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.
(a) 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?
3.6*10^-3
Gotcha cheater
To find the value of the "b" coefficient in the differential equation d2V/dt2 + a*dV/dt + b*V = 0, we need to use the given values of R, L, and C in the series RLC circuit.
Given: R = 251 Ω, L = 0.07 H, C = 2e-06 F
From the given information, we know that the circuit is underdamped, meaning R^2 < 4LC. In this case, we have R^2 = (251 Ω)^2 = 63001 Ω^2, 4LC = 4 * (0.07 H) * (2e-06 F) = 5.6e-06 Ω^2.
Since 63001 Ω^2 > 5.6e-06 Ω^2, we can conclude that R^2 is greater than 4LC.
Therefore, the "b" coefficient in the differential equation can be determined as follows:
b = 1 / (2 * sqrt(LC))
Let's substitute the given values into the formula:
b = 1 / (2 * sqrt(0.07 H * 2e-06 F))
Simplifying:
b = 1 / (2 * sqrt(0.07 * 2 * 1e-06))
b = 1 / (2 * sqrt(0.14e-06))
b = 1 / (2 * 0.0118321596)
b ≈ 1 / 0.0236643192
b ≈ 42.2303618
Therefore, the value of "b" in the differential equation is approximately 42.2303618.