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Chapter 2: Problem 7

Suppose we measure the speed of light in a laboratory, with the light traveling a path of 10 \(\mathrm{m}\). How accurately do you have to time the light travel time to measure \(c\) to eight signifi cant figures?

Short Answer

Expert verified
The timing accuracy needed is 3.34 \( \times \ 10^{-16}\) seconds.

Step by step solution

01

- Understand the Speed of Light

The speed of light in a vacuum is approximately 299,792,458 meters per second (m/s). This value is usually denoted by the symbol \(c\).
02

- Path Length

The problem states that the light travels a path of 10 meters. Denote this distance as \(d = 10 \mathrm{m}\).
03

- Calculate the Time for Light to Travel 10 Meters

Using the formula \(t = \frac{d}{c}\), where \(t\) is the time, \(d\) is the distance and \(c\) is the speed of light, calculate the time it takes light to travel 10 meters: \[t = \frac{10 \mathrm{m}}{299,792,458 \frac{\mathrm{m}}{s}} = 3.33564 \times 10^{-8} \mathrm{s}\]
04

- Determine the Required Timing Accuracy

For eight significant figures, we need to account for a precision of 1 part in 10^8. The required timing accuracy is then: \[ \backslash frac{3.33564 \times 10^{-8} s}{10^{8}} = 3.33564 \times 10^{-16} s \]

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Key Concepts

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

Significant Figures
Significant figures are the digits in a number that carry meaningful information about its precision. When dealing with measurements, significant figures help to indicate how accurate and precise the measured value is. For instance, the speed of light is often measured as 299,792,458 meters per second, which has nine significant figures. In this exercise, when we talk about measuring the speed of light to eight significant figures, we mean that our measurement must be accurate to one part in 10^8. This high level of precision mandates that each part of the measurement, including distance and time, must be equally precise.
Distance and Time Relationship
The relationship between distance and time is a fundamental concept in physics, described by the formula: \[ c = \frac{d}{t} \] Here, \( c \) represents the speed of light, \( d \) is the distance traveled, and \( t \) is the time taken to travel that distance. This formula shows that to measure the speed of light accurately, we need precise measurements of both distance and time. In our problem, the distance \( d \) is fixed at 10 meters. Thus, the accuracy of measuring \( c \) largely depends on precisely timing how long it takes for light to travel this distance. Any error in timing directly affects the calculated speed of light.
Scientific Notation Accuracy
Scientific notation is a way to express very large or very small numbers conveniently. It helps in maintaining clarity about the precision of measurements by focusing on significant figures. For example, the small time interval it takes for light to travel 10 meters is calculated as: \[ t = 3.33564 \times 10^{-8} \text{ seconds} \] This notation indicates that our time measurement is accurate to five significant figures. Accuracy in scientific notation is crucial because it ensures that the precision of the measurement is clear and easy to read, reducing the chance of error when performing calculations or comparisons.
Timing Precision
To measure the speed of light accurately, timing precision is critical. In our exercise, we've calculated that the light takes \( 3.33564 \times 10^{-8} \text{ seconds} \) to travel 10 meters. For eight significant figures accuracy, required timing accuracy is: \[ 3.33564 \times 10^{-16} \text{ seconds} \] This tiny fraction demonstrates the level of precision needed. In practical terms, achieving such precise timing requires highly sophisticated instruments, like atomic clocks, which can measure extremely short durations with high accuracy. Precision in timing is essential because any deviation, even at a minuscule level, can vastly affect the calculated speed of light.

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Most popular questions from this chapter

Suppose two objects have energy fluxes, \(f\) and \(f+\Delta f,\) where \(\Delta f \ll f\). Derive an approximate expression for the magnitude difference \(\Delta m\) between these objects. Your expression should have \(\Delta m\) proportional to \(\Delta f\). (Hint: Use the fact that \(\ln (1+x) \cong x\) when \(x \ll 1 .\)

What is the distance to a star whose parallax is 0.1 arc sec?

(a) How does the absolute bolometric magnitude vary with the temperature of a star (assuming the radius stays constant)? (b) Does the absolute visual magnitude vary in the same way?

For a star of radius \(R\), whose radiation follows a blackbody spectrum at temperature \(T\), derive an expression for the bolometric correction.

As we will see in Chapter 21 , the universe is filled with blackbody radiation at a temperature of \(2.7 \mathrm{K}\). (a) At what wavelength does the spectrum of that radiation peak? (b) What part of the electromagnetic spectrum is this?
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