Problem 46 - University Physics with Modern Physics

Question

A \(0.250-\mathrm{H}\) inductor carries a time-varying current given by the expression \(i=(124 \mathrm{mA}) \cos [(240 \pi / \mathrm{s}) t] .\) (a) Find an expression for the induced emf as a function of time. Graph the current and induced emf as functions of time for \(t=0\) to \(t=\frac{1}{60} \mathrm{s}\) . (b) What is the maximum emf? What is the current when the induced emf is a maximum? (c) What is the maximum current? What is the induced emf when the current is a maximum?

Step-by-Step Solution

Answer

(a) \( \epsilon(t) = 7.464\pi \sin(240\pi t) \). Max emf: \( 23.42 \, \text{V} \), current at max emf: 0 A. Max current: 0.124 A, emf at max current: 0 V.

Step 1: Understand the Inductor and Current Function

The current through the inductor is given by the function \( i(t) = (124 \, \text{mA}) \cos[(240 \pi / \text{s}) t] \). Converted to amperes, this is \( i(t) = 0.124 \cos(240 \pi t) \). We need to differentiate this to find the induced emf which follows the formula \( \epsilon(t) = -L \frac{di}{dt} \), where \( L \) is the inductance (0.250 H).

Step 2: Differentiate Current to Find emf

Differentiate the current function with respect to time, \( \frac{di}{dt} = -0.124 \times (240 \pi) \sin(240 \pi t) \). Substitute into the emf expression: \( \epsilon(t) = -0.250 \times (-0.124 \times 240 \pi \times \sin(240 \pi t)) \). Simplifying gives \( \epsilon(t) = 7.464 \pi \sin(240 \pi t) \).

Step 3: Identify Maximum Values

The expressions \( \cos \) and \( \sin \) have maximum values of 1. Thus, maximum emf occurs when \( \sin(240 \pi t) = 1 \), giving \( \epsilon_{max} = 7.464 \pi \approx 23.42 \, \text{V} \). Maximum current is \( i_{max} = 0.124 \, \text{A} \).

Step 4: Determine Current at Maximum emf

Since \( \sin(240 \pi t) = 1 \) corresponds to \( \cos(240 \pi t) = 0 \), the current \( i \) at maximum emf is zero.

Step 5: Validate Current and emf Alignment

The expression reveals at maximum current \( \cos(240 \pi t) = 1 \) and \( \sin(240 \pi t) = 0 \), hence induced emf is zero at maximum current. The phase shift of \( \frac{\pi}{2} \) between \( \sin \) and \( \cos \) functions confirms this period difference.

Key Concepts

Time-Varying Current

When dealing with circuits, a time-varying current is one that changes over time, as opposed to a steady, constant current. In this exercise, the current is expressed by the function \( i(t) = (124 \, \text{mA}) \cos[(240 \pi / \text{s}) t] \). This function tells us that the current oscillates as a cosine wave, which is a common waveform for alternating current (AC) circuits. The amplitude here is 124 mA, which is the peak value of the current. The term \((240 \pi / \text{s})\) inside the cosine function is the angular frequency, illustrating how fast the current oscillates per unit time. Understanding these components is essential, as they help in predicting how the current behaves over time. Pay attention to the units: converting between milliamperes (mA) and amperes (A) is often necessary in calculations for real-world applications.

Induced EMF

In circuits, particularly those involving inductors, an electromotive force (emf) can be induced when the current through the inductor changes. The induced emf can be found using the formula \( \epsilon(t) = -L \frac{di}{dt} \). Here, \( L \) represents the inductance, which measures how effectively an inductor can create emf in response to a change in current. For our example, the inductance is \(0.250 \text{ H}\). The negative sign in this formula is based on Lenz's law, indicating that the direction of the induced emf opposes the change in current that created it. To find the induced emf as a function of time, you would differentiate the current function \( i(t) \) with respect to time. In our problem, this gives the expression for \( \epsilon(t) = 7.464 \pi \sin(240 \pi t) \), showing that the emf varies sinusoidally and is phase-shifted by \( 90^{\circ} \) or \( \pi/2 \) radians relative to the current.

Differentiation of Trigonometric Functions

Differentiation is a mathematical process used to determine how a function changes at any given point. When we deal with trigonometric functions, the differentiation rules become crucial. For cosine functions, the derivative follows the pattern \( \frac{d}{dt}[\cos(kt)] = -k \sin(kt) \), where \( k \) is a constant. This rule is pivotal in our solution since it allows us to compute the rate at which the current changes with time, which directly affects the induced emf. In our problem, differentiating the time-varying current \( i(t) = 0.124 \cos(240 \pi t) \) results in \( \frac{di}{dt} = -0.124 \times (240 \pi) \sin(240 \pi t) \). The pattern '\( \cos \)' to '\( -\sin \)' and including the chain rule for \( k \)'s coefficient shows how variation in trigonometric inputs resounds through its differentiation. This concept is applicable beyond physics, being widely used in engineering and mathematical studies where periodic changes are analyzed.