We are given a simple series RLC circuit with the following information:

• $V_C(t) = 4 \cos(105t)$ V (capacitor voltage)
• Inductance $L = 2 \, \text{mH} = 2 \times 10^{-3} \, \text{H}$
• Capacitance $C = 80 \, \text{nF} = 80 \times 10^{-9} \, \text{F}$
• We are asked to find $V_S(t)$, the source voltage.

Step 1: Analyze the circuit

Since this is an RLC series circuit, the total voltage $V_S(t)$ is the sum of the voltage across the inductor $V_L(t)$, the capacitor $V_C(t)$, and any resistor (though there doesn't seem to be a resistor in this case, we assume the resistance is negligible).

The relationship between the current and voltages in an RLC circuit is given by:

• Voltage across the inductor: $V_L(t) = L \frac{dI(t)}{dt}$
• Voltage across the capacitor: $V_C(t) = \frac{1}{C} \int I(t) \, dt$
• The current through the inductor and capacitor is the same, as they are in series.

Step 2: Obtain the angular frequency

From the given capacitor voltage $V_C(t) = 4 \cos(105t)$, we see that the angular frequency $\omega = 105 \, \text{rad/s}$.

Step 3: Find the impedance of the inductor and capacitor

• The impedance of the inductor $Z_L = j\omega L = j(105)(2 \times 10^{-3}) = j0.21 \, \Omega$
• The impedance of the capacitor $Z_C = \frac{1}{j\omega C} = \frac{1}{j(105)(80 \times 10^{-9})} \approx -j119.05 \, \Omega$

Step 4: Calculate the total impedance

The total impedance $Z_{total}$ is the sum of the inductor and capacitor impedances:

$Z_{total} = Z_L + Z_C = j0.21 - j119.05 = -j118.84 \, \Omega$

Step 5: Relate the