Answer to Question #107018 in Differential Equations for Gayatri Yadav

Question #107018
A series RLC circuit with R = 6 ohm, C = 0.02 Farad and L = 0.1 has no applied voltage. Find the subsequent current in the circuit if the initial charge, on the capacitor is q0 and the initial current is zero.
1
Expert's answer
2020-04-02T13:05:52-0400

The magnitude of the subsequent current can be obtained from the equation of damped oscillations : "\\ddot{q}+2\\beta\\dot{q}+\\omega_0^2q=0" , where "\\beta=\\frac{R}{2L},\\omega_0=\\frac{1}{\\sqrt{LC}}"

In our case "\\omega_0^2-\\beta^2<0" then is a case of aperiodic damping. The equation solution is

"q=c_1e^{-\\alpha_1t}+c_2e^{-\\alpha_2t}" where "\\alpha_1=\\beta+\\sqrt{\\beta^2-\\omega_0^2}" , "\\alpha_2=\\beta-\\sqrt{\\beta^2-\\omega_0^2}"

From the conditions "q(0)=q_0=c_1+c_2" and "\\dot{q}(0)=I(0)=0=\\alpha_1c_1+\\alpha_2c_2" we find

"c_1=-\\frac{\\alpha_2q_0}{\\alpha_1-\\alpha_2}" ,"c_2=\\frac{\\alpha_1q_0}{\\alpha_1-\\alpha_2}" then "\\dot{q}(t)=I(t)=q_0\\frac{\\alpha_1\\alpha_2}{\\alpha_1-\\alpha_2}(e^{-\\alpha_1t}-e^{-\\alpha_2t})" . As "\\alpha_1=50, \\alpha_2=10" then

"I(t)=12.5q_0(e^{-50t}-e^{-10t})" A


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