Problem 21 - Physics

Question

A laser emits a narrow beam of light. The radius of the beam is \(1.0 \times 10^{-3} \mathrm{~m}\), and the power is \(1.2 \times 10^{-3} \mathrm{~W}\). What is the intensity of the laser beam?

Step-by-Step Solution

Answer

The intensity of the laser beam is approximately 382 W/m².

Step 1: Understand Intensity Formula

The intensity of a beam of light is defined as the power per unit area. The formula for intensity \( I \) is given by \( I = \frac{P}{A} \) where \( P \) is the power of the laser, and \( A \) is the cross-sectional area of the beam.

Step 2: Calculate the Area of the Beam

The beam has a circular cross-section, so we use the formula for the area of a circle: \( A = \pi r^2 \). The radius \( r \) given is \( 1.0 \times 10^{-3} \) m. Substituting the value of \( r \), we get:\[ A = \pi (1.0 \times 10^{-3})^2 = \pi \times 1.0 \times 10^{-6} = 3.14 \times 10^{-6} \text{ m}^2 \].

Step 3: Calculate the Intensity

Now, substitute the values for power \( P = 1.2 \times 10^{-3} \) W and area \( A = 3.14 \times 10^{-6} \text{ m}^2 \) into the intensity formula:\[ I = \frac{1.2 \times 10^{-3}}{3.14 \times 10^{-6}} \approx 382 \text{ W/m}^2 \].

Step 4: Conclusion

The intensity of the laser beam, which measures how much power is distributed over the area of the beam, is approximately \( 382 \text{ W/m}^2 \).

Key Concepts

Power of Laser

The power of a laser represents the rate at which energy is emitted by the laser in the form of light. Power is measured in watts (W) and can be thought of as the total amount of energy transferred per second. In the context of lasers, understanding the power allows us to determine how much energy the beam is delivering over time. For example, in our exercise, the laser has a power of \(1.2 \times 10^{-3}\) W, indicating a relatively low-power beam. This is typical of many laser applications where precision is more critical than raw power.

Area of Beam

When discussing laser beams, the area of the beam pertains to the cross-sectional area through which the beam travels. This is crucial since it helps determine how the power is distributed across the space the beam covers. The beam in our exercise is given a circular cross-section, so we use the formula for the area of a circle, \( A = \pi r^2 \).

This formula requires knowing the radius of the circle, which is provided as \(1.0 \times 10^{-3}\) m. Plugging this into our formula, we calculate \( A = \pi (1.0 \times 10^{-3})^2 = 3.14 \times 10^{-6} \text{ m}^2 \). This small area indicates a tightly focused beam, a common feature of laser technology.

Intensity Formula

The intensity formula is a fundamental equation used to calculate how much power a beam of light carries per unit area. It's given by the formula \( I = \frac{P}{A} \), where \( I \) stands for intensity, \( P \) is the power of the laser, and \( A \) is the area over which the power is spread.

• This formula helps in understanding how densely packed the energy of the beam is.
• Intensity is measured in watts per square meter (W/m²), indicating how much power is hitting each square meter of the area.

By using this formula, we can gain insights into the strength of the laser's effect on different materials or surfaces it encounters.

Calculating Intensity

Calculating the intensity of a laser beam involves using the intensity formula discussed earlier. Here, the power of the laser (\( P = 1.2 \times 10^{-3} \) W) and the area of the beam (\( A = 3.14 \times 10^{-6} \text{ m}^2 \)) are substituted into the formula.

These values result in: \[ I = \frac{1.2 \times 10^{-3}}{3.14 \times 10^{-6}} \approx 382 \text{ W/m}^2 \]. This calculation shows how efficiently the laser's power is concentrated in the beam's cross-section. Calculating intensity is vital for applications requiring precise power levels, such as in optical fiber communications or laser cutting technologies. By knowing the intensity, you can predict how the laser will interact with materials and adjust accordingly.