Answer:
Step-by-step explanation:
To find the current I(t) in the RLC circuit, we can use the differential equation that describes the circuit behavior. The equation is given by:
L dI/dt + RI + Q/C = E(t)
where:
L is the inductance (4 H)
R is the resistance (32 Ω)
C is the capacitance (0.01 F)
Q is the charge on the capacitor (initially Q(0) = 3.1 C)
E(t) is the input voltage (10 V)
First, let's differentiate the charge Q(t) with respect to time to obtain the current I(t):
dQ/dt = C dV/dt
Since V(t) = Q(t)/C, we can rewrite the equation as:
dQ/dt = C d^2Q/dt^2
Substituting these expressions into the differential equation and rearranging terms, we get:
L dI/dt + RI + Q/C = E(t)
L d^2Q/dt^2 + R dQ/dt + Q/C = E(t)
Now, we can solve this second-order linear homogeneous ordinary differential equation with constant coefficients to find Q(t), and then calculate I(t) by differentiating Q(t) with respect to time.
Given the initial conditions I(0) = 0 and Q(0) = 3.1 C, we can solve for the values of Q(t) and I(t).
However, the input voltage E(t) is not specified, so we need additional information about the input voltage waveform or function to solve the differential equation and obtain the complete expression for I(t).