Question

A low-power laser has a power of \(0.50 \mathrm{~mW}\) and a beam diameter of \(3.0 \mathrm{~mm}\) a) Calculate the average light intensity of the laser beam. b) Compare the value from part (a) to the intensity of light from a \(100-\mathrm{W}\) light bulb viewed from \(2.0 \mathrm{~m}\).

Short answer

Answer: The average light intensity of the laser beam is approximately 70.7 W/m^2, while the intensity of light from a 100 W light bulb viewed from 2.0 meters away is approximately 1.99 W/m^2. Therefore, the laser beam has a significantly greater intensity compared to the light from the light bulb.

Step-by-step solution

Convert the diameter of the laser beam to radius

First, we need to convert the diameter of the laser beam into radius and convert the units to meters: Radius = (Diameter / 2) = (3.0 mm) / 2 = 1.5 mm = 0.0015 m

Calculate the area of the laser beam

Next, we can calculate the area of the laser beam using the formula for the area of a circle: Area = π × (radius)^2 Area = π × (0.0015 m)^2 ≈ 7.07 × 10^(-6) m^2

Calculate the average light intensity of the laser beam

Now, we can calculate the average light intensity of the laser beam using the formula: Intensity = Power / Area Intensity = (0.50 mW) / (7.07 × 10^(-6) m^2) First, convert the power from milliwatts to watts: 0.50 mW = 0.0005 W Now, substitute the values and calculate the intensity: Intensity ≈ (0.0005 W) / (7.07 × 10^(-6) m^2) ≈ 70.7 W/m^2

Calculate the area of the sphere centered on the light bulb

To find the intensity of light from a 100 W light bulb viewed from 2.0 m, we need to find the area of the sphere centered on the light bulb with radius 2.0 m: Sphere Area = 4 × π × (radius)^2 Sphere Area = 4 × π × (2.0 m)^2 = 16 × π ≈ 50.27 m^2

Calculate the intensity of the light bulb

Next, we can calculate the intensity of light from a 100 W light bulb viewed from 2.0 m by dividing its power by the sphere area we found in the previous step: Intensity = Power / Sphere Area Intensity = (100 W) / (50.27 m^2) ≈ 1.99 W/m^2

Compare the intensities

From our calculations, we found that: - The average light intensity of the laser beam is ≈ 70.7 W/m^2 - The intensity of light from a 100 W light bulb viewed from 2.0 m is ≈ 1.99 W/m^2 The average light intensity of the laser beam is significantly greater than the intensity of light from a 100 W light bulb viewed from 2.0 m.

Key concepts

Laser Beam Intensity

When discussing laser beam intensity, we are referring to the power distributed over the area through which the laser light is spread. Intensity is a crucial factor in lasers, determining how much energy is deposited on a given surface area in applications ranging from cutting materials to medical treatments. Understanding laser intensity not only helps in evaluating a laser's effectiveness for a specific application but also in ensuring safe usage.

To grasp this concept clearly, imagine a laser pointer that emits a very focused beam; its intensity is high because the light's energy is concentrated over a small area. Conversely, a laser with the same power but a wider beam would have a lower intensity. Higher intensity means a greater amount of energy is being delivered to a specific area, which can have varying effects depending on the material and the duration of exposure.

Power to Intensity Conversion

The power to intensity conversion is indispensable when working with laser beams and calculating how much energy hits a surface. To convert the laser power (in watts) to intensity (in watts per square meter), one must divide the laser's power by the area over which the beam is distributed.

Formula for Intensity

The formula looks like this:

\[Intensity = \frac{Power}{Area}\]
Assuming power is given in watts and area in square meters, the intensity will result in watts per square meter. For example, if a laser emits 0.50 milliwatts (which is 0.0005 watts) over an area of 7.07 x 10^-6 square meters, the intensity would be roughly 70.7 watts per square meter. This calculation is essential in measuring how powerfully a laser can act upon a point of contact, such as in precise cutting or medical procedures.

Spherical Area Calculation

The spherical area calculation is key when determining the intensity of light from a source radiating in all directions, such as a light bulb. The surface area of a sphere is calculated using the formula:

\[Sphere \ Area = 4 \times \pi \times (radius)^2\]
For a sphere with a radius of 2.0 meters, the area is 4 times pi times 4 meters squared, totaling approximately 50.27 square meters. Knowledge of this area allows us to calculate the intensity of light at a given distance from the bulb. This spherical area calculation is not only important in lighting, but it also plays a significant role in fields like astronomy and telecommunications, where understanding the spread of a signal or illumination over a spherical surface is critical.