We know the Current formula in L/R circut
I(t) = Io [1 - e-t/τ ]
, where Io = ε/R is the maximum current
and the time constant τ = L/R
if current reaches 99 % of steady state, then
[1 - e-t/τ ]
= 0.99 or e-t/τ
= 0.01 = 10-2 .
Hence t/τ = 2 × ln10 = 4.6
if current reaches 99.9 % of steady state, then
[1 - e-t/τ ]
= 0.999 or e-t/τ
= 0.001 = 10-3 .
Hence t/τ = 3 × ln10 = 6.9