Self Induction

Step 1: Understanding the Concept:
This question deals with self-inductance, which is the property of a coil to induce an EMF in itself when the current flowing through it changes. The magnitude of this induced EMF is proportional to the rate of change of the current.

Step 2: Key Formula or Approach:
The average EMF ( $%5Cmathcal%7BE%7D$ ) induced in a coil due to self-inductance ( $L$ ) is given by: $%5Cmathcal%7BE%7D%20%3D%20-L%20%5Cfrac%7B%5CDelta%20I%7D%7B%5CDelta%20t%7D$ where $%5CDelta%20I$ is the change in current and $%5CDelta%20t$ is the time interval over which the change occurs. We are interested in the magnitude, so we use: $%7C%5Cmathcal%7BE%7D%7C%20%3D%20L%20%5Cleft%7C%20%5Cfrac%7B%5CDelta%20I%7D%7B%5CDelta%20t%7D%20%5Cright%7C%20%5Cimplies%20L%20%3D%20%5Cfrac%7B%7C%5Cmathcal%7BE%7D%7C%20%5Ccdot%20%5CDelta%20t%7D%7B%7C%5CDelta%20I%7C%7D$

Step 3: Detailed Explanation:
Given common data:
Average induced EMF, $%7C%5Cmathcal%7BE%7D%7C%20%3D%202%20%5C%2C%20%5Ctext%7BV%7D$ .
Time interval, $%5CDelta%20t%20%3D%200.4%20%5C%2C%20%5Ctext%7Bs%7D$ .
Analysis of Statement (A):
The current changes from $I_%7Binitial%7D%20%3D%205%20%5C%2C%20%5Ctext%7BA%7D$ to $I_%7Bfinal%7D%20%3D%202%20%5C%2C%20%5Ctext%7BA%7D$ . The change in current is $%5CDelta%20I%20%3D%20I_%7Bfinal%7D%20-%20I_%7Binitial%7D%20%3D%202%20-%205%20%3D%20-3%20%5C%2C%20%5Ctext%7BA%7D$ . The magnitude of the change is $%7C%5CDelta%20I%7C%20%3D%203%20%5C%2C%20%5Ctext%7BA%7D$ . Now, let's calculate the self-inductance $L$ : $L%20%3D%20%5Cfrac%7B2%20%5C%2C%20%5Ctext%7BV%7D%20%5Ctimes%200.4%20%5C%2C%20%5Ctext%7Bs%7D%7D%7B3%20%5C%2C%20%5Ctext%7BA%7D%7D%20%3D%20%5Cfrac%7B0.8%7D%7B3%7D%20%5C%2C%20%5Ctext%7BH%7D%20%5Capprox%200.267%20%5C%2C%20%5Ctext%7BH%7D$ Converting to millihenries (mH): $0.267%20%5C%2C%20%5Ctext%7BH%7D%20%3D%20267%20%5C%2C%20%5Ctext%7BmH%7D$ . Statement (A) says the self-inductance is 0.266 mH. Our calculated value is approximately 267 mH. There is a factor of 1000 difference. Thus, statement (A) is incorrect.
Analysis of Statement (B):
The current changes from $I_%7Binitial%7D%20%3D%204%20%5C%2C%20%5Ctext%7BA%7D$ to $I_%7Bfinal%7D%20%3D%20-4%20%5C%2C%20%5Ctext%7BA%7D$ (4 A in the opposite direction). The change in current is $%5CDelta%20I%20%3D%20I_%7Bfinal%7D%20-%20I_%7Binitial%7D%20%3D%20-4%20-%204%20%3D%20-8%20%5C%2C%20%5Ctext%7BA%7D$ . The magnitude of the change is $%7C%5CDelta%20I%7C%20%3D%208%20%5C%2C%20%5Ctext%7BA%7D$ . Now, let's calculate the self-inductance $L$ : $L%20%3D%20%5Cfrac%7B2%20%5C%2C%20%5Ctext%7BV%7D%20%5Ctimes%200.4%20%5C%2C%20%5Ctext%7Bs%7D%7D%7B8%20%5C%2C%20%5Ctext%7BA%7D%7D%20%3D%20%5Cfrac%7B0.8%7D%7B8%7D%20%5C%2C%20%5Ctext%7BH%7D%20%3D%200.1%20%5C%2C%20%5Ctext%7BH%7D$ Converting to millihenries (mH): $0.1%20%5C%2C%20%5Ctext%7BH%7D%20%3D%20100%20%5C%2C%20%5Ctext%7BmH%7D$ . Statement (B) says the self-inductance is 0.10 mH. Our calculated value is 100 mH. Again, there is a factor of 1000 difference. Thus, statement (B) is incorrect.

Step 4: Final Answer:
Since both statements (A) and (B) are incorrect, the correct option is (D).

Step 1: Understanding the Concept:
This problem involves the concept of self-induction in an inductor (coil). When the current through an inductor changes, it induces an electromotive force (EMF) that opposes this change. The magnitude of this induced EMF is proportional to the rate of change of the current.

Step 2: Key Formula or Approach:
The average EMF ( $%5Cmathcal%7BE%7D$ ) induced in a coil with self-inductance $L$ is given by Faraday's law of induction: $%5Cmathcal%7BE%7D_%7Bavg%7D%20%3D%20-L%20%5Cfrac%7B%5CDelta%20I%7D%7B%5CDelta%20t%7D$ where $%5CDelta%20I$ is the change in current ( $I_%7Bfinal%7D%20-%20I_%7Binitial%7D$ ) and $%5CDelta%20t$ is the time taken for this change. We are interested in the magnitude of the EMF.

Step 3: Detailed Explanation:
Part 1: Calculate the initial steady current.
When the switch is closed for a long time, a constant (steady) current flows. In this DC steady state, the inductor behaves like a simple wire, and the current is determined by the battery voltage $V$ and the resistance $R$ . $I_%7Binitial%7D%20%3D%20%5Cfrac%7BV%7D%7BR%7D%20%3D%20%5Cfrac%7B12%20%5C%2C%20%5Ctext%7BV%7D%7D%7B6%20%5C%2C%20%5COmega%7D%20%3D%202%20%5C%2C%20%5Ctext%7BA%7D$ Part 2: Determine the change in current.
When the switch is opened, the circuit is broken, and the current drops to zero. $I_%7Bfinal%7D%20%3D%200%20%5C%2C%20%5Ctext%7BA%7D$ The change in current is: $%5CDelta%20I%20%3D%20I_%7Bfinal%7D%20-%20I_%7Binitial%7D%20%3D%200%20%5C%2C%20%5Ctext%7BA%7D%20-%202%20%5C%2C%20%5Ctext%7BA%7D%20%3D%20-2%20%5C%2C%20%5Ctext%7BA%7D$ Part 3: Calculate the average induced EMF.
Given data:
Inductance, $L%20%3D%2010%20%5C%2C%20%5Ctext%7BmH%7D%20%3D%2010%20%5Ctimes%2010%5E%7B-3%7D%20%5C%2C%20%5Ctext%7BH%7D$ .
Time interval, $%5CDelta%20t%20%3D%201%20%5C%2C%20%5Ctext%7Bms%7D%20%3D%201%20%5Ctimes%2010%5E%7B-3%7D%20%5C%2C%20%5Ctext%7Bs%7D$ .
Using the formula for the magnitude of the average induced EMF: $%7C%5Cmathcal%7BE%7D_%7Bavg%7D%7C%20%3D%20%5Cleft%7C%20-L%20%5Cfrac%7B%5CDelta%20I%7D%7B%5CDelta%20t%7D%20%5Cright%7C%20%3D%20L%20%5Cleft%7C%20%5Cfrac%7B%5CDelta%20I%7D%7B%5CDelta%20t%7D%20%5Cright%7C$ $%7C%5Cmathcal%7BE%7D_%7Bavg%7D%7C%20%3D%20(10%20%5Ctimes%2010%5E%7B-3%7D%20%5C%2C%20%5Ctext%7BH%7D)%20%5Ctimes%20%5Cleft%7C%20%5Cfrac%7B-2%20%5C%2C%20%5Ctext%7BA%7D%7D%7B1%20%5Ctimes%2010%5E%7B-3%7D%20%5C%2C%20%5Ctext%7Bs%7D%7D%20%5Cright%7C$ The $10%5E%7B-3%7D$ terms in the numerator and denominator cancel out. $%7C%5Cmathcal%7BE%7D_%7Bavg%7D%7C%20%3D%2010%20%5Ctimes%202%20%3D%2020%20%5C%2C%20%5Ctext%7BV%7D$

Step 4: Final Answer:
The average EMF induced across the coil would be 20 V.

High-Yield Trend

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2 MCQs

Official Solution

Official Solution

Self Induction

High-Yield Trend

Chapter Questions 2 MCQs

Official Solution

Official Solution

Chapter Questions
2 MCQs