Problem 35 - University Physics with Modern Physics

Question

The microwaves in a certain microwave oven have a wavelength of \(12.2 \mathrm{cm} .\) (a) How wide must this oven be so that it will contain five antinodal planes of the electric field along its width in the standing wave pattern? (b) What is the frequency of these microwaves? (c) Suppose a manufacturing error occurred and the oven was made 5.0 \(\mathrm{cm}\) longer than specificd in part (a). In this case, what would have to be the frequency of the microwaves for there still to be five antinodal planes of the electric field along the width of the oven?

Step-by-Step Solution

Answer

(a) 48.8 cm (b) 2.46 GHz (c) 2.23 GHz

Step 1: Understanding Antinodal Planes

Antinodal planes occur where the amplitude of the standing wave is maximum. For a given width of the oven, five antinodal planes require four complete wavelengths across the width of the oven.

Step 2: Calculate Required Width of the Oven

Since five antinodal planes correspond to four complete cycles of the wave, the width of the oven must be four times the wavelength of the microwaves. Given the wavelength as \(12.2\, \text{cm}\), the required oven width is: \[ \text{Width} = 4 \times 12.2\, \text{cm} = 48.8\, \text{cm} \]

Step 3: Calculate the Frequency of Microwaves

The frequency \(f\) of a wave is given by the formula \(f = \frac{c}{\lambda}\), where \(c\) is the speed of light \((3 \times 10^8\, \text{m/s})\) and \(\lambda\) is the wavelength \((12.2\, \text{cm} = 0.122\, \text{m})\). Thus, \[ f = \frac{3 \times 10^8\, \text{m/s}}{0.122\, \text{m}} = 2.46 \times 10^9\, \text{Hz} \]

Step 4: Adjust Oven Width Due to Manufacturing Error

The oven width was made 5.0 cm longer than calculated in Step 2. Therefore, the new width of the oven is: \[ \text{New Width} = 48.8\, \text{cm} + 5.0\, \text{cm} = 53.8\, \text{cm} \]

Step 5: Calculate New Frequency for Modified Oven

To maintain five antinodal planes, with the new width of 53.8 cm, the microwave wavelength must be such that four wavelengths fit in this width. Letting \(\lambda'\) be the new wavelength, \[ 4\lambda' = 53.8\, \text{cm} \Rightarrow \lambda' = \frac{53.8}{4} = 13.45\, \text{cm} \]Converting to meters, \(\lambda' = 0.1345\, \text{m}\). The new frequency \(f'\) is then: \[ f' = \frac{3 \times 10^8\, \text{m/s}}{0.1345\, \text{m}} = 2.23 \times 10^9\, \text{Hz} \]

Key Concepts

Standing Wave

Standing waves are quite fascinating. They form when two waves of the same frequency and amplitude travel in opposite directions and interfere with each other. This results in the creation of fixed points, called nodes and antinodes.
A node is where the wave has zero amplitude, while an antinode is where the wave reaches its maximum amplitude.
Standing waves can occur in a variety of physical contexts, such as strings, air columns, and microwave ovens.

• In a microwave oven, the microwaves reflect off the walls, setting up standing wave patterns inside the oven.
• This allows for certain predictable points in space where the energy concentration, or intensity, is strongest.

When cooking food, it's these antinodal regions where energy is intensely focused, allowing food to cook evenly if placed correctly. Understanding this helps in designing the oven dimensions to match specific standing wave patterns.

Wavelength

The wavelength of a wave is the distance between consecutive crests or troughs. It's a core concept in understanding wave behavior and is crucial in applications like microwaves.
The wavelength is symbolized by the Greek letter \( \lambda \) and typically measured in meters.

• In the context of a microwave, shorter wavelengths mean more waves fit in the same space, leading to different standing wave patterns.
• In our exercise, with microwaves having a wavelength of \(12.2\, \text{cm}\), the aim is to fit complete wavelengths—as many as needed—across the width of the oven.

For practical purposes like designing a microwave oven, knowing the number of wavelengths required for certain antinodal configurations helps determine the ideal size of the cavity that will efficiently heat the food.

Frequency Calculation

Frequency is another essential property of waves, indicating how many wave cycles pass a point per second. It is measured in hertz (Hz). The equation \( f = \frac{c}{\lambda} \) links frequency \( f \) with the speed of light \( c \) (approximately \(3 \times 10^8\, \text{m/s}\) for microwaves) and wavelength \( \lambda \).

To calculate frequency:

• First, identify the wavelength of the microwaves, which will be in meters for calculations.
• With \( c \) and \( \lambda \) known, use the formula to find \( f \).

In the exercise solution, the given wavelength is \(0.122\, \text{m}\), so the frequency becomes \( f = \frac{3 \times 10^8\, \text{m/s}}{0.122\, \text{m}} = 2.46 \times 10^9\, \text{Hz} \).
This frequency determines how much energy the microwaves carry and consequently how effective they are at heating food in the oven. Understanding this principle is crucial for designing equipment that relies on microwave technology.