A series circuit contains a resistor with R = 40Ω, an inductor with L = 2 H, a capacitor with
0.0025 F , and a 12 − V battery. The initial charge is Q = 0.001 C and the initial current is
0. Using the method of undetermined coefficients, find the charge at time t
To find the charge at time t, we need to solve the differential equation that describes the behavior of the circuit. The differential equation for a series RLC circuit is given by:
L * d^2Q/dt^2 + R * dQ/dt + (1/C) * Q = 0
where:
- L is the inductance of the inductor (2 H in this case)
- R is the resistance of the resistor (40 Ω in this case)
- C is the capacitance of the capacitor (0.0025 F in this case)
- Q is the charge at time t
To solve this equation using the method of undetermined coefficients, we assume a solution of the form:
Q(t) = A * cos(rt) + B * sin(rt)
where A and B are constants to be determined, and r is the natural frequency of the circuit, given by:
r = sqrt(1/(LC) - (R/(2L))^2)
To determine the values of A and B, we need to plug our assumed solution into the differential equation and solve for A and B.
First, we need to take the first and second derivatives of Q(t) with respect to t:
dQ/dt = -A * r * sin(rt) + B * r * cos(rt)
d^2Q/dt^2 = -A * r^2 * cos(rt) - B * r^2 * sin(rt)
Next, we substitute these derivatives into the differential equation:
L * (-A * r^2 * cos(rt) - B * r^2 * sin(rt)) + R * (-A * r * sin(rt) + B * r * cos(rt)) + (1/C) * (A * cos(rt) + B * sin(rt)) = 0
Now, we can simplify the equation by grouping the terms with the same trigonometric functions:
(-r^2LC * A * cos(rt) + rLC * B * sin(rt)) + (-RrL * A * sin(rt) - RLC * B * cos(rt)) + (1/C) * (A * cos(rt) + B * sin(rt)) = 0
Next, we collect the terms with the same trigonometric functions:
[A * ((1/C) - r^2LC) + B * (-RrL + (1/C))] * cos(rt) + [A * (-RrL - (1/C)) + B * ((1/C) - RLC)] * sin(rt) = 0
Since the equation must hold for all values of t, the coefficients multiplying the cos(rt) and sin(rt) terms must both be zero. This gives us a system of equations to solve for A and B:
(1/C - r^2LC) * A - (RrL - 1/C) * B = 0
-RrL * A + (1/C - RLC) * B = 0
Now we have a system of linear equations, which we can solve to find the values of A and B. Once we have determined A and B, we plug them back into our assumed solution Q(t) = A * cos(rt) + B * sin(rt) to obtain the final expression for the charge Q at time t.