Radiation pressure: A microwave oven operates with sinusoidal microwaves at a frequency of 2400 MHz. The height of the oven cavity is 25 cm and the base measures 30 cm by 30 cm. Assume that microwave energy is generated uniformly on the upper surface of the cavity and propagates directly downward toward the base. The base is lined with a material that completely absorbs microwave energy. The total microwave energy content of the cavity is 0.50 µJ. What magnitude force does the microwave beam exert on the base of the oven? (c = 3.00 × 10⁸ m/s, μ₀ = 4π × 10^-7 T ∙ m/A, ε₀ = 8.85 × 10^-12 C²/N ∙ m²) A) 1.6 µN B) 2.0 µN C) 2.5 µN D) 3.0 µN E) 3.5 µN

Question

Quizplus · Accepted Answer

The force exerted by the microwave beam on the base of the oven can be calculated using the formula for radiation pressure $P = \frac{S}{c}$, where $S$ is the Poynting vector representing the power per unit area (W/m$^2$) and $c$ is the speed of light. The total power $P_{total}$ can be found from the total energy content ($E$) and the time ($t$) it takes for the microwaves to travel from the top to the base of the oven, but since we're dealing with a static scenario, we focus on the energy content and its relation to force. The force ($F$) exerted is related to the pressure ($P$) by the area ($A$) of the base, $F = P \cdot A$. Given that the energy is completely absorbed, the energy content directly relates to the force through $E = F \cdot d$, where $d$ is the distance the wave travels (which in this case is not directly needed due to the absorption scenario simplifying the calculation).Given that the energy content is $0.50 \mu J$ and it is completely absorbed by the base, the force exerted can be calculated by considering the energy to force conversion in the context of radiation pressure, where the absorption implies a momentum change that directly relates to the force. The calculation involves understanding that the radiation pressure is the energy density ($u$) times $c$, and since the energy is fully absorbed, this results in a force over the area of the base.Without needing to explicitly calculate the time or the detailed propagation dynamics, the key is understanding that the force exerted by the radiation is a function of the total energy absorbed and the area over which it is absorbed. Given the parameters, the correct calculation leads to a force magnitude that matches choice B) 2.0 µN, which is derived from the principles of radiation pressure and energy absorption in a confined space.

Radiation Pressure: a Microwave Oven Operates with Sinusoidal Microwaves at a Frequency