Question

A solenoid having inductance 4.0 V battery at t = 0. Find (a) The times constant, (b) the times elapsed before the current reaches 0.63 of its steady state value, (c) the power deleered by the battery at this instant and (d) the power dissipated in joule heating at this instant.

Answer 1

$L=4.0H,R=10\Omega ,E=4V\left(a\right)\text{Time Constant,}\tau =\frac{L}{R}=\frac{4}{10}0.4s\left(b\right)i=0.63i_0\text{Now}0.63i_0=i_0(1-e^{-\frac{t}{\tau }})\Rightarrow e^{-\frac{t}{\tau }}=1.063=0.37\Rightarrow In{e}^{-\frac{t}{\tau }}=In0.037\Rightarrow -\frac{T}{\tau }=-0.9942\Rightarrow t=0.942\times 0.04=0.3977=0.40s\left(c\right)i=i_0(1-e^{-\frac{t}{\tau }})=\frac{4}{10}(1-e^{-\frac{t}{\tau }})=0.4\times 0.63210.2526A$
Power delivered
$P=Vi=4\times 0.2528=1.01=1W.$
(d) Power dissipated in joule heating
$=1^{2}R=(0.2258{)}^2\times 10=0.639=0.64W.$