An LCR circuit connected in series has a resistance of 5hms, an inductance of 0.05 henry, acapacitor of 4×104 farad, and an applied alternating emf of 200 cos 100 tvolts. Find an expressionfor the current flowing through this circuit if the initial current and the initial charge on thecapacitor are both zero
Question
An LCR circuit connected in series has a resistance of 5hms, an inductance of 0.05 henry, acapacitor of 4×104 farad, and an applied alternating emf of 200 cos 100 tvolts. Find an expressionfor the current flowing through this circuit if the initial current and the initial charge on thecapacitor are both zero
Solution
The current in an LCR circuit connected in series is given by the equation:
I(t) = E(t) / Z
where:
- I(t) is the current at time t,
- E(t) is the applied emf at time t, and
- Z is the impedance of the circuit.
The impedance Z of an LCR circuit in series is given by the equation:
Z = sqrt(R^2 + (XL - XC)^2)
where:
- R is the resistance,
- XL is the inductive reactance, given by the equation XL = ωL,
- XC is the capacitive reactance, given by the equation XC = 1/(ωC), and
- ω is the angular frequency of the applied emf.
Given that R = 5 ohms, L = 0.05 henry, C = 4×10^-4 farad, and E(t) = 200 cos(100t) volts, we can find ω from the emf equation:
ω = 100 rad/s
We can now calculate XL and XC:
XL = ωL = 100 * 0.05 = 5 ohms XC = 1/(ωC) = 1/(100 * 4×10^-4) = 2500 ohms
We can now calculate Z:
Z = sqrt(R^2 + (XL - XC)^2) = sqrt((5)^2 + (5 - 2500)^2) = sqrt(25 + 2495^2) = 2495 ohms
Finally, we can find the expression for the current I(t):
I(t) = E(t) / Z = (200 cos(100t)) / 2495 = 0.08016 cos(100t) A
This is the expression for the current flowing through the circuit at time t, given that the initial current and the initial charge on the capacitor are both zero.
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