Consider the series RLC circuit in which R2<4L/C (underdamped).
Assume that at t=0 , the charge on the capacitor has its maximum value.
The differential equation obeyed by the potential across the capacitor can be written in the following form
d2V/dt2+a*dV/dt+b*V=0
In the circuit above, R=251 Ohm, L=0.07 H, C=2e-06 F.
(b) What is the decay constant (in seconds) according to which the charge on the capacitor is decaying?
(c) The times at which the total energy stored in the RLC circuit is exclusively of electric nature can be written as
t= a+bn n=0,1,2,3,…
What is b?
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To find the decay constant (a) in the given series RLC circuit, we first need to determine the values of a and b in the differential equation:
d^2V/dt^2 + a * dV/dt + b * V = 0
We know that the decay constant is given by the formula:
a = (R / 2L)
Given that R = 251 Ohms and L = 0.07 H, we can calculate a:
a = (251 / (2 * 0.07)) = 1792.8571
Therefore, the decay constant (a) is approximately 1792.8571 seconds.
To find b, we need to calculate the value of (b * C) in the differential equation:
b * C = (1 / (4L * C - R^2))
Given that R = 251 Ohms, L = 0.07 H, and C = 2e-06 F, we can calculate (b * C):
(b * C) = (1 / (4 * 0.07 * 2e-06 - (251^2)))
(b * C) = (1 / (5.6e-05 - 63001))
(b * C) = (1 / -62999.99994)
(b * C) = -1.5873e-08
So, the value of (b * C) is approximately -1.5873e-08.
Finally, since the equation for the times at which the total energy stored in the RLC circuit is exclusively of electric nature is given as:
t = a + bn
We can determine b by comparing the given equation with the form above. It can be observed that b is the coefficient multiplied by n. Therefore, b is equal to (b * C) multiplied by the capacitance (C):
b = (b * C) * C
b = (-1.5873e-08) * (2e-06)
b = -3.1746e-14
Therefore, the value of b is approximately -3.1746e-14.