Critical damping occurs in a series RLC circuit when the resistance R satisfies which condition in terms of L and C?

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Multiple Choice

Critical damping occurs in a series RLC circuit when the resistance R satisfies which condition in terms of L and C?

Explanation:
In a series RLC circuit, the behavior of the natural (homogeneous) response is governed by the characteristic equation from L d^2i/dt^2 + R di/dt + (1/C) i = 0. Try i ~ e^{st}, which gives L s^2 + R s + 1/C = 0. Critical damping happens when the two roots are real and equal, meaning the discriminant is zero: R^2 − 4L(1/C) = 0. Solving gives R^2 = 4L/C, so R = 2√(L/C) (taking the positive resistance). At this value, the system returns to equilibrium without oscillating and as quickly as possible. If R is smaller, you get underdamping with oscillations; if larger, overdamping with slower, non-oscillatory decay.

In a series RLC circuit, the behavior of the natural (homogeneous) response is governed by the characteristic equation from L d^2i/dt^2 + R di/dt + (1/C) i = 0. Try i ~ e^{st}, which gives L s^2 + R s + 1/C = 0. Critical damping happens when the two roots are real and equal, meaning the discriminant is zero: R^2 − 4L(1/C) = 0. Solving gives R^2 = 4L/C, so R = 2√(L/C) (taking the positive resistance). At this value, the system returns to equilibrium without oscillating and as quickly as possible. If R is smaller, you get underdamping with oscillations; if larger, overdamping with slower, non-oscillatory decay.

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