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A nearly cylindrical laserbeam impinges normally on a perfectly absorbing surface. The irradiance of the beam (assuming it to be uniform over its cross section) is \(40 \mathrm{W} / \mathrm{cm}^{2}\). If the diameter of the beam is \(2.0 / \sqrt{\pi} \mathrm{cm}\) how much energy is absorbed per minute?

Short Answer

Expert verified
The energy absorbed per minute is 2400 Joules.

Step by step solution

01

Understand the Problem

We have a laser beam with a known irradiance and diameter impacting a surface. We need to calculate the energy absorbed by this surface over a period of time (1 minute in this case).
02

Calculate the Area of the Beam's Cross Section

The diameter of the beam is given as \( \frac{2.0}{\sqrt{\pi}} \) cm. The radius \( r \) is half of the diameter, so \( r = \frac{1.0}{\sqrt{\pi}} \) cm. The area \( A \) of a circle is calculated using the formula \( A = \pi r^2 \). Substitute the radius to find \( A \).\[A = \pi \left(\frac{1.0}{\sqrt{\pi}}\right)^2 = \pi \cdot \frac{1.0^2}{\pi} = 1.0 \, \text{cm}^2\]
03

Calculate Energy per Second

The given irradiance is \( 40 \, \text{W/cm}^2 \). Multiply the irradiance by the area of the beam’s cross-section to get the power (energy per second) absorbed by the surface.\[\text{Power} = 40 \, \text{W/cm}^2 \times 1.0 \, \text{cm}^2 = 40 \, \text{W}\]
04

Calculate Energy Absorbed per Minute

Energy absorbed per second is 40 W. Since power is energy per unit time and 1 watt equals 1 joule per second, to find the total energy absorbed per minute, we multiply the power by 60 seconds.\[\text{Energy per minute} = 40 \, \text{W} \times 60 \, \text{s} = 2400 \, \text{J}\]
05

Conclusion

The total energy absorbed by the surface per minute is 2400 Joules, considering the uniform irradiance and the beam’s diameter.

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Key Concepts

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

Laser Beam
A laser beam is a concentrated source of light. It stands out because it emits light that is focused into a narrow and precise beam. All the photons (light particles) travel in the same direction, which makes laser beams highly effective for cutting, precision measurements, and medical procedures. This characteristic also allows lasers to maintain intensity over long distances without spreading out like ordinary light sources.
Laser beams are composed of coherent light, meaning the light waves align with one another, both spatially and temporally. This coherence is what gives them their unique ability to stay tightly focused. They are used in numerous applications, from simple laser pointers to complex surgical instruments.
One important property of lasers is their beam diameter, which impacts how concentrated the light is. A smaller diameter means more concentration and potential intensity, while a larger diameter might disperse the light more.
Irradiance
Irradiance measures the power of electromagnetic radiation per unit area. It's commonly expressed in watts per square centimeter (W/cm²) and is a critical concept in understanding lasers. In the context of a laser beam interacting with a surface, irradiance tells us how much light energy is hitting a specific area of that surface.
For our example, the laser's irradiance is 40 W/cm². This figure represents the amount of power hitting each square centimeter of the surface. Higher irradiance implies more energy being transferred, which can lead to greater effects like heating or cutting.
The uniformity of irradiance over the laser beam's cross-sectional area is important for predictable interactions with materials. If it weren’t uniform, some parts might absorb more energy than others, leading to uneven effects.
Energy Absorption
Energy absorption refers to how a material takes in energy from an incident light source, like a laser beam. Materials that can absorb energy effectively are often used in applications involving lasers. When energy from a laser is absorbed, it can manifest in various forms, such as heat.
In our laser beam problem, the surface perfectly absorbs the irradiance from the laser, meaning all incoming energy is converted into heat. This is significant for applications like laser cutting, where precise energy absorption is crucial for accurate results.
Total energy absorbed depends not only on the irradiance but also on the duration of exposure. In this case, the absorbed energy was calculated over one minute, demonstrating how energy accumulates over time, leading to 2400 Joules absorbed by the surface.
Cross-sectional Area
The cross-sectional area of a laser beam is a crucial factor in determining how much energy a surface will absorb. The area is simply the slice of space that the laser covers when it meets a surface. For laser beams, which are cylindrical, this area is circular.
Calculating the cross-sectional area involves determining the beam's diameter. In our example, this is given by the diameter formula involving \(/\sqrt{\pi}\), which simplifies to a neat 1 cm² for the area. This size affects how spread out the beam's energy is across the surface.
The consistency of energy delivery across the entire cross-sectional area means that if the irradiance is uniform, as in our scenario, each square centimeter absorbs the same power. Understanding this concept helps in analyzing the efficiency and impact of the laser beam on various materials and applications.

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