Discussion Forum

In the figure  below , the switches S₁ and S₂  are closed simultaneously  at t=0 and  a current starts to flow in the circuit.  Both the batteries have  the same magnitude of the electromotive force (emf) and the polarities are as indicated in the figure. Ignore mutual inductance between the inductors. The current I in the middle wires reaches its maximum magnitude I_max at time $t=\tau$ . which of the following statements is (are) true ?

Answer:

Explanation:

$I_{1}=\frac{V}{R}(1-e^-{\frac{tR}{L}}$

$I_{2}=\frac{V}{R}(1-e^-{\frac{tR}{2L})}$

  From principle of superposition,

$I=I_{1}-I_{2}\Rightarrow I=\frac{V}{R}e^{-\frac{tR}{2L}}(1-e^{-\frac{tR}{2L}})$        .......(i)

I is maximum when $\frac{\text{d}I}{\text{d}t}=0$ , which gives

$e^{-\frac{IR}{2L}}=\frac{1}{2}$   or  $t =\frac{2L}{R}In 2$

 Substituting this time in Eq. (i), we get

      $I_{max}=\frac{V}{4R}$