Final answer:
The current i(t) in the RLC circuit with the given parameters is 2√2 sin(wt - 30°), which does not match any of the provided options.
Step-by-step explanation:
Finding the Current i(t) in an RLC Circuit
In the electrical circuit described, with a voltage source U = 10√2 sin(wt+30°), resistance R = 3 ohms, inductive reactance XL = 8 ohms, and capacitive reactance XC = 4 ohms, we're asked to determine the value of the current i(t). To solve this, we must consider the impedance and the phase difference φ between the voltage and the current.
The total impedance Z in an RLC circuit is given by Z = √(R^2 + (XL - XC)^2). Here, XL > XC, indicating that the circuit is inductively dominant. The net reactance is XL - XC = 4 ohms, and hence the impedance Z = √(3^2 + 4^2) = 5 ohms. The amplitude of the current I0 can be found by dividing the voltage amplitude by the impedance, I0 = Vo/Z = (10√2)/5 = 2√2 A.
To find the phase angle φ, we use the fact that tan(φ) = (XL - XC)/R. Since XL - XC = 4 ohms and R = 3 ohms, φ = arctan(4/3), which is approximately 53°. However, because the voltage leads the current by 30° and the inductive circuit causes the current to lag, the phase angle φ of the current behind the voltage is 30°. Therefore, the correct expression for the current is i(t) = I0 sin(wt - φ), or i(t) = 2√2 sin(wt - 30°), which matches none of the options provided in the multiple-choice question, indicating a possible error in the options.