The steady state output (V_out), of the circuit shown below, will

The correct option is saturate to -V_EE.

Let's analyze the given circuit diagram step-by-step:

1. Identify the Circuit Configuration:
The circuit represents an op-amp integrator. The non-inverting terminal (+) is connected to ground through a resistor $R_2$, and a feedback capacitor $C_1$ is connected between the inverting input terminal (-) and the output terminal $V_{\text{out}}$. The input source of $0.1\text{V}$ is applied through resistor $R_1$ to the inverting input terminal.

2. Analyze the Terminal Voltages:
Since no current enters the non-inverting input terminal of an ideal operational amplifier, the voltage at the non-inverting terminal is:

$V_{+}=0\text{V}$

Assuming the op-amp is operating in its linear region initially, we can apply the virtual short concept:

$V_{-}=V_{+}=0\text{V}$

3. Determine the Current and Output Voltage:
The current entering the node at the inverting terminal through resistor $R_1$ is:

$I_{\text{in}}=\frac{0.1-V_{-}}{R_1}=\frac{0.1}{R_1}$

Because the input impedance of the op-amp is infinite, this current cannot enter the inverting terminal and must flow entirely through the feedback capacitor $C_1$ towards the output:

$I_c=I_{\text{in}}=\frac{0.1}{R_1}$

The current through the capacitor is related to the voltage across it by:

$I_c=C_1\frac{d(V_{-}-V_{\text{out}})}{dt}=-C_1\frac{d{V}_{\text{out}}}{dt}$

Equating the two expressions for current:

$-C_1\frac{d{V}_{\text{out}}}{dt}=\frac{0.1}{R_1}$

$\frac{d{V}_{\text{out}}}{dt}=-\frac{0.1}{R_1{C}_1}$

Integrating both sides with respect to time $t$:

$V_{\text{out}}\left(t\right)=V_{\text{out}}\left(0\right)-\frac{0.1}{R_1{C}_1}t$

4. Analyze the Steady State Behavior:
The equation shows that the output voltage $V_{\text{out}}$ decreases linearly over time because the input voltage is a constant positive DC value ($0.1\text{V}$). As time $t$ approaches infinity ($t\to \infty$), the output voltage continues to decrease until it hits the negative power supply limit of the op-amp. Since the negative supply voltage rail is $-V_{\text{EE}}$, the output voltage cannot fall below this limit. Therefore, in the steady state, the output will saturate to -V_EE.