Final answer:
To find I_L and V_C for the given RLC circuit with initial conditions, more information is required, such as the configuration (series or parallel) and the specific time frame (transient or steady state). For transient analysis, expressions involve exponential functions of time and the time constant.
Step-by-step explanation:
To find the current I_L through an inductor L and the voltage drop V_C across a capacitor C in an RLC circuit with initial conditions, we use the formulas derived from the differential equations governing such circuits. Given that R = 5 ohms, C = 2.5 farads, L = 0.5 henry, an initial current of 0 amp, and an initial voltage of 12 volts, we would typically need to solve these differential equations. However, we need to consider whether the circuit is in series or parallel, as well as the context of the question like whether it's referring to an initial transient period or after a long time has passed (steady state). Without this specific information, and the fact the question text including examples refers to various circuits and components, we cannot provide a precise formula for I_L and V_C.
Generally for an RLC circuit, the current through the inductor I_L after a switch is opened can be described using the formula I_L(t) = I_0e-t/τ_L, where τ_L is the time constant and I_0 is the initial current. The time constant τ_L is given by L/R. For the voltage across the capacitor, the formula is usually more complex, involving an expression that depends on both the charge and discharge rates determined by RC time constants.
Since more information is required to provide an exact formula for this specific RLC circuit, we recommend the student to clarify whether the circuit is series or parallel, and if the analysis is for the transient or steady state condition.