If the intensity of a laser beam is 600 W/m2, what is the root mean square value for the electric field in the path of the beam? Recall that the root mean square value for the electric field is the amplitude of the electric field wave divided by the square root of 2.
The correct answer and explanation is:
To find the root mean square (RMS) value of the electric field for a laser beam with intensity I=600 W/m2I = 600 \, \text{W/m}^2, we use the relationship between intensity and the electric field in an electromagnetic wave: I=12cε0E02I = \frac{1}{2} c \varepsilon_0 E_0^2
Where:
- II is the intensity,
- c=3.00×108 m/sc = 3.00 \times 10^8 \, \text{m/s} is the speed of light,
- ε0=8.85×10−12 C2/N\cdotpm2\varepsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2/\text{N·m}^2 is the vacuum permittivity,
- E0E_0 is the peak (maximum) electric field amplitude.
Step 1: Solve for E0E_0
Rearrange the intensity formula: E0=2Icε0E_0 = \sqrt{\frac{2I}{c \varepsilon_0}}
Substitute the known values: E0=2×600(3.00×108)(8.85×10−12)E_0 = \sqrt{\frac{2 \times 600}{(3.00 \times 10^8)(8.85 \times 10^{-12})}} E0=12002.655×10−3=4.52×105E_0 = \sqrt{\frac{1200}{2.655 \times 10^{-3}}} = \sqrt{4.52 \times 10^5} E0≈672 V/mE_0 \approx 672 \, \text{V/m}
Step 2: Find the RMS value
Erms=E02=6722≈6721.414≈475 V/mE_{\text{rms}} = \frac{E_0}{\sqrt{2}} = \frac{672}{\sqrt{2}} \approx \frac{672}{1.414} \approx 475 \, \text{V/m}
Final Answer:
475 V/m\boxed{475 \, \text{V/m}}
Explanation (300 words):
The electric field in a laser beam is part of the electromagnetic wave that propagates through space. The intensity of the wave tells us how much power passes through a unit area. In physics, the relationship between the intensity of an electromagnetic wave and the electric field amplitude is given by: I=12cε0E02I = \frac{1}{2} c \varepsilon_0 E_0^2
This equation comes from electromagnetic wave theory, where energy is equally shared between electric and magnetic fields. Here, E0E_0 is the peak (maximum) value of the oscillating electric field, which varies sinusoidally.
In practical applications, we often use the root mean square (RMS) value of the electric field, because it gives the average effective value over time. The RMS value is related to the peak value by: Erms=E02E_{\text{rms}} = \frac{E_0}{\sqrt{2}}
This is analogous to RMS voltage in AC circuits, and it simplifies power calculations. For a laser beam with intensity 600 W/m2600 \, \text{W/m}^2, we first solve for the peak electric field using the given formula. Once the peak value is found, dividing it by 2\sqrt{2} gives the RMS value.
This RMS value represents the effective strength of the electric field component of the laser, and it’s crucial for understanding how the beam interacts with matter, such as exerting forces on charged particles or inducing currents in materials.