**(1)** A rectangular loop has a sliding connector *PQ *of length *l *and resistance *R *Ω and it is moving with a speed *v *as shown. The set-up is placed in a uniform magnetic field going into the plane of the paper. The three currents *I*_{1}, *I*_{2} and *I *are

(1) *I*_{1} = *I*_{2} = *Blv*/6*R*, *I* = *Blv*/3*R*

(2) *I*_{1} = – *I*_{2} = *Blv*/*R*, *I* = 2*Blv*/*R*

(3) *I*_{1} = *I*_{2} = *Blv*/3*R*, *I* = 2*Blv*/3*R*

*I*

_{1}=

*I*

_{2}=

*I*=

*Blv*/

*R*

The emf induced in the sliding connector *PQ* is ε* *= *Blv*, which is shown as a battery of *internal* resistance *R* in the adjoining figure.

The two *external* resistances *R* and *R* appear in parallel with this battery. The effective external resistance is therefore *R*/2 and the battery has to drive the current *I* through a total resistance *R* +* *(*R*/2), which is 3*R*/2

Therefore, I = ε/(3*R*/2) = 2*Blv*/3*R*.

The current *I* gets divided *equally* between the two identical branches and hence *I*_{1} = *I*_{2} = *Blv*/3*R*.

**(2)**In the circuit shown below, the key

*K*is closed at

*t*= 0. The current through the battery is

(1) *V*(*R*_{1 }+ *R*_{2})/*R*_{1}*R*_{2} at *t = *0 and *V/R*_{2} at *t *= ∞

(2) *VR*_{1}*R*_{2}/√(*R*_{1}^{2}+ *R*_{2}^{2}) at *t = *0 and *V/R*_{2} at *t *= ∞

(3) *V/R*_{2} at *t *= 0 and *V*(*R*_{1 }+ *R*_{2})/*R*_{1}*R*_{2} at *t = *∞

(4) *V/R*_{2} at *t *= 0 and *VR*_{1}*R*_{2}/√(*R*_{1}^{2}+ *R*_{2}^{2}) at *t = *∞

The current through the inductance branch is zero at the instant the circuit is switched on. [Remember the exponential growth of current in an LR circuit]

Therefore, at the instant* t* = 0 the current is limited by R_{2} alone and is equal to *V/R*_{2}.

At *t *= ∞ the current has become steady, there is no induced voltage to oppose the flow of current and the inductor functions as a piece of conductor. The current is now limited by *R*_{1 }and *R*_{2} which are in *parallel* with the battery. The current in this case is *V/*[*R*_{1}*R*_{2}/(*R*_{1 }+ *R*_{2})] = *V*(*R*_{1 }+ *R*_{2})/*R*_{1}*R*_{2}, as given in option (3).

You will find similar useful multiple choice question (with solution) here.