Consider an electrical circuit containing a two way switch $$S$$. Initially $$S$$ is open and then $$T_1$$ is connected to $$T_2$$. As the current in $$R = 6 \, \Omega$$ attains a maximum value of steady-state level, $$T_1$$ is disconnected from $$T_2$$ and immediately connected to $$T_3$$. Potential drop across $$r = 3 \, \Omega$$ resistor immediately after $$T_1$$ is connected to $$T_3$$ is _________ V. (Round off to the Nearest Integer)

We need to find the potential drop across the resistor $$r = 3\ \Omega$$ immediately after the switch $$S$$ is flipped from terminal $$T_2$$ to terminal $$T_3$$.

1. Analyze the Initial Steady-State Condition (Switch at $$T_2$$)

Initially, terminal $$T_1$$ is connected to $$T_2$$ for a long time until the circuit reaches a steady state:

• In a DC steady state, an ideal inductor ($$L$$) acts as a perfect short circuit (zero resistance).
• The total resistance in the active loop is simply the single resistor $$R = 6\ \Omega$$.
• The source voltage is $$V = 6\text{ V}$$.

Using Ohm's Law, we calculate the maximum steady-state current ($$I_0$$) flowing through the inductor:

$$I_0 = \frac{V}{R} = \frac{6\text{ V}}{6\ \Omega} = 1\text{ A}$$

2. Analyze the Circuit Immediately After Switching (Switch at $$T_3$$)

At $$t = 0$$, terminal $$T_1$$ is disconnected from $$T_2$$ and instantly connected to $$T_3$$:

• An inductor opposes sudden changes in the current passing through it. Therefore, the current running through the inductor immediately after switching ($$I(0^+)$$) must equal its value just before switching ($$I(0^-)$$).
• This means the current through the new closed loop containing the inductor $$L$$ and the resistor $$r$$ is:

  $$I(0^+) = I_0 = 1\text{ A}$$

3. Calculate the Potential Drop Across $$r$$

Now, we can find the instantaneous potential drop ($$V_r$$) across the resistor $$r = 3\ \Omega$$ using Ohm's Law:

$$V_r = I(0^+) \times r$$

$$V_r = 1\text{ A} \times 3\ \Omega = 3\text{ V}$$

Conclusion

The potential drop across the $$r = 3\ \Omega$$ resistor immediately after the connection change is 3 V.