As shown in the figure, in steady state, the charge stored in the capacitor is _____ $$\times 10^{-6}$$ C.

We need to find the charge stored in the capacitor under steady-state conditions for the given circuit.

1. Understand the Steady-State Condition

In a DC circuit, once a capacitor is fully charged and reaches a steady state, it acts as an open circuit (infinite resistance). This means:

• No current flows through the branch containing the capacitor ($$C$$) and the resistor ($$R'$$).
• Current only flows through the closed loop containing the battery ($$E$$), its internal resistance ($$r$$), and the parallel resistor ($$R$$).

2. Calculate the Circuit Current ($$I$$)

Using the values given in the circuit diagram 

• EMF of the battery ($$E$$) = $$10\text{ V}$$
• Internal resistance ($$r$$) = $$10\ \Omega$$
• External resistance ($$R$$) = $$100\ \Omega$$

The total current flowing through the main loop is:

$$I = \frac{E}{R + r} = \frac{10}{100 + 10} = \frac{10}{110} = \frac{1}{11}\text{ A}$$

3. Determine the Potential Difference Across the Capacitor ($$V_C$$)

Since no current flows through the branch with $$R' = 200\ \Omega$$, there is no potential drop across it ($$V_{R'} = 0$$). Therefore, the potential difference ($$V_C$$) across the capacitor plates is exactly equal to the potential difference across the parallel resistor ($$R$$):

$$V_C = I \times R$$

$$V_C = \frac{1}{11}\text{ A} \times 100\ \Omega = \frac{100}{11}\text{ V}$$

4. Calculate the Stored Charge ($$Q$$)

The capacitance value given is $$C = 1.1\ \mu\text{F} = 1.1 \times 10^{-6}\text{ F}$$. Using the formula for charge stored in a capacitor ($$Q = C \times V_C$$):

$$Q = 1.1 \times 10^{-6}\text{ F} \times \frac{100}{11}\text{ V}$$

We can rewrite $$1.1$$ as $$\frac{11}{10}$$ to simplify the expression easily:

$$Q = \left(\frac{11}{10} \times 10^{-6}\right) \times \frac{100}{11}$$

$$Q = \frac{100}{10} \times 10^{-6}\text{ C} = 10 \times 10^{-6}\text{ C}$$

Conclusion

The charge stored in the capacitor in steady state is 10 $$\times 10^{-6}\text{ C}$$.