consider the circuit part a what is the current through the battery?
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Current through the battery after a long time,
I=ReqE=R1R2E(R1+R2) as then inductor will not hinder.
When the circuit comes in steady state, that is after a long times, the current through the inductor is
And, the time constant is τ=R1L since only R1 affects the current in the inductor, otherwise potential difference is always E across that branch.