To determine the exact voltages at points a and b, we need to use complex number analysis and phasor techniques since we're dealing with AC circuits in sinusoidal steady state. Here's a breakdown of how to find the voltages:

🪵Rough Schematic

1. Define the circuit parameters in the frequency domain:

• Current source: Let's assume the current source is given by $is(t) = I_s * cos(ωt)$, where $I_s$ is the amplitude of the current and ω = 2πf is the angular frequency. In phasor form, this becomes $I_s$.
• Resistors: Resistors remain the same in the frequency domain: 10 Ω and 1 Ω.
• Capacitor: The impedance of the capacitor is $Z_C = 1/(jωC) = 1/(jω)$.
• Inductor: The impedance of the inductor is $Z_L = jωL = jω$.

2. Analyze the circuit using nodal analysis:

• Node a: Let's denote the voltage at node a as $V_a$.
• Node b: Let's denote the voltage at node b as $V_b$.

3. Apply Kirchhoff's Current Law (KCL) at node a:

The sum of currents entering node a must be equal to the sum of currents leaving node a.

• Current from the current source: $I_s$
• Current through the 10 Ω resistor: $(I_s - V_a) / 10$
• Current through the capacitor: $V_a / Z_C = jωV_a$
• Current through the 1 Ω resistor and inductor: $(V_a - V_b) / (1 + jω)$

Applying KCL:

$$ I_s = (I_s - V_a) / 10 + jωV_a + (V_a - V_b) / (1 + jω) $$

4. Express $V_b$ in terms of $V_a$:

The voltage at node b is simply the voltage drop across the inductor and the 1 Ω resistor: