Final answer:
To find the current i(t) in the given RLC circuit connected to a DC source using Laplace transforms, we calculate the total impedance in the s-domain, use Laplace transform of the circuit equations, and apply inverse Laplace to determine i(t).
Step-by-step explanation:
To analyze the response of the given series RLC circuit using Laplace transforms, we first note that this is a transient analysis problem where the circuit is connected to a DC voltage source of 10V. The values for resistance (R), inductance (L), and capacitance (C) are provided. Given that R=2 ohms, L=0.5H, and C=10μF, we aim to determine the current i(t) through the circuit.
When representing the circuit in the s-domain, the impedance of an inductor is sL, the impedance of a capacitor is 1/(sC), and the resistance remains unchanged. Therefore, the total impedance Z(s) in the s-domain is given by Z(s) = R + sL + 1/(sC). The initial current through the circuit is zero since the capacitor will initially act as an open circuit to the DC source.
By using the Laplace transform of the circuit equations and applying the initial conditions, we can solve for I(s), which is the Laplace transform of i(t). After simplifying and finding I(s), we can take the inverse Laplace transform to get i(t), which will show the transient behavior of the current over time.