A stream of photons (electromagnetic ra- diation) is directed along the positive x-axis. The stream of photons has an average inten- sity 243 W/m
. Find the peak electric field,
. The speed of light is
m/s, the permeabil- ity of free space is
H/m and the permittivity of free space is
F/m. 1.
N/C 2.
N/C 3. 1.13579 N/C 4. 302.56 N/C 5.
N/C 6. 427.885 N/C 7. 0.803125 N/C 8. 91542.6 N/C
The Correct Answer and Explanation is:
To determine the peak electric field EpE_p in an electromagnetic wave, the following relationship between intensity and field amplitude is utilized:
I=12cϵ0Ep2I = \frac{1}{2} c \epsilon_0 E_p^2
where:
- I=243 W/m2I = 243 \text{ W/m}^2 (given intensity),
- c=2.99792×108 m/sc = 2.99792 \times 10^8 \text{ m/s} (speed of light),
- ϵ0=8.854×10−12 F/m\epsilon_0 = 8.854 \times 10^{-12} \text{ F/m} (permittivity of free space).
Rearranging for EpE_p,
Ep=2Icϵ0E_p = \sqrt{\frac{2I}{c \epsilon_0}}
Substituting values,
Ep=2×243(2.99792×108)×(8.854×10−12)E_p = \sqrt{\frac{2 \times 243}{(2.99792 \times 10^8) \times (8.854 \times 10^{-12})}}
Calculating,
Ep=4862.654×10−3E_p = \sqrt{\frac{486}{2.654 \times 10^{-3}}}
Ep=1.831×105E_p = \sqrt{1.831 \times 10^5}
Ep≈427.885 N/CE_p \approx 427.885 \text{ N/C}
Thus, the correct answer is 427.885 N/C.
Explanation
The intensity of an electromagnetic wave is proportional to the square of its electric field amplitude. This relationship comes from energy conservation in wave propagation. The intensity quantifies the average energy per unit area per second carried by the wave.
In free space, the electric and magnetic fields are interrelated through Maxwell’s equations. The energy transported by photons is distributed among these fields. The equation I=12cϵ0Ep2I = \frac{1}{2} c \epsilon_0 E_p^2 ensures consistency with energy density formulations.
Electromagnetic waves propagate through space by continuously transferring energy between electric and magnetic components. The permittivity ϵ0\epsilon_0 characterizes how the electric field interacts with free space, while the speed of light cc ensures proper scaling in dynamic environments.
The result Ep=427.885 N/CE_p = 427.885 \text{ N/C} demonstrates that even a moderate-intensity electromagnetic wave carries substantial electric field strength. Higher intensities would yield stronger field amplitudes, crucial in applications like wireless communication and laser physics.
