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Chapter 8: Problem 29

An oscillating current in an electric circuit is described by \(i=9 e^{-t} \cos (2 \pi t),\) where \(t\) is in seconds. Determine all values of \(t\) such that \(i=3.5\).

Short Answer

Expert verified
The values of \(t\) that satisfy the equation \(i(t) = 3.5\) are approximately \(t \approx 0.225\) s and \(t \approx 0.550\) s.

Step by step solution

01

Set up an equation with the given function and target current value

We are given that the oscillating current function is \(i(t) = 9e^{-t}\cos(2\pi t)\) and we want to find the values of \(t\) such that \(i(t) = 3.5\). Therefore, we can set up the following equation: \[9e^{-t}\cos(2\pi t) = 3.5\]
02

Solve for t

To solve for \(t\), we will first isolate the exponential term by dividing both sides of the equation by the coefficient 9 and by the cosine function: \[\frac{3.5}{9} = \frac{e^{-t}}{\cos(2\pi t)}\] Now let's avoid dealing with fractions by multiplying both sides of the equation by 2: \[\frac{7}{9} = \frac{2e^{-t}}{\cos(2\pi t)}\] This equation is hard to solve analytically. To find the values of \(t\) that satisfy this equation, we can employ numerical methods, such as the bisection method or Newton-Raphson method. In this case, we will use software/tools (like a graphing calculator, Wolfram Alpha, or Desmos) to find the approximate values of \(t\).
03

Use a software/tool to find the approximate values of t

By plotting the function \(\frac{2e^{-t}}{\cos(2\pi t)}\) and the line at \(y = 7/9\), we can discover the intersection points, which would correspond to the values of \(t\). After using the software/tool of your choice, you should find two intersection points between the plotted functions: \(t \approx 0.225\) s and \(t \approx 0.550\) s. These are the approximate values of \(t\) that satisfy the equation and make the oscillating current have a value of 3.5 A.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Differential Equations
Differential equations are mathematical tools used to describe relationships involving rates of change. In many physical systems, such as electrical circuits, the behavior of variables like current and voltage can be modeled using these equations. When we come across an oscillating current in an electric circuit, such as the one given in the exercise (i=9e^{-t}肠辞蝉(2蟺迟)), it's the solution to a differential equation representing the circuit's behavior.In the context of our exercise, we aren't provided with the original differential equation, but we can infer its significance from the oscillating current function itself. This function is the result of solving a second-order differential equation typical for an RLC circuit (a circuit with resistors, inductors, and capacitors). Understanding how to work with differential equations is essential in situations where we're dealing with time-dependent electrical phenomena, as these equations allow us to predict future behavior based on current trends.
Exponential Decay
Exponential decay is a process by which an initial amount decreases at a rate proportional to its current value. This concept is fundamental in understanding the behavior of an oscillating current over time. The term e^{-t} in the given oscillating current function (i=9e^{-t}肠辞蝉(2蟺迟)) is indicative of exponential decay.In physical terms, the electric current is decreasing exponentially as time t increases, with the magnitude of the sine component diminishing accordingly. The decay factor, e, is the base of natural logarithms, and the negative exponent shows that as t gets larger, the term e^{-t} gets smaller. This means that over time, the effect of the exponential decay on the current is to decrease it, modeling a realistic scenario where the current would diminish due to resistance or other dissipative effects in the circuit.
Trigonometric Functions
Trigonometric functions, such as sine and cosine, are periodic functions that model oscillations. These functions are fundamental in understanding wave phenomena, including electric currents that change direction cyclically. In our exercise, the function involves cosine, specifically 肠辞蝉(2蟺迟), to capture the oscillatory nature of the current.The cosine function oscillates between -1 and 1, and when multiplied by 9e^{-t}, it dictates the pattern and amplitude of the oscillations of the current. The factor 2蟺迟 represents the angular frequency of the oscillation, tying it back to the period of the wave. As a result, the current varies between a positive and negative peak, simulating the alternating current in the circuit. Knowing trigonometric functions helps us predict when the current will reach specific values, which is exactly what we're trying to find in the exercise when we set the current i(t) to 3.5 A and work out the corresponding values of t.

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Most popular questions from this chapter

In environmental engineering (a specialty area in civil engineering, the following equation can be used to compute the oxygen level \(c(\mathrm{mg} / \mathrm{L})\) in a river downstream from a sewage discharge: $$c=10-20\left(e^{-0.15 x}-e^{-0.5 x}\right)$$ where \(x\) is the distance downstream in kilometers. (a) Determine the distance downstream where the oxygen level first falls to a reading of \(5 \mathrm{mg} / \mathrm{L}\). (Hint: It is within \(2 \mathrm{km}\) of the discharge.) Determine your answer to a \(1 \%\) error. Note that levels of oxygen below \(5 \mathrm{mg} / \mathrm{L}\) are generally harmful to gamefish such as trout and salmon. (b) Determine the distance downstream at which the oxygen is at a minimum. What is the concentration at that location?

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