Chapter 30: Problem 42
An inductor is connected to the terminals of a battery that has an emf of 16.0 V and negligible internal resistance. The current is 4.86 mA at 0.940 ms after the connection is completed. After a long time, the current is 6.45 mA. What are (a) the resistance \(R\) of the inductor and (b) the inductance \(L\) of the inductor?
Short Answer
Step by step solution
Understand the Problem Scenario
Identify Key Equations
Calculate the Resistance \( R \)
Verify Initial Condition Equation
Solve for Time Constant \( \tau \)
Calculate the Inductance \( L \)
Summary of Results
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Inductor resistance
- resistance is caused by the inherent imperfections and material properties of the inductor coil.
- It is always considered when calculating the maximum current in an RL circuit.
When dealing with circuits over time, resistive heating can lead to inefficiencies, affecting total circuit behavior.
Inductance calculation
- Inductance is measured in Henries (H).
- A larger inductance means a greater capacity to oppose changes in current.
In practical applications, calculating inductance is essential for designing filters, transformers, and tuning circuits.
The precise calculation ensures the components behave correctly in dynamic scenarios, such as during power surges or load changes.
Time constant in circuits
- Mathematically, it is represented as \( \tau = \frac{L}{R} \).
- A shorter time constant means the circuit reacts quickly to changes in voltage.
- Conversely, a longer time constant means the circuit responds more slowly.
In our example, solving for \( \tau \) involves setting the exponential factor such that it accounts for the observed current at a specific time, giving insights into how fast the circuit stabilizes.
Understanding the time constant helps in controlling the transient behavior of circuits during initial startup or sudden power increases, a critical aspect in both the analysis and design of time-sensitive electronic systems.