ. 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.]

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