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30. Induction and Inductance

LR Circuits

Physics

30. Induction and Inductance

LR Circuits

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8:52 minutes

Problem 69b

Textbook Question

A 50 cm solenoid with 1000 turns has an inductance of 20 mH. Flipping a switch disconnects the inductor from the battery and connects it to a resistor. What is the value of the resistance if the magnetic field decreases by 50% in 150 μs?

Verified step by step guidance

Step 1: Understand the problem. The solenoid is disconnected from the battery and connected to a resistor, causing the magnetic field to decrease. The problem involves calculating the resistance using the time constant of the RL circuit. The time constant (τ) is related to the inductance (L) and resistance (R) by the formula:

τ = \frac{L}{R}

Step 2: Recall the relationship between the time constant and the decay of the magnetic field. The magnetic field decreases exponentially in an RL circuit, following the equation:

B = B

₀e^-tτ, where

B

₀ is the initial magnetic field,

B

is the magnetic field at time

t

, and

τ

is the time constant.

Step 3: Use the given information to find the time constant. The magnetic field decreases by 50% in 150 μs, meaning

B = \frac{1}{2} B

₀ at

t = 150 \times 10^- 6 s

. Substitute this into the exponential decay formula and solve for

τ

\frac{1}{2} = e^- \frac{150}{τ}

. Take the natural logarithm of both sides to isolate

τ

Step 4: Relate the time constant to the resistance. Once

τ

is calculated, use the formula

τ = \frac{L}{R}

to solve for

R

. The inductance

L

is given as 20 mH, or

20 \times 10^- 3 H

. Rearrange the formula to find

R = \frac{L}{τ}

Step 5: Substitute the values into the formula. Use the calculated value of

τ

and the given inductance

L

to find the resistance

R

. Ensure the units are consistent throughout the calculation.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Inductance

Inductance is a property of an electrical component, typically a coil or solenoid, that quantifies its ability to store energy in a magnetic field when an electric current flows through it. It is measured in henries (H) and is defined as the ratio of the induced electromotive force (emf) to the rate of change of current. In this question, the solenoid's inductance of 20 mH indicates how effectively it can store magnetic energy.

Time Constant

The time constant, denoted by τ (tau), is a measure of the time it takes for the current in an inductor to change significantly when connected to a resistor. It is calculated as τ = L/R, where L is the inductance and R is the resistance. In this scenario, the time constant will help determine how quickly the magnetic field decreases when the inductor is disconnected from the battery and connected to the resistor.

Magnetic Field Decay

When an inductor is disconnected from a power source and connected to a resistor, the magnetic field associated with the inductor begins to decay. The rate of this decay is exponential, characterized by the time constant. In this question, the magnetic field decreases by 50% in 150 μs, which can be used to find the resistance by relating the decay time to the time constant of the circuit.

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Related Practice

Textbook Question

(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 $I = I_{0} e^{- t / τ^{'}}$ $I = I_{0} e^{- t / τ^{'}}$ . 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.]

426

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Textbook Question

Use a phasor diagram to analyze the RL circuit of FIGURE P32.49. In particular, Find an expression for the crossover frequency ω_c.

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Multiple Choice

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?

The switch in FIGURE P30.76 has been open for a long time. It is closed at t = 0 s. What is the current through the 20 Ω resistor immediately after the switch is closed?

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Textbook Question

The switch in FIGURE P30.76 has been open for a long time. It is closed at t = 0 s. What is the current through the 20 Ω resistor after the switch has been closed a long time?

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Textbook Question

The switch in FIGURE P30.77 has been open for a long time. It is closed at t = 0 s. After the switch has been closed for a long time, what is the current in the circuit? Call this current I₀.

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Textbook Question

The switch in FIGURE P30.77 has been open for a long time. It is closed at t = 0 s. Find an expression for the current I as a function of time. Write your expression in terms of I₀, R, and L.

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