Problem 45 - Master Resource Book in JEE Main Physics

Question

The number of turns of primary and secondary coils of a transformer are 5 and 10 respectively and mutual inductance of the transformer is \(25 \mathrm{H}\). Now, number of turns in primary and secondary are made 10 and 5 respectively. Mutual inductance of transformer will be (a) \(25 \mathrm{H}\) (b) \(12.5 \mathrm{H}\) [c) \(50 \mathrm{H}\) (d) 6. \(25 \mathrm{H}\)

Step-by-Step Solution

Answer

The mutual inductance remains 25 H after switching the turns.

Step 1: Understanding Mutual Inductance Formula

The mutual inductance \( M \) of a transformer is given by the formula \( M = k \sqrt{L_1 L_2} \), where \( L_1 \) and \( L_2 \) are the self-inductances of the primary and secondary coils, respectively, and \( k \) is the coupling coefficient. In an ideal transformer, \( k \) is 1.

Step 2: Calculate Initial Inductances

The inductance is proportional to the number of turns squared, i.e., \( L \propto N^2 \). For the initial setup, with \( N_1 = 5 \) and \( N_2 = 10 \), \( L_1 \propto 5^2 \) and \( L_2 \propto 10^2 \). Therefore, \( L_1 = 25x \) and \( L_2 = 100x \) for some proportionality constant \( x \).

Step 3: Initial Mutual Inductance

With \( M = 25 \mathrm{H} \), from \( M = \sqrt{L_1 L_2} \), we have \( 25 = \sqrt{(25x)(100x)} \). Thus, \( 25 = \sqrt{2500x^2} \), giving \( 2500x^2 = 625 \), hence \( x^2 = 0.25 \) or \( x = 0.5 \).

Step 4: New Inductance Calculation

For the new configuration, \( N_1 = 10 \) and \( N_2 = 5 \), the inductances are \( L_1' = 100x \) and \( L_2' = 25x \). Since \( x = 0.5 \), \( L_1' = 100 \times 0.5 = 50 \) and \( L_2' = 25 \times 0.5 = 12.5 \).

Step 5: Calculate New Mutual Inductance

With the new setup, the mutual inductance is \( M' = \sqrt{L_1' L_2'} = \sqrt{50 \times 12.5} = \sqrt{625} = 25 \mathrm{H} \).

Step 6: Conclusion

From the calculations, we see that the mutual inductance of the transformer remains unchanged even after switching the number of turns. The answer to the problem is (a) 25 H.

Key Concepts

Self-Inductance

Self-inductance occurs in a coil when a change in electric current induces a voltage in the same coil. This property of a coil is key in transformers as it relates to how efficiently they can transfer electromagnetic energy. When a coil has more turns, it has more self-inductance because each additional loop increases the overall induced magnetic field.

Self-inductance is represented by the symbol \( L \) and has the unit of henrys (H). The self-inductance of a coil is determined by factors like the number of turns, coil area, and the type of substance through which the coil is wound. For instance:

• A larger number of turns leads to higher self-inductance as shown in the phrase \( L \propto N^2 \).
• The inductance is directly related to the square of the number of turns.

In the given exercise, self-inductance of the primary and secondary coils are parts of the calculation to find the mutual inductance. Calculating the self-inductance initially helps in determining how effectively the transformer will operate. Changes in the number of turns in both coils affects their self-inductance, yet interestingly does not affect the mutual inductance in this scenario.

Coupling Coefficient

The coupling coefficient, denoted as \( k \), measures how well two inductors (such as coils in a transformer) are linked through their mutual magnetic field. It ranges between 0 and 1, where 1 means perfect coupling and all magnetic fields are shared between coils.

In our context:

• The coupling coefficient helps to determine the mutual inductance using the formula \( M = k \sqrt{L_1 L_2} \).
• For an ideal transformer, \( k \) is often assumed to be 1, implying near perfect coupling.

In practical transformers, the coefficient might be slightly less than 1 due to imperfect alignment or other inefficiencies, but in calculation exercises like this, it simplifies to perfect coupling. Thus, focusing on coil turns and how they influence \( L_1 \) and \( L_2 \) becomes significant. Understanding this improves comprehension of how different turns' configurations can affect not transformation efficiency, but simply the magnitude of energy transferred.

Transformer Turns Ratio

The transformer turns ratio is a crucial concept in understanding how transformers operate. This ratio, defined as the number of turns in the primary coil relative to the number of turns in the secondary coil, determines the voltage transformation that the transformer will perform.

• When the number of secondary turns is higher, the voltage in the secondary coil increases compared to the primary voltage.
• Conversely, fewer secondary turns result in a lower secondary voltage.

The initial problem describes the turns ratio changing from \(5:10\) to \(10:5\). Despite altering the turns and thus the potential voltages, the mutual inductance remains constant at 25 H, indicating that mutual inductance in such setups can remain unaffected by these changes.

This demonstrates the principle that while voltages can significantly change with turns described by the ratio, the inherent link provided by magnetic flux (as illustrated by the exercise) remains constant if the effective turns product coupled by magnetic fields stays equivalent. Understanding this principle helps in electric circuit applications, especially when designing transformers to adjust voltage levels effectively.