Question

Electric Circuit The diagram shows a simple electric circuit consisting of a power source, a resistor, and an inductor. A model of the current \(I,\) in amperes \((\mathrm{A}),\) at time \(t\) is given by the first-order differential equation $$L \frac{d I}{d t}+R I=E(t)$$ where \(E(t)\) is the voltage \((\mathrm{V})\) produced by the power source, \(R\) is the resistance, in ohms \((\Omega),\) and \(L\) is the inductance, in henrys \((\mathrm{H}) .\) Suppose the electric circuit consists of a \(24-\mathrm{V}\) power source, a \(12-\Omega\) resistor, and a \(4-\mathrm{H}\) inductor. (a) Sketch a slope field for the differential equation. (b) What is the limiting value of the current? Explain.

Short answer

The limiting value of the current is 2 Amperes.

Step-by-step solution

Plugging values into equations

We begin by substituting the given values into the first order differential equation. This gives: \[ 4 \frac{d I}{d t} + 12I = 24 \]This can be rearranged as \[ \frac{d I}{d t} = \frac{24 - 12I }{4} \]

Sketching the slope field

The slope field is a graphical representation of the slope of the solution curves at various points in the ty-plane. It is drawn by computing the right-hand side of the rearranged equation for different values of I (current) and t (time). The slope field will provide a visual representation of the behavior of the circuit over time. We will use a graphing tool to plot this curve.

Finding the limiting value

To find the limiting value of the current, we set the derivative \frac{dI}{dt} to 0, as when time tends to infinity, the rate of change of the current will approach 0. Hence, this gives us:\[ 0 = \frac{24 - 12I }{4} \]Solving for I, \[ I = 2 A \]This is the limiting or steady-state value of the current.

Key concepts

Electric Circuits

Understanding electric circuits is foundational to exploring a range of subjects in physics and engineering. An electric circuit typically includes a power source, such as a battery or generator, and one or more components like resistors and inductors that manage the flow of electricity. The flow of electric charge through a circuit is what we call an electric current, which is quantified in amperes (A).

When we analyze circuits, we often use differential equations to model the change in current over time. Intricate behavior in electrical systems, such as how quickly a current builds up or diminishes, is influenced by the electrical properties of the circuit components—particularly, resistance and inductance. By applying mathematical models, we can predict circuit behavior under different conditions and gain insights into design and stability.

Slope Field

A slope field, also known as a direction field, is a visual tool used in differential equations to understand how a function behaves. It consists of many short line segments or arrows over a plane, typically the t-y plane where 't' represents time and 'y' represents another variable such as current. Each segment has a slope calculated from the differential equation at that point.

Visualizing Change over Time

For an electric circuit, slope fields allow us to see how the current changes over time. By considering various points '(t, I)' and drawing line segments with slopes given by \( \frac{dI}{dt} \), we can visualize how the current might increase or decrease at different moments. This graphical representation is particularly useful for understanding the qualitative behavior of circuits without solving the equation explicitly.

Limiting Value of Current

As time progresses, electric circuits usually reach a state where the current no longer changes, known as the steady-state current. At this point, the change in current with respect to time, \( \frac{dI}{dt} \), is equal to zero. This concept is essential in understanding the limiting value of the current.

To find this limit, we set the derivative to zero and solve for the current 'I'. In mathematical terms for our given exercise, we solved \( 0 = \frac{24 - 12I }{4} \) and found that \( I = 2 \ A \). This value represents the maximum current that will eventually flow through the circuit once all transient behaviors have disappeared and stability is achieved.

Inductance

Inductance is a property of an electrical conductor by which a change in current in the conductor creates a voltage (electromotive force) in both the conductor itself (self-inductance) and in any nearby conductors (mutual inductance). It is quantified in units of henrys (H).

Role of Inductors in Circuits

Inductors are components that add inductance to a circuit. They are commonly used to store energy in their magnetic field or to filter out unwanted frequencies. The higher the inductance, the more the current lags behind the voltage due to the energy being stored in the magnetic field. This time-dependent behavior can be modeled with differential equations, as seen in our exercise where an inductor plays a crucial role in the dynamic response of the circuit.

Resistance

Resistance is another fundamental property of electric circuits, measuring how much a material opposes the flow of electric current. It is expressed in ohms (\(\Omega\)).

Resistors are elements within a circuit designed to provide a specific resistance. The resistance affects how much current will flow for a given voltage, according to Ohm's Law \( V = IR \), where 'V' is voltage, 'I' is current, and 'R' is resistance. In our given exercise, we can see its influence: the resistor determines how the current builds up over time and ultimately the limiting value of current that the circuit will allow, which is essential for controlling and protecting electrical components.