Question $2$

$2.$ Consider the circuit shown in Figure $2\mathrm{.}1.$

In this question "the circuit values" referred to below are: $\backslash (\backslash$boldsymbol$\{$V$\}\_\{\backslash$mathbf$\{1\}\},\backslash$boldsymbol$\{$V$\}\_\{\backslash$mathbf$\{2\}\},\backslash$mathbf$\{$R$\},\backslash$mathbf$\{$L$\}\backslash ).$

$($a$)$ For the circuit shown in Figure $2\mathrm{.}1,$ find a formula in terms of the circuit values for the time constant, $\backslash (\backslash$boldsymbol$\{\backslash$tau$\}\backslash ),$ that will characterise the change in current through the inductor for time, $\backslash (\backslash$boldsymbol$\{$t$\}>\backslash$mathbf$\{0\}\backslash ).$

$[8$ marks$]$

$($b$)$ For the circuit shown in Figure $2\mathrm{.}1,$ find a formula in terms of the circuit values for the initial current flow through the inductor at time, $\backslash (\backslash$boldsymbol$\{$t$\}=0^\{+\}\backslash ).$

$[8$ marks$]$

$($c$)$ For the circuit shown in Figure $2\mathrm{.}1,$ find a formula for the final current flow through the inductor at time, $\backslash (\backslash$boldsymbol$\{$t$\}\backslash$gg $\backslash$boldsymbol$\{\backslash$tau$\}\backslash ).$

$[10$ marks$]$

$($d$)$ Hence obtain a formula that will give the current flow through the inductor, $\backslash (\backslash$boldsymbol$\{$i$\}\_\{\backslash$boldsymbol$\{$L$\}\}(\backslash$boldsymbol$\{$t$\})\backslash ),$ for any time, $\backslash (\backslash$boldsymbol$\{$t$\}>0\backslash ),$ in terms of the circuit values, simplifying the expression obtained as appropriate.