Answer
a. 2.3 time constants.
b. 4.6 time constants.
c. 6.9 time constants.
Work Step by Step
The current through an LR circuit is given by the following equation.
$$I= I_{max}(1-e^{-t/\tau})$$
Solve for t.
$$ e^{-t/\tau}=1-\frac{I}{ I_{max}}$$
$$t=-\tau ln(1-\frac{I}{ I_{max}}) $$
a. $I=0.90I_{max}$
$$t=-\tau ln(1-\frac{I}{ I_{max}})=-\tau ln(1-0.90)=2.3\tau $$
b. $I=0.990I_{max}$
$$t=-\tau ln(1-\frac{I}{ I_{max}})=-\tau ln(1-0.990)=4.6\tau $$
c. $I=0.999I_{max}$
$$t=-\tau ln(1-\frac{I}{ I_{max}})=-\tau ln(1-0.999)=6.9\tau $$
