https://file.aiquickdraw.com/mathbot/file/1723777608020woiumrqs.jpg

For the circuit above, assume that:

𝑉=2𝑐𝑜𝑠(1𝑡)𝑉
𝑅1=10Ω
𝑅2=9Ω
𝐿1=3𝐻
𝐶1=1/3𝐹

Calculate the current 𝑖𝑡 flowing through the circuit, expressing your answer in the form: 𝐴𝑐𝑜𝑠(𝜔𝑡+𝜙), where 𝜙 is in degrees.

Please round both𝐴 and 𝑝ℎ𝑖 to 3 sig figs, as otherwise the algebra-checker gets confused and awards you no points.

To solve for the current \( i(t) \) flowing through the circuit, we will follow these steps:

### Step 1: Circuit Analysis
The circuit consists of a series combination of:
- Resistor \( R_1 \)
- Inductor \( L_1 \)
- Resistor \( R_2 \)
- Capacitor \( C_1 \)

Given:
- \( V(t) = 2\cos(1t) \) V
- \( R_1 = 10 \, \Omega \)
- \( R_2 = 9 \, \Omega \)
- \( L_1 = 3 \, \text{H} \)
- \( C_1 = \frac{1}{3} \, \text{F} \)

### Step 2: Calculate the Impedances
- The impedance of the inductor \( L_1 \) is \( Z_{L1} = j\omega L_1 \), where \( \omega = 1 \, \text{rad/s} \).
- The impedance of the capacitor \( C_1 \) is \( Z_{C1} = \frac{1}{j\omega C_1} \).

Substitute the values:
- \( Z_{L1} = j(1)(3) = 3j \, \Omega \)
- \( Z_{C1} = \frac{1}{j(1)\left(\frac{1}{3}\right)} = 3j \, \Omega \)

### Step 3: Total Impedance Calculation
The total impedance \( Z_{\text{total}} \) is the sum of the impedances:
\[
Z_{\text{total}} = R_1 + Z_{C1} + R_2 + Z_{L1} 
= 10 + 3j + 9 + 3j 
= 19 + 6j \, \Omega
\]

### Step 4: Calculate the Current
The current \( i(t) \) is given by Ohm's Law:
\[
i(t) = \frac{V(t)}{Z_{\text{total}}}
\]
Substituting the given \( V(t) \) and \( Z_{\text{total}} \):
\[
i(t) = \frac{2\cos(1t)}{19 + 6j}
\]

### Step 5: Convert to Polar Form
Convert the impedance to polar form:
\[
Z_{\text{total}} = \sqrt{19^2 + 6^2} \angle \tan^{-1}\left(\frac{6}{19}\right)
\]
\[
|Z_{\text{total}}| = \sqrt{361 + 36} = \sqrt{397} \approx 19.9 \, \Omega
\]
\[
\theta_Z = \tan^{-1}\left(\frac{6}{19}\right) \approx 17.5^\circ
\]

The voltage is \( V(t) = 2\cos(1t) = 2 \angle 0^\circ \).

So the current in polar form:
\[
i(t) = \frac{2 \angle 0^\circ}{19.9 \angle 17.5^\circ} = \frac{2}{19.9} \angle -17.5^\circ
\]
\[
i(t) \approx 0.1005 \cos(1t - 17.5^\circ)
\]

### Final Answer
The current \( i(t) \) is approximately:
\[
i(t) \approx 0.101 \cos(1t - 17.5^\circ) \, \text{A}
\]

### Questions
Do you need any further details or have any questions? Here are some related problems:

1. How do you convert impedance from rectangular to polar form?
2. What is the significance of the phase angle in AC circuits?
3. How does the inductance of a coil affect the impedance at different frequencies?
4. What is the effect of capacitance on the phase of current in an AC circuit?
5. How would the current change if the frequency of the source was increased?
6. What happens to the total impedance if another resistor was added in series?
7. How can you calculate the resonant frequency of an LC circuit?
8. What would happen to the phase angle if the resistance was increased?

**Tip:** Remember that in AC analysis, phase angles can shift the current and voltage waveforms, which affects power calculations and resonance.

Learn how to calculate the current flowing through an AC circuit consisting of resistors, an inductor, and a capacitor. Step-by-step solution provided for determining current in the form Acos(ωt + φ), rounding to 3 significant figures.

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