A series circuit consist of a resister with R =200 ohm, an inductor with L=1H, a capacitor with C = 0.002F, and a 12V battery.If the initial charge and current are both 0, find the charge and current at time t.

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To find the charge (Q) and current (I) at time (t) in a series circuit with a resistor, inductor, capacitor, and a battery, we need to solve the differential equation that describes the circuit behavior.

The equation for the series circuit is:

L(dI/dt) + (1/C)q + RI = V

Where:
L is the inductance (1H)
C is the capacitance (0.002F)
R is the resistance (200 ohm)
V is the battery voltage (12V)
q is the charge on the capacitor (C)

In this problem, we know that the initial charge (q0) and current (I0) are both zero. So, we can rewrite the equation as:

L(dI/dt) + (1/C)q + RI = V

Now, we need to solve this differential equation to find I(t) and q(t).

Step 1: Rearrange the equation

L(dI/dt) = - (1/C)q - RI + V

Step 2: Differentiate both sides of the equation with respect to t

L(d^2I/dt^2) = - (1/C)dq/dt - R(dI/dt)

Step 3: Substitute q = CV into the equation

L(d^2I/dt^2) = - (1/C)(CdV/dt) - R(dI/dt)

Step 4: Simplify the equation

L(d^2I/dt^2) = - (dV/dt) - R(dI/dt)

Step 5: Substitute V = 12V into the equation

L(d^2I/dt^2) = - 12 - R(dI/dt)

Step 6: Solve the second-order homogeneous linear differential equation using the characteristics equation

The characteristics equation is Lm^2 + Rm + 12 = 0

Solve this equation to find the roots m1 and m2.

Step 7: Write the general solution for the differential equation

I(t) = Ae^(m1t) + Be^(m2t)

Where A and B are constants determined by the initial conditions.

Step 8: Substitute the initial condition I0 = 0 into the general solution

0 = Ae^(m1*0) + Be^(m2*0)

Simplify this equation to find the relationship between A and B.

Step 9: Substitute the initial condition q0 = 0 into the equation q = CV

0 = C(Ve^(m1*0) + Ve^(m2*0))

Simplify this equation to find the relationship between A and B.

Step 10: Solve the initial conditions equations to find the values of A and B.

Step 11: Substitute the values of A and B into the general solution

I(t) = Ae^(m1t) + Be^(m2t)

Step 12: Substitute the values of m1 and m2 into the general solution

I(t) = Ae^(m1t) + Be^(m2t)

Step 13: Substitute the values of A and B into the general solution

I(t) = Ae^(m1t) + Be^(m2t)

Now, with the specific values of L, C, R, and V, you can solve the differential equation using the given initial conditions to find the charge (Q) and current (I) at time (t).

To find the charge and current at time t in a series circuit, we need to calculate the values using the principles of circuit analysis.

First, let's find the time constant (τ) of the circuit. The time constant is the amount of time it takes for the voltage or current in a circuit to reach 63.2% of the way towards its final value.

The time constant (τ) of an RC circuit is given by the formula:
τ = R * C

In this case, the resistance (R) is 200 ohms and the capacitance (C) is 0.002F. Therefore, the time constant (τ) is:
τ = 200 ohms * 0.002F
τ = 0.4 seconds

Next, we can use the time constant to find the charge (Q(t)) in the circuit at time t. The charge in a series RC circuit is given by the formula:
Q(t) = Q_max * (1 - e^(-t/τ))

In this formula, Q_max is the maximum charge, which initially starts at zero in this case. So we can simplify the equation to:
Q(t) = Q_max * (1 - e^(-t/τ))
= Q_max * (1 - e^(-t/0.4))

Since the initial charge (Q(0)) is zero, we can find the charge at time t using the above formula.

To find the current (I(t)) at time t, we'll use Ohm's Law and the voltage across the circuit components. The total voltage in the circuit is provided as 12V by the battery, and it is shared between the resistor, inductor, and capacitor:

Ohm's Law for the resistor:
V_R = I(t) * R

Ohm's Law for the inductor:
V_L = L * dI(t)/dt

Ohm's Law for the capacitor:
V_C = (1/C) * ∫[0,t] (I(t)) dt

Since we know the voltage across each component and the total voltage, we can solve these equations simultaneously to find the current in the circuit at time t.

I hope this explanation helps you understand the steps involved in finding the charge and current at time t in a series circuit.