Question

Find \(i(t)\) in the following circuit, given \( R = \frac{1}{3} \, \Omega \), \( L = \frac{1}{4} \, H \), \( C = 3 \, F \), and \( v(t) = \sin 2t \) 

• A. \( 5 \sin(2t + 53.1^\circ) \)
• B. \( 5 \sin(2t - 53.1^\circ) \)
• C. \( 25 \sin(2t + 53.1^\circ) \)
• D. \( 25 \sin(2t - 53.1^\circ) \)

Hint:

For AC circuits with RLC components, the phase shift in the current is determined by the impedance of the circuit, which depends on the resistance, inductance, and capacitance.

Answer 1

The Correct Option is C

In an RLC circuit with sinusoidal voltage, the impedance can be expressed as: \[ Z = R + j\left( \omega L - \frac{1}{\omega C} \right) \] where \( \omega = 2 \) rad/s (angular frequency), \( R = \frac{1}{3} \, \Omega \), \( L = \frac{1}{4} \, H \), and \( C = 3 \, F \). The impedance will give us the current, which is: \[ I(t) = \frac{V(t)}{Z} \] Substituting the given values into the formula, we get the current as \( 25 \sin(2t + 53.1^\circ) \), where the phase angle comes from the argument of the impedance.
Final Answer: \[ \boxed{25 \sin(2t + 53.1^\circ)} \]