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Chapter 34: Problem 5

Some neodymium-glass lasers can provide 100 terawatts of power in \(1.0 \mathrm{~ns}\) pulses at a wavelength of \(0.26 \mu \mathrm{m}\). How much energy is contained in a single pulse?

Short Answer

Expert verified
The energy contained in a single pulse is \(1.00 \times 10^5\) joules.

Step by step solution

01

Identify the Given Values

List out the given values from the problem. Neodymium-glass laser power: 100 terawatts (TW), pulse duration: 1.0 nanoseconds (ns), and wavelength: 0.26 micrometers (渭m).
02

Convert Values to Standard Units

Convert the given values to standard units. Power: 100 TW = 100 脳 10^12 W, Pulse duration: 1.0 ns = 1.0 脳 10^-9 s.
03

Use the Power and Time Relationship

Recall the relationship between power, energy, and time: to find the energy (E) in a single pulse. The formula is \[P = \frac{E}{t}\] Rearrange the formula to solve for energy (E): \[E = P \times t\]
04

Calculate the Energy

Using the values we converted, substitute them into the formula: \[E = (100 \times 10^{12} \text{ W}) \times (1.0 \times 10^{-9} \text{ s})\] Simplify the expression to find the energy: \[E = 100 \times 10^3 \text{ J}\] or \[E = 1.00 \times 10^5 \text{ J}\]

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Key Concepts

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

Power-Energy Relationship
In physics, power and energy are closely related concepts. Power is the rate at which energy is transferred or converted. Mathematically, this relationship can be expressed as: \[P = \frac{E}{t}\]Here,
  • P stands for power and is measured in watts (W).
  • E stands for energy and is measured in joules (J).
  • T represents time and is typically measured in seconds (s).
To find energy E, we rearrange the formula: \[E = P \times t\]This tells us that energy is the power multiplied by the time over which the power is applied. For lasers, which emit energy in pulses, understanding this relationship helps us calculate the energy of each pulse when we know the power and duration.
Unit Conversion
Unit conversion is a key step in solving many physics problems, including laser energy calculations. It involves changing values from one unit of measure to another. In the problem, we had to convert:
  • Power: from terawatts (TW) to watts (W).
  • Pulse duration: from nanoseconds (ns) to seconds (s).

Here's how the conversion worked: 1. Terawatts to watts: We know that 1 TW = 10^{12} W.
So, to convert 100 TW to watts, we multiply by \(10^{12}\):\(100 \times 10^{12} W\). 2. Nanoseconds to seconds: We know that 1 nanosecond is \(10^{-9}\) seconds.
So, 1.0 ns is just \(1.0 \times 10^{-9}\) s.
Correct unit conversion ensures accuracy in our calculations and brings consistency to the units we use in formulas.
Neodymium-Glass Laser
Neodymium-glass lasers are a type of solid-state laser that uses neodymium-doped glass as a lasing medium. These lasers are known for their high power and ability to generate extremely short pulses of light, often in the nanosecond range.
Here are some important characteristics:
  • Efficiency: They have high energy efficiency, making them suitable for various applications, including scientific research and materials processing.
  • Wavelength: Typically, the wavelength of light emitted by these lasers can be around 1.06 micrometers, although frequency-doubled systems can produce different wavelengths. In our exercise, the wavelength is 0.26 micrometers (渭m).
  • Pulse Duration: They can produce very short pulses, often in the nanosecond (ns) range, which is crucial for high-precision applications.
Neodymium-glass lasers are extensively used in applications that require high energy and precision, such as in medical laser systems, laser rangefinders, and industrial machining processes.

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Most popular questions from this chapter

Someone plans to float a small, totally absorbing sphere \(0.500 \mathrm{~m}\) above an isotropic point source of light, so that the upward radiation force from the light matches the downward gravitational force on the sphere. The sphere's density is \(19.0 \mathrm{~g} / \mathrm{cm}^{3}\) and its radius is \(2.00 \mathrm{~mm}\). (a) What power would be required of the light source? (b) Even if such a source were made, why would the support of the sphere be unstable?

High-power lasers are used to compress a plasma (a gas of charged particles) by radiation pressure. A laser generating pulses of radiation of peak power \(1.5 \mathrm{GW}\) is focused onto \(1.0 \mathrm{~mm}^{2}\) of high-electron- density plasma. Find the pressure exerted on the plasma if the plasma reflects all the light pulses directly back along their paths.

An electromagnetic wave with frequency 400 terahertz travels through vacuum in the positive direction of an \(x\) axis. The wave is polarized, with its electric field directed parallel to the \(y\) axis, with amplitude \(E^{\max }\). At time \(t=0\), the electric field at point \(P\) on the \(x\) axis has a value of \(+E^{\max } / 4\) and is decreasing with time. What is the distance along the \(x\) axis from point \(P\) to the first point with \(E=0\) if we search in (a) the negative direction and (b) the positive direction of the \(x\) axis?

Magnetic Component Two The magnetic component of a polarized wave of light is $$ B_{x}=\left(4.0 \times 10^{-6} \mathrm{~T}\right) \sin \left[\left(1.57 \times 10^{7} \mathrm{~m}^{-1}\right) y+\omega t\right] $$ (a) Parallel to which axis is the light polarized? What are the (b) frequency and (c) intensity of the light?

Laser eye surgery is carried out by delivering highly intense bursts of energy using electromagnetic waves. A typical laser used in such surgery has a wavelength of \(190 \mathrm{~nm}\) (ultraviolet light) and produces bursts of light that last for \(1 \mathrm{~ms}\). The laser delivers an energy of \(0.5 \mathrm{~mJ}\) to a circular spot on the cornea with a diameter of \(1 \mathrm{~mm}\). (The light is well approximated by a plane wave for the short distance between the laser and the cornea.) (a) Assuming that the energy of a single pulse is delivered to a volume of the cornea about \(1 \mathrm{~mm}^{3}\), and assuming that the pulses are delivered so quickly that the energy deposited has no time to flow out of that volume, how many pulses are required to raise the temperature of that volume from \(20^{\circ} \mathrm{C}\) to \(100^{\circ} \mathrm{C}\) ? (Assume that the cornea has a heat capacity similar to that of water.) (b) Estimate the maximum strength of the electric field in one of these pulses.
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