An electric circuit consists of an inductance 0.1 henry, a resistance of 20ohms and a condenser of capacitance 25 microfarads. Calculate the charge and the current in the circuit at time any time, given that initial current and charge are 0 and 0.05 respectively
Question
An electric circuit consists of an inductance 0.1 henry, a resistance of 20ohms and a condenser of capacitance 25 microfarads. Calculate the charge and the current in the circuit at time any time, given that initial current and charge are 0 and 0.05 respectively
Solution
The given circuit is an LCR circuit (Inductor, Resistor, Capacitor). The charge q(t) and current i(t) in the circuit can be found using the following formulas:
- q(t) = Qe^(-Rt/2L)cos(wd*t + φ)
- i(t) = dq(t)/dt
Where:
- Q is the maximum charge (initial charge)
- R is the resistance
- L is the inductance
- t is the time
- wd is the damped angular frequency = sqrt((1/LC) - (R^2/4L^2))
- φ is the phase constant = arctan((Rsqrt(C/L))/(1 - R^2sqrt(C/4L)))
Given:
- Q = 0.05 C
- R = 20 Ω
- L = 0.1 H
- C = 25 μF = 25*10^-6 F
Let's calculate wd and φ:
wd = sqrt((1/(0.12510^-6)) - ((20^2)/(4*0.1^2))) = sqrt((10^6) - (10^4)) = 990 rad/s
φ = arctan((20sqrt(2510^-6/0.1))/(1 - 20^2sqrt(2510^-6/(4*0.1)))) = arctan((50)/(1 - 500)) = -1.57 rad
Now, we can substitute these values into the equations for q(t) and i(t):
q(t) = 0.05e^(-20t/2*0.1)cos(990t - 1.57) i(t) = dq(t)/dt = derivative of q(t) with respect to t
To find the derivative of q(t), you can use the chain rule and product rule of differentiation. The derivative of e^(-10t) is -10e^(-10t), and the derivative of cos(990t - 1.57) is -990sin(990t - 1.57). Applying the product rule gives:
i(t) = -10*0.05e^(-10t)cos(990t - 1.57) - 0.05e^(-10t)*990sin(990t - 1.57)
So, the charge and current in the circuit at any time t are given by q(t) and i(t) respectively.
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