Question

The parameters of an RLC circuit with an input voltage E(t) are given. Find the current I(t) and the charge on the capacitor. Use the appropriate equation LI′′ + RI′ + (1/C)I = E′(t). (a) R =

Answer 1

The final answer cannot be provided without the specific value of the resistance (R) in the RLC circuit.

Step-by-step explanation:

The given second-order linear differential equation for the RLC circuit is
`\(LI'' + RI' + (1)/(C)I = E'(t)\), where \(I(t)\) is the current, \(L\)` is the inductance,
`\(R\)`is the resistance,
`\(C\) is the capacitance, and \(E(t)\)` is the input voltage. To find the current
`(\(I(t)\))`, one needs to solve this differential equation, but the solution depends on the specific values of
`\(R\), \(L\), and \(C\).`

In general, solving such equations involves finding the complementary function and particular integral. The complementary function captures the behavior of the circuit without the external input
`(\(E(t)\)`, while the particular integral represents the response to the input. The complete solution is the sum of these two components.

The charge on the capacitor is related to the current by the equation
`\(Q(t) = \int I(t) dt\). So, once \(I(t)\)` is determined, the charge on the capacitor can be found by integrating the current with respect to time.

Without the specific value for
`\(R\)`, it's not possible to provide the numerical solution for
`\(I(t)\)` or the charge on the capacitor. However, this explanation outlines the general approach to solving the differential equation and finding the current and charge in an RLC circuit with the given parameters.

Answer 2

To determine the current I(t) and charge on the capacitor in the RLC circuit, the value of R is needed. Without this specific value, calculating the current and charge isn't feasible using the provided equation.

Step-by-step explanation:

In an RLC circuit, the behavior of current and charge on the capacitor is governed by the second-order linear differential equation LI′′ + RI′ + (1/C)I = E′(t), where I(t) represents the current, L is the inductance, R is the resistance, C is the capacitance, and E(t) is the input voltage.

To solve for current (I(t)) and the charge on the capacitor, it's crucial to have the resistance value (R). Without it, a specific solution cannot be derived. The differential equation is solvable with known values of L, R, and C, considering the input voltage E(t) as well.

For a complete solution, one would use various methods like Laplace transforms or the method of undetermined coefficients, depending on the nature of the input signal and initial conditions. Given the equation's second-order nature, specifying initial conditions (such as initial current or charge on the capacitor) would further aid in determining the unique solution for I(t) and the capacitor's charge over time.

Hence, without the value of R, a specific solution for the current (I(t)) and the charge on the capacitor cannot be derived, as it's a crucial parameter in solving the differential equation governing the circuit's behavior.