Question

If the eye receives an average intensity greater than 1.0 \(\times\) 10\(^2\) W/m\(^2\), damage to the retina can occur. This quantity is called the \(damage\) \(threshold\) of the retina. (a) What is the largest average power (in mW) that a laser beam 1.5 mm in diameter can have and still be considered safe to view head-on? (b) What are the maximum values of the electric and magnetic fields for the beam in part (a)? (c) How much energy would the beam in part (a) deliver per second to the retina? (d) Express the damage threshold in W/cm\(^2\).

Short answer

(a) 0.1767 mW, (b) 274.57 V/m (electric), 9.15 × 10⁻⁷ T (magnetic), (c) 0.1767 mJ/s, (d) 0.01 W/cm².

Step-by-step solution

Calculate Cross-sectional Area

First, we calculate the cross-sectional area of the laser beam. The diameter is given as 1.5 mm, which is 0.0015 meters. The radius is half of that: \( r = \frac{0.0015}{2} \text{ m} = 0.00075 \text{ m} \). The area \( A \) is given by the formula for the area of a circle: \( A = \pi r^2 = \pi (0.00075)^2 \approx 1.767 \times 10^{-6} \text{ m}^2 \).

Calculate Maximum Power (a)

The intensity (given in the problem as the damage threshold) is power per unit area. Therefore, the maximum power \( P \) is given by \( P = I \times A \), where \( I = 1.0 \times 10^2 \text{ W/m}^2 \) and \( A = 1.767 \times 10^{-6} \text{ m}^2 \). Hence, \( P = 1.0 \times 10^2 \times 1.767 \times 10^{-6} = 1.767 \times 10^{-4} \text{ W} \). To convert to milliwatts (mW), multiply by 1000: \( 1.767 \times 10^{-4} \times 1000 = 0.1767 \text{ mW} \).

Calculate Maximum Electric Field (b)

The average intensity \( I \) is related to the maximum electric field \( E_{max} \) by the equation \( I = \frac{1}{2} \varepsilon_0 c E_{max}^2 \), where \( \varepsilon_0 = 8.85 \times 10^{-12} \text{ F/m} \) (the permittivity of free space) and \( c = 3.00 \times 10^8 \text{ m/s} \) (the speed of light). Solving for \( E_{max} \), we get: \( E_{max} = \sqrt{\frac{2I}{\varepsilon_0 c}} = \sqrt{\frac{2 \times 1.0 \times 10^2}{8.85 \times 10^{-12} \times 3.00 \times 10^8}} \approx 274.57 \text{ V/m} \).

Calculate Maximum Magnetic Field (b)

The maximum magnetic field \( B_{max} \) is related to the maximum electric field \( E_{max} \) by \( B_{max} = \frac{E_{max}}{c} \). Using \( E_{max} = 274.57 \text{ V/m} \) and \( c = 3.00 \times 10^8 \text{ m/s} \), we find \( B_{max} = \frac{274.57}{3.00 \times 10^8} \approx 9.15 \times 10^{-7} \text{ T} \).

Calculate Energy Delivered Per Second to Retina (c)

The energy delivered per second (power) to the retina is the same as the calculated maximum power from Step 2. Therefore, the energy delivered per second is 0.1767 mW, which equals 0.1767 millijoules per second, since 1 watt equals 1 joule per second.

Convert Damage Threshold to W/cm² (d)

Convert the given intensity from W/m² to W/cm². We know that 1 m² = 10,000 cm², hence \( 1.0 \times 10^2 \text{ W/m}^2 = \frac{1.0 \times 10^2}{10,000} \text{ W/cm}^2 = 0.01 \text{ W/cm}^2 \).

Key concepts

Laser Safety

Laser safety is a critical aspect when dealing with lasers, especially in environments like laboratories and industries. Lasers produce concentrated beams of light that can be harmful, particularly to the eyes.
The damage threshold of the retina, which is the eye's sensitivity limit to laser intensity, is a key concept in laser safety. By determining the damage threshold, industries can ensure that laser devices operate within safe limits and do not cause harm to human eyes.
For instance, if a laser emits light above 100 W/m², it surpasses the damage threshold for the retina. This is when safety measures, like protective eyewear and regulated exposure times, become essential. It's crucial to ensure that the power of the laser beam remains below this threshold to avoid permanent eye damage.

Intensity Calculation

Intensity is a measure of how much power is distributed over a particular area. It is calculated as power per unit area, typically expressed in watts per square meter (W/m²).
In laser applications, understanding intensity is vital for determining whether a laser beam is safe. For example, by calculating the laser's cross-sectional area and knowing the acceptable intensity threshold, one can determine the maximum safe power output.
This is done using the formula:

• Intensity, I = Power, P / Area, A

This calculation helps in converting units and adjusting laser setups to adhere to safety guidelines. If the calculated power exceeds the threshold when the intensity formula is applied, adjustments must be made to prevent potential hazards.

Electromagnetic Fields

Electromagnetic fields (EMFs) are fundamental to understanding how lasers work. Lasers emit coherent light that comprises electric and magnetic fields oscillating perpendicular to each other.
In the context of determining laser safety levels, we often need to calculate the maximum electric and magnetic fields a laser beam can produce. The electric field (E) primarily affects how the intensity is perceived and is directly linked to the potential energy delivered.
Using relationships like:

• Maximum electric field, E_max = \( \sqrt{\frac{2I}{\varepsilon_0 c}} \)
• Maximum magnetic field, B_max = E_max / c

Where \( \varepsilon_0 \) (epsilon naught) is the permittivity of free space and \( c \) is the speed of light, we can assess the electromagnetic properties associated with a laser beam.
These calculations help in adjusting the laser systems to comply with safety standards, ensuring that neither the electric nor the magnetic fields exceed safe levels for human exposure.