Ch. 25 - Electric Current and Resistance

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 25 - Electric Current and Resistance

Problem 81b

Chapter 24, Problem 81b

Suppose a current is given by the equation I = 1.40 sin 210t, where I is in amperes and t in seconds. What is the rms value of the current?

Verified step by step guidance

The given current is expressed as a sinusoidal function: I = 1.40 sin(210t), where the amplitude of the current is 1.40 A. The root mean square (rms) value of a sinusoidal current is related to its amplitude.

The formula for the rms value of a sinusoidal current is:

I_{rms} = \frac{I}{\sqrt{2}}

, where

I

is the peak (amplitude) current.

Substitute the amplitude of the current,

I = 1.40

A, into the formula:

I_{rms} = \frac{1.40}{\sqrt{2}}

Simplify the expression to calculate the rms value. Note that

\sqrt{2}

is approximately 1.414, but leave the result in terms of the fraction for now.

The final expression for the rms value of the current is:

I_{rms} = \frac{1.40}{\sqrt{2}}

. You can calculate the numerical value if needed.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Root Mean Square (RMS) Value

The RMS value of an alternating current (AC) is a measure of the effective value of the current, which is equivalent to a direct current (DC) that would produce the same power in a resistive load. It is calculated by taking the square of the instantaneous values, averaging them over a complete cycle, and then taking the square root of that average.

Sine Wave Function

The sine wave function describes how the current varies with time in AC circuits. In the equation I = 1.40 sin(210t), the sine function indicates that the current oscillates between positive and negative values, with a peak amplitude of 1.40 A and a frequency determined by the coefficient of t, which affects the rate of oscillation.

Frequency and Angular Frequency

Frequency refers to the number of cycles a wave completes in one second, measured in hertz (Hz). Angular frequency, denoted by the term inside the sine function (in this case, 210), is related to frequency by the equation ω = 2πf, where ω is the angular frequency in radians per second. This concept is crucial for determining the time period and behavior of the current over time.

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