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The problem involves solving for the steady-state current \( I(t) \) in an RLC circuit, given values for the components: \( R = 10 \, \Omega \), \( L = 1 \, \text{H} \), \( C = 0.1 \, \text{F} \), and a periodic voltage \( E(t) \) with a period of \( 2\pi \). The voltage function \( E(t) \) is defined as:
\[
E(t) = 
\begin{cases} 
-50t^2 & \text{if } -\pi < t < 0 \\
50t^2 & \text{if } 0 < t < \pi 
\end{cases}
\]
This piecewise function implies that \( E(t) \) is an even function due to its symmetry, and it's periodic with period \( 2\pi \).

To find the steady-state current \( I(t) \), we need to solve the differential equation derived from Kirchhoff's Voltage Law (KVL) for an RLC circuit:
\[
L \frac{d^2 I}{dt^2} + R \frac{dI}{dt} + \frac{1}{C} I = \frac{dE}{dt}.
\]
Since \( E(t) \) is given, \( \frac{dE}{dt} \) will also be a piecewise function based on the derivative of \( 50t^2 \) within each interval. After obtaining \( \frac{dE}{dt} \), the Fourier series expansion can be used to express \( E'(t) \) as a sum of sines and cosines. The solution for \( I(t) \) will involve calculating the response for each harmonic.

Would you like the full solution including the Fourier series computation and the first four partial sums?

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1. How do we determine the differential equation for an RLC circuit?
2. What is the Fourier series of \( E'(t) \) for a piecewise quadratic function?
3. Why is the steady-state response focused on harmonics instead of transient solutions?
4. What method can we use to solve the non-homogeneous differential equation with periodic forcing?
5. How does symmetry in \( E(t) \) simplify Fourier series calculations?

**Tip:** For periodic functions, Fourier series expansions are powerful for obtaining steady-state solutions in RLC circuits.

Find the steady-state current I(t) in the RLC circuit where R = 10 Ω, L = 1 H, C = 10^(-1) F, and with periodic E(t) as defined in the image.

Solve for the steady-state current I(t) in an RLC circuit with given values of R, L, and C, and a periodic voltage function E(t). Learn how to use differential equations and Fourier series to find the current response in this electrical circuit problem.

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