A coil with a certain number of turns has a specified time constant. If the number of turns is doubled, its time constant woulda)remain unaffectedb)become doublec)become four-foldd)get halvedCorrect answer is option 'B'. Can you explain this answer?

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Understanding Time Constant in Inductors
The time constant of an inductor is a crucial concept in understanding how it responds to changes in current. It is defined as the time taken for the current to rise to approximately 63.2% of its final value when a voltage is suddenly applied.
Key Factors Affecting Time Constant
- Inductance (L): The inductance of a coil is directly proportional to the number of turns (N). More turns mean greater inductance.
- Resistance (R): The resistance of the coil also affects the time constant, but here we focus on how changes in the number of turns influence the inductance.
Relation Between Inductance and Number of Turns
- When you double the number of turns in a coil, the inductance (L) increases according to the formula:
L ∝ N².
- Therefore, if the number of turns is doubled, the new inductance becomes four times the original inductance (L' = 4L).
Time Constant Formula
- The time constant (τ) is given by the formula:
τ = L/R.
- If the inductance increases by a factor of four due to doubling the number of turns, the new time constant becomes:
τ' = L'/R = (4L)/R = 4τ.
Conclusion
- Thus, when the number of turns is doubled, the time constant becomes four-fold, confirming that the correct answer is indeed option 'B'.
This relationship highlights the significant impact that the number of turns in a coil has on its inductance and consequently its time constant, which is essential for applications in circuits involving inductors.

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