Math Problem Statement
an RL circuit has an emf given (in volts) by 3sin2t, a resistance of 10 ohms, an inductance of 0.5 henry, and an initial current of 6 amperes. fine the current in the circuit at any time t. give correct solution and answer
Solution
We are given an RL circuit with the following parameters:
- Voltage (emf):
- Resistance
- Inductance
- Initial current
The equation that governs the RL circuit is given by the first-order linear differential equation derived from Kirchhoff's voltage law:
Substituting the given values:
Step 1: Solve the homogeneous equation
First, solve the homogeneous differential equation:
This simplifies to:
This is a separable differential equation. Solving it:
Integrating both sides:
Exponentiating both sides:
where is a constant to be determined later using the initial condition.
Step 2: Solve the particular solution
We now find a particular solution to the non-homogeneous equation. We try a solution of the form:
Substitute into the original non-homogeneous equation:
Simplifying:
Group the terms involving and :
Equating coefficients of and :
Step 3: Solve for constants and
From the second equation, . Substituting this into the first equation:
Now, substitute into :
Thus, the particular solution is:
Step 4: General solution
The general solution is the sum of the homogeneous and particular solutions:
Step 5: Apply the initial condition
We are given . Substituting into the general solution:
Solving for :
Final Solution
The current in the circuit at any time is:
This is the complete solution for the current as a function of time.
Would you like any further details or explanations?
Here are 5 related questions:
- How does the inductance affect the behavior of an RL circuit?
- What would happen to the solution if the resistance was increased?
- How does the particular solution represent the steady-state response of the circuit?
- Can you explain the difference between transient and steady-state behavior in this context?
- What role does the initial condition play in determining the total solution?
Tip: In RL circuits, the transient response is often dominated by the inductor, while the steady-state response is determined by the resistor.
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Math Problem Analysis
Mathematical Concepts
Differential Equations
RL Circuit
Kirchhoff's Voltage Law
Transient and Steady-State Behavior
Formulas
Kirchhoff's Voltage Law: L(dI/dt) + RI = V(t)
General Solution: I(t) = I_h(t) + I_p(t)
Homogeneous Solution: I_h(t) = C_2 * e^(-20t)
Particular Solution: I_p(t) = A * cos(2t) + B * sin(2t)
Theorems
Separation of Variables
Method of Undetermined Coefficients
Suitable Grade Level
Undergraduate Electrical Engineering or Physics
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