but we had to find v(t) accross capacitor 1/20 F

My apologies for the oversight. Let's determine the voltage \( v_C(t) \) across the capacitor \( \frac{1}{20} \, \text{F} \).

### Step 1: Recalculate the Current \( I(t) \)
We already calculated the current in the circuit as:

\[
I(t) = 0.958 \cos(10t + 58.3^\circ) \, \text{A}
\]

### Step 2: Calculate the Voltage Across the Capacitor
The voltage across the capacitor is given by:

\[
v_C(t) = I(t) \cdot Z_C
\]

Where the impedance of the capacitor \( Z_C \) is:

\[
Z_C = -j2 \, \Omega
\]

So,

\[
v_C(t) = 0.958 \cos(10t + 58.3^\circ) \times (-j2)
\]

### Step 3: Simplify the Expression
Now let's calculate \( v_C(t) \) using the current:

1. Convert the current into its phasor form:
   \[
   I(t) = 0.958 \angle 58.3^\circ \, \text{A}
   \]
2. The voltage across the capacitor will then be:
   \[
   V_C = I \cdot Z_C = 0.958 \angle 58.3^\circ \times (-j2)
   \]

   Multiplying by \( -j \) is equivalent to rotating the phasor by \( -90^\circ \):

   \[
   V_C = 0.958 \times 2 \angle (58.3^\circ - 90^\circ) = 1.916 \angle -31.7^\circ \, \text{V}
   \]

### Step 4: Convert Back to Time Domain
Finally, convert the phasor \( V_C \) back to the time domain:

\[
v_C(t) = 1.916 \cos(10t - 31.7^\circ) \, \text{V}
\]

### Final Answer:
The voltage across the capacitor \( v_C(t) \) is:

\[
v_C(t) = 1.916 \cos(10t - 31.7^\circ) \, \text{V}
\]

Would you like further details or have any questions?

---

Here are some related questions to consider:
1. How would the voltage across the capacitor change if the capacitance were halved?
2. What is the phase difference between the source voltage and the capacitor voltage?
3. Can you find the voltage across the inductor as well?
4. How does the frequency of the source affect the impedance of the capacitor?
5. What would be the effect on the circuit if the resistor value is increased?

**Tip:** When dealing with capacitors and inductors, remember that their impedances are frequency-dependent, and they introduce phase shifts in the circuit, which is crucial for accurate analysis.

Learn how to calculate the voltage across a capacitor in an electric circuit with a detailed step-by-step solution. This problem involves finding the voltage using current and impedance, incorporating phasor analysis. Suitable for advanced high school and college-level students.

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