RL time constant change when the number of turns is doubled A coil in an RL circuit is rewound so that the number of turns is doubled, with similar wire and form factor so that resistance scales roughly with length. If the initial time constant is T, what is the new time constant?

Introduction / Context:
The RL time constant τ controls current rise and decay dynamics: τ = L / R. When a coil is modified (e.g., changing turns), both inductance and resistance change. Understanding how τ scales is vital in electromagnet design and transient analysis.

Given Data / Assumptions:

• Single-layer coil, same geometry aside from turn count.
• Wire gauge and packing similar so coil length increases with turns.
• Inductance L ∝ n^2 for fixed core/geometry; resistance R ∝ length ∝ n.

Concept / Approach:
Start from τ = L/R. If n → 2n, then L_new ≈ 4L (square law) and R_new ≈ 2R (linear law). Therefore τ_new = (4L)/(2R) = 2 (L/R) = 2T.

Step-by-Step Solution:
Initial τ = T = L/R.n doubles → L_new = 4L.n doubles → R_new = 2R.τ_new = L_new / R_new = 4L / 2R = 2T.

Verification / Alternative check:
Using solenoid formula L = μ n^2 A / l and R = ρ ℓ / A_w, with ℓ roughly proportional to n, yields the same scaling.

Why Other Options Are Wrong:
T or T/2 ignores simultaneous changes in L and R; 4T assumes R unchanged, which is unrealistic when turns are doubled; T/4 is contrary to both scalings.

Common Pitfalls:
Holding R constant when changing turns; forgetting L ∝ n^2 rather than L ∝ n.

Final Answer:
2T