Question
Question: In the given figure, after switch \(S\) is closed at time \[t = 0\], the emf of the source is automa...
In the given figure, after switch S is closed at time t=0, the emf of the source is automatically adjusted to maintain a constant current i through S.
(a) Find the current through the inductor as a function of time.
(b) At what time is the current through the resistor equal to the current in the inductor?
Solution
In the figure, we have a Resistor and inductor in the circuit and the current flowing through LR circuit is given as i=i0(1−eL−Rt) where i0=RE which is called maximum current of the circuit, Junction rules we will find current in the inductor as a function of time.
Complete step by step answer:
(a) Let us suppose the current i1 flows in the resistor in downward direction and current i2 flows in the inductor in downward direction when the switch is closed at t=0. Total current through Current source says i.Then we have,
i=i1+i2
Now, the voltages across resistor is Ri1 and voltage across inductor is given by,
−Ldtdi2
So, Ri1−Ldtdi2=0
Ldtdi1+i1R=0
Above equation is similar to the general equation E=Ldtdi+Ri
Whose solution is given by i=i0eL−Rt
So we get the solution as,
i1=ieL−Rt
Since, at t=0 i1=0 and maximum current is i0=i
So we get,
i2=i−i1
∴i2=i(1−eL−Rt) Which is the current through the inductor.
Hence, current through the inductor is given by i2=i(1−eL−Rt).
(b) Since from part (i) we have,
i1=ieL−Rt
⇒i2=i(1−eL−Rt)
Equate both equations we get,
(1−eL−Rt)=eL−Rt
Taking Natural logarithm on both sides:
(LRt)=ln2
∴t=RLln2
Hence, the time at which current through inductor and resistor is t=RLln2.
Note: Remember, when the switch is just closed the charges start to flow in the circuit as circuit gets completed and the inductor starts to build a magnetic field around it as the inductor reaches the maximum point of producing magnetic field it acts as a short circuit.The time at which inductor is fully charged is known as its time constant.