To solve this problem, we need to analyze the steady-state current $I(t)$ in the $RLC$-circuit given the parameters and the form of $E(t)$ as a piecewise function. The following steps outline the approach:

Problem Breakdown

1. Given Parameters:

   • Resistance $R = 10 \, \Omega$
   • Inductance $L = 1 \, \text{H}$
   • Capacitance $C = 0.1 \, \text{F}$
   • Forcing function $E(t)$ is defined as: -50t^2 & \text{if } -\pi < t < 0, \\ 50t^2 & \text{if } 0 < t < \pi. \end{cases}$$
   • $E(t)$ is periodic with period $2\pi$.

2. Differential Equation: Since this is an $RLC$-circuit, the governing differential equation for the current $I(t)$ (with $E'(t)$ as the driving function) is: $L \frac{d^2 I}{dt^2} + R \frac{dI}{dt} + \frac{1}{C} I = E'(t).$ Substituting the values of $L$, $R$, and $C$, the equation becomes: $\frac{d^2 I}{dt^2} + 10 \frac{dI}{dt} + 10 I = E'(t).$

3. Fourier Series for $E(t)$: Since $E(t)$ is a periodic function with period $2\pi$, we can express it as a Fourier series. We need to determine the Fourier coefficients, which involve calculating:

   • The constant term $a_0$.
   • The coefficients $a_n$ and $b_n$ for the cosine and sine terms, respectively.

4. Steady-State Solution: We will find the steady-state solution $I(t)$ by solving the differential equation for each Fourier component of $E'(t)$ separately. Each term in the Fourier series of $E'(t)$ will produce a corresponding response in $I(t)$.

5. Sketch the First Four Partial Sums: After determining the Fourier components, we can construct and sketch the first four partial sums to approximate $I(t)$.

Solution Outline

1. Compute $E'(t)$: Find $E'(t)$ by differentiating $E(t)$ piecewise.

2. Fourier Series of $E(t)$: Since $E(t)$ is piecewise-defined, calculate the Fourier coefficients by integrating over each interval.

3. Solve the Differential Equation for Each Fourier Term: Substitute each Fourier component of $E'(t)$ into the differential equation and solve for the corresponding term in $I(t)$.

4. Construct the Partial Sums: Combine the solutions for each term to build the partial sums.

Would you like me to proceed with a detailed calculation for each of these steps?

Further Related Questions:

1. How do we compute the Fourier series coefficients for a piecewise quadratic function?
2. How does the periodicity of $E(t)$ affect the steady-state solution $I(t)$?
3. What role does each circuit parameter $R$, $L$, and $C$ play in shaping the response $I(t)$?
4. How would the solution change if the function $E(t)$ were defined differently?
5. Can we use a Laplace transform to solve this differential equation instead?

Tip:

For $RLC$-circuit problems with a periodic input, decomposing the input into its Fourier series is a powerful method for finding steady-state solutions.