A coil having 90 turns is connected to a 120 V, 60 Hz source. If the effective value of the magnetizing current is 4 A. Calculate the inductance of the coil. 

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  1. 79.6 mH
  2. 121.2 mH 
  3. 32.2 mH
  4. 83.16 mH 

Answer (Detailed Solution Below)

Option 1 : 79.6 mH
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Detailed Solution

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Explanation:

Problem Statement:

A coil with 90 turns is connected to a 120 V, 60 Hz source, and the effective value of the magnetizing current is given as 4 A. The task is to calculate the inductance of the coil.

Solution:

To determine the inductance of the coil, we use the formula for inductive reactance and the relationship between inductance, current, and voltage in an AC circuit:

Formula:

Inductive reactance (XL) is given by:

XL = 2πfL

Where:

  • XL: Inductive reactance (in ohms)
  • f: Frequency of the AC source (in Hz)
  • L: Inductance of the coil (in henrys)

The current in the coil is related to the inductive reactance and voltage as:

I = V / XL

Where:

  • I: Current through the coil (in amperes)
  • V: Voltage across the coil (in volts)

By substituting XL = 2πfL into the current equation, we get:

I = V / (2πfL)

Rearranging the equation to solve for L:

L = V / (2πfI)

Step-by-Step Calculation:

Given:

  • V = 120 V
  • f = 60 Hz
  • I = 4 A

Substituting the values into the formula:

L = 120 / (2 × π × 60 × 4)

First, calculate the denominator:

2 × π × 60 × 4 = 1507.96

Now divide the numerator by the denominator:

L = 120 / 1507.96

L = 0.0796 H

Convert henrys to millihenrys:

L = 79.6 mH

Thus, the inductance of the coil is 79.6 mH.

Correct Option: Option 1 (79.6 mH)

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 2: 121.2 mH

This value is incorrect because it does not match the calculated inductance. If we mistakenly used an incorrect frequency or current value, or made an error in the denominator during computation, we might arrive at this result. However, the correct computation yields 79.6 mH.

Option 3: 32.2 mH

This value is also incorrect. A result of 32.2 mH would imply either a significant error in the formula application or incorrect substitution of given values. The inductance calculated using the correct formula and values is 79.6 mH.

Option 4: 83.16 mH

While closer to the correct answer, this value is slightly off. This discrepancy might arise from rounding errors or using approximated constants (such as π = 3.14 instead of 3.1416). The precise calculation yields 79.6 mH.

Conclusion:

The correct inductance of the coil, based on the given parameters and using accurate calculations, is 79.6 mH. Understanding the relationship between inductive reactance, frequency, voltage, and current is crucial for solving such problems correctly. Analyzing other options highlights the importance of accurate computation and proper substitution of values in formulas.

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