Final answer:
To find the amplitude of the electric field in the laser beam, we first calculate the intensity, and then use the relationship between intensity and the electric field. The calculated amplitude of the electric field for a 2.45 kW CO2 laser with a 2.99 mm diameter beam is approximately 8.27 × 10 V/m.
Step-by-step explanation:
To calculate the amplitude of the electric field in the beam of a 2.45 kW carbon dioxide laser with a beam diameter of 2.99 mm, we need to first determine the beam's intensity. The intensity (I) is the power per unit area and is given by:
I = Power/Area
Considering a circular cross-section for the beam, the area (A) can be determined using the formula for the area of a circle (A = πr²), where 'r' is the radius of the beam:
A = π (d/2)²
Since the diameter (d) is 2.99 mm, we convert it to meters to maintain SI units:
d = 2.99 mm = 2.99 × 10⁻³ m
Therefore, the radius (r) is:
r = d/2 = (2.99 × 10⁻³ m) / 2 = 1.495 × 10⁻³ m
So, the area (A) is:
A = π (1.495 × 10⁻³ m)² ≈ 7.03 × 10⁻⁶ m²
The intensity can now be found:
I = (2.45 kW) / (7.03 × 10⁻⁶ m²) ≈ 3.48 × 10⁵ W/m²
The relationship between the intensity and the amplitude of the electric field E is given by the formula:
I = ½ ε0 c E²
Where:
- ε0 is the permittivity of free space (ε0 = 8.854 × 10⁻±² F/m)
- c is the speed of light in a vacuum (c = 3 × 10⁸ m/s)
Rearranging for E, we get:
E = √(2I / (ε0 c))
Plugging in the values for I, ε0, and c:
E = √(2 × 3.48 × 10⁵ W/m² / (8.854 × 10⁻±² F/m × 3 × 10⁸ m/s))
After calculation, the amplitude of the electric field in the laser beam is found to be:
E ≈ 8.27 × 10 V/m