A series RLC circuit, with R = 200ohms, L = 0.1H and C = 10µF, has an initial charge on the capacitor of Q = 2.67mC. A switch is closed at t=0 allowing the capacitor to discharge through the R and L mentioned earlier.The circuit is1. Overdamped2. Underdamped3. Critically dampedIf one of the integration constants (A1 and A2) that come out of the solution to the differential equation is 0, then the other is Answer (Type your answer rounded off to nearest integer, and ignore the negative sign, if any)
Question
A series RLC circuit, with R = 200ohms, L = 0.1H and C = 10µF, has an initial charge on the capacitor of Q = 2.67mC. A switch is closed at t=0 allowing the capacitor to discharge through the R and L mentioned earlier.The circuit is1. Overdamped2. Underdamped3. Critically dampedIf one of the integration constants (A1 and A2) that come out of the solution to the differential equation is 0, then the other is Answer (Type your answer rounded off to nearest integer, and ignore the negative sign, if any)
Solution 1
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Solution 2
To solve this problem, we first need to determine the type of damping in the circuit. This is determined by the damping factor, which is given by the formula:
Damping factor = R / (2 * sqrt(L/C))
Substituting the given values:
Damping factor = 200 / (2 * sqrt(0.1/10e-6)) = 1
A damping factor of 1 indicates that the circuit is critically damped.
In a critically damped RLC circuit, the charge Q on the capacitor as a function of time t is given by the solution to the differential equation:
Q(t) = (A1 + A2t) * e^(-Rt/2L)
Given that the initial charge on the capacitor is Q = 2.67mC, we can solve for the integration constants A1 and A2. At t=0, Q(0) = A1 = 2.67mC.
If one of the integration constants is 0, then the other is equal to the initial charge on the capacitor. Therefore, if A2 = 0, then A1 = 2.67mC.
However, the question asks for the answer in integer form and without a negative sign. The charge is typically measured in coulombs (C), but in this case it's given in milli-coulombs (mC). To convert to coulombs, we divide by 1000:
A1 = 2.67mC / 1000 = 0.00267C
Rounding to the nearest integer gives 0. Therefore, the answer is 0.
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