Final answer:
The time constant of an L-R circuit will double when both the length and the number of turns in the coil are doubled, because inductance increases by a factor of four while resistance doubles.
Step-by-step explanation:
When you double the length and the number of turns in the coil of an L-R circuit, the time constant of the circuit will change.
The time constant (τ) in an L-R circuit is defined as the time it takes for the current to change significantly, specifically until it reaches about 63.2% of its final value, and is given by the equation τ = L/R, where L is the inductance and R is the resistance. Doubling the number of turns in the coil increases the inductance by a factor of four, because inductance is proportional to the square of the number of turns (L ∼ N2). Doubling the length of the wire would double the resistance because resistance is proportional to length (R ∼ L). As a result, the new time constant is Lnew / Rnew = (4L) / (2R) = 2L/R = 2τ, meaning the time constant will double.