Question

The age of the universe is about \(5 \times 10^{17} \mathrm{~s} ;\) the shortest light pulse produced in a laboratory (1990) lasted for only \(6 \times\) \(10^{-15} \mathrm{~s}\) (see Table \(1-3\) ). Identify a physically meaningful time interval approximately halfway between these two on a logarithmic scale.

Short answer

The physically meaningful time interval approximately halfway between the age of the universe and the shortest light pulse on a logarithmic scale is about 55.4 seconds.

Step-by-step solution

Convert values to logarithmic scale

The first step is to convert the given values from a normal scale to a logarithmic scale. This can be accomplished by taking the logarithm to base 10 of each value. Thus, for the age of the universe \( \log(5 \times 10^{17}) \approx 17.7\) and for the shortest light pulse \( \log(6 \times 10^{-15}) \approx -14.22\).

Calculate average in logarithmic scale

The next step is to calculate the average of these two numbers on a logarithmic scale. This includes summing up the two logarithmic numbers and then divide by 2. \((17.7 - 14.22) / 2 \approx 1.74\).

Find the power of 10 for the average

The final step is to convert this logarithmic average back into the regular scale by taking it as the exponent of 10. The result of this conversion will be \(10^{1.74} \approx 55.4\) seconds, giving us the required time interval.

Key concepts

Age of the Universe

The age of the universe is an incredibly vast measurement that represents the time elapsed since the Big Bang, which is considered the beginning of our universe. Scientists peg this age at about 13.8 billion years, but in terms of seconds, it's approximately \(5 \times 10^{17}\) seconds. This colossal number emphasizes just how ancient the universe is. To appreciate its enormity:

• The universe outdates any current celestial bodies, including stars and galaxies.
• It's beyond the human capacity to observe directly; rather, we infer it through astronomical measurements and cosmic microwave background radiation.
• The age helps us understand the universe's development stages, from the formation of elements to the birth of galaxies.

Converting this age into different units, like seconds, offers a way to compare it to much smaller time intervals, such as the shortest light pulse, using a logarithmic scale.

Shortest Light Pulse

The shortest light pulse ever produced is a testament to modern technology, recorded in 1990. It lasted only \(6 \times 10^{-15}\) seconds. That's a fraction of a second, demonstrating:

• The unique ability of lasers to generate incredibly brief pulses for precise measurements.
• The extreme agility of light used in experimental physics to observe fast processes on a femtosecond scale.
• Significance in applications that require observing and manipulating processes at the atomic level, such as in chemical reactions and materials research.

Such advancements have paved the way for innovations in fields like optics and communications. By comparing this pulse to cosmic timescales using a logarithmic scale, we see just how staggeringly different these time spans are.

Time Interval Calculation

Calculating a midpoint on a logarithmic scale between two vastly different time intervals, such as the age of the universe and the shortest light pulse, provides a meaningful comparative tool. The key steps are:

• Converting both time values, \(5 \times 10^{17}\) seconds and \(6 \times 10^{-15}\) seconds, to the logarithmic scale using base 10: \(\log(5 \times 10^{17}) \approx 17.7\) and \(\log(6 \times 10^{-15}) \approx -14.22\).
• Finding the average of these logs, which involves adding them and dividing by two to get \(1.74\).
• Finally, translating this result back to regular time by raising 10 to the power of the average, \(10^{1.74} \approx 55.4\) seconds.

This process demonstrates how logarithmic scales help make large and small numbers comparable. Finding a meaningful midpoint helps us better understand the spectrum of time, offering perspectives on both astronomical and microscopic scales.