# In the circuit shown below, the key K is closed at $\dpi{100} t=0$ . The current through the battery is Option 1) $\frac{V(R_{1}+R_{2})}{R_{1}R_{2}}at\; t=0\; and\; \frac{V}{R_{2}}at\; t=\infty$ Option 2) $\frac{VR_{1}R_{2}}{\sqrt{R_{1}^{2}+R_{2}^{2}}}\, at\; t=0\; and\; \frac{V}{R_{2}}\, at\; t=\infty$ Option 3) $\frac{V}{R_{2}}\, \, at\,t=0\; and\; \, \frac{V(R_{1}+R_{2})}{R_{1}R_{2}}at\; t=\infty$ Option 4) $\frac{V}{R_{2}}at\; t=0\; and\;\frac{VR_{1}R_{2}}{\sqrt{R_{1}^{2}+R_{2}^{2}}}\, at\; t=\infty$

As we learnt in

Self Inductance -

An emf is induced in the coil or the circuit which oppose the change that causes it. Which is also known back  emf.

- wherein

at t=0, there is no current through inductor

$\therefore$ I=0

as $t \rightarrow \infty$

$I=\frac{V}{R_{eq}}$

$=\frac{V/R_{1}R_2}{R_1+R_2}$

$=\frac{V(R_1+R_2)}{R_1+R_2}$

Option 1)

$\frac{V(R_{1}+R_{2})}{R_{1}R_{2}}at\; t=0\; and\; \frac{V}{R_{2}}at\; t=\infty$

Incorrect option

Option 2)

$\frac{VR_{1}R_{2}}{\sqrt{R_{1}^{2}+R_{2}^{2}}}\, at\; t=0\; and\; \frac{V}{R_{2}}\, at\; t=\infty$

Incorrect option

Option 3)

$\frac{V}{R_{2}}\, \, at\,t=0\; and\; \, \frac{V(R_{1}+R_{2})}{R_{1}R_{2}}at\; t=\infty$

Correct option

Option 4)

$\frac{V}{R_{2}}at\; t=0\; and\;\frac{VR_{1}R_{2}}{\sqrt{R_{1}^{2}+R_{2}^{2}}}\, at\; t=\infty$

Incorrect option

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