Chapter 31: Problem 38
When a solenoid is connected to a 48.0-V dc battery that has negligible internal resistance, the current in the solenoid is 5.50 A. When this solenoid is connected to an ac source that has voltage amplitude 48.0 V and angular frequency 20.0 rad/s, the current in the solenoid is 3.60 A. What is the inductance of this solenoid?
Short Answer
Step by step solution
Identify Given Values
Calculate Resistance Using DC Circuit
Use AC Circuit to Find Impedance
Relate Impedance to Inductive Reactance
Solve for Inductance
Finalize
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Ohm's Law
In the context of our exercise, we utilized Ohm’s Law to find the resistance of the solenoid using the given DC circuit values. By substituting the known voltage (48.0 V) and current (5.50 A) into the formula, we calculated the resistance to be 8.727 ohms. This resistance is an essential step in further calculations involving the RL circuit.
Impedance
In the context of RL circuits, impedance is defined as \( Z = \sqrt{R^2 + (\omega L)^2} \). In the exercise, we calculated impedance using the AC voltage amplitude (48.0 V) and current (3.60 A). The impedance was found to be 13.333 ohms. Calculating impedance was key to eventually determining the inductance of the solenoid because it incorporates both the resistive and inductive elements of the circuit.
RL Circuit
In our problem, we dealt with an RL series circuit. The inductance was determined by solving an equation derived from the impedance formula \( Z = \sqrt{R^2 + (\omega L)^2} \). By substituting the known values of impedance (13.333 ohms), resistance (8.727 ohms), and angular frequency (20.0 rad/s), we solved for the inductance, establishing a basis for understanding how RL circuits manage electrical currents.
Inductive Reactance
For our exercise, we combined this concept with the impedance formula to find the inductance of the solenoid. We rearranged the impedance equation to solve for the inductive reactance and eventually the inductance. Inductive reactance plays a crucial role in determining how much the inductor in an RL circuit resists the flow of alternating current, which directly contributes to calculating the inductance value needed to solve the exercise.