Question

The microwaves in a certain microwave oven have a wavelength of 12.2 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 cm longer than specified 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?

Short answer

(a) 30.5 cm. (b) 2.46 GHz. (c) 2.11 GHz.

Step-by-step solution

Understand the Standing Wave Condition

In a standing wave pattern, antinodes are points of maximum amplitude. For a wave to have 5 antinodal planes across the breadth of the oven, we know that an antinodal plane occurs at every half-wavelength. Therefore, for 5 antinodal planes, the oven's width must contain 2.5 wavelengths. Let $\lambda$ be the wavelength. The total width of the oven $W$ is given by $\frac{5}{2}\lambda$.

Calculate the Oven's Width

Given that the wavelength $\lambda$ is 12.2 cm: $$W=\frac{5}{2}\times 12.2\text{ cm}=30.5\text{ cm}$$Thus, the oven must be 30.5 cm wide.

Determine the Frequency of the Microwaves

The frequency $f$ of the microwaves can be determined using the speed of light $c$ and the relationship $c=\lambda f$. For microwaves, $c=3.00\times {10}^8\text{ m/s}$. First, convert the wavelength to meters:$$\lambda =12.2\text{ cm}=0.122\text{ m}$$Then solve for $f$:$$f=\frac{c}{\lambda }=\frac{3.00\times {10}^8\text{ m/s}}{0.122\text{ m}}\approx 2.46\times {10}^9\text{ Hz}$$ The frequency is approximately $2.46\times {10}^9\text{ Hz}$.

Consider the Manufacturing Error

If the oven is made 5.0 cm longer, the new width $W^{\prime }$ is: $$W^{\prime }=30.5\text{ cm}+5.0\text{ cm}=35.5\text{ cm}$$

Recalculate the Required Microwave Frequency

To still have 5 antinodal planes across the new width, equate $W^{\prime }$ to $\frac{5}{2}{\lambda }^{\prime }$ and solve for ${\lambda }^{\prime }$:$$\frac{5}{2}{\lambda }^{\prime }=35.5\text{ cm}\Rightarrow {\lambda }^{\prime }=\frac{35.5}{2.5}=14.2\text{ cm}=0.142\text{ m}$$Now, calculate the new frequency $f^{\prime }$:$$f^{\prime }=\frac{c}{{\lambda }^{\prime }}=\frac{3.00\times {10}^8\text{ m/s}}{0.142\text{ m}}\approx 2.11\times {10}^9\text{ Hz}$$The new frequency is approximately $2.11\times {10}^9\text{ Hz}$.

Key concepts

Standing Wave

In wave physics, standing waves are a fascinating concept, often seen when a wave reflects back and interferes with itself, creating a pattern. Think of a standing wave as a wave that oscillates in place rather than traveling from one point to another. This happens due to reflections that reinforce the wave's amplitude at certain points, called antinodes, and cancel out at nodes (points of no displacement).
This phenomenon is crucial for understanding how microwaves in an oven can pattern themselves to create maximum points of energy concentration (antinodal planes). In the case of microwaves, these standing waves ensure that food heats evenly when the antinodal planes distribute the energy efficiently through the oven.

Microwave Frequency

Frequency is vital in wave physics as it tells us how many oscillations happen over a time unit. For microwaves, frequency determines the energy these waves transmit. The frequency is inversely proportional to the wavelength, meaning as the wavelength decreases, the frequency increases, leading to greater energy transfer.

• The frequency of electromagnetic waves like microwaves is linked with their ability to heat substances since higher frequency waves can impart more energy onto particles.
• Using the known speed of light and wavelength, frequency can be found through the equation: $$f=\frac{c}{\lambda }$$ where $c$ is the speed of light.

In microwave ovens, knowing this frequency helps to optimize the heating effect by ensuring energy focuses on the antinodal planes.

Wavelength Calculation

Wavelength is another core concept of wave physics, indicating the distance over which the wave's shape repeats. For standing wave patterns in microwave ovens, calculating the wavelength is key to ensuring waves fit within the oven's dimensions.
In practical scenarios, such as designing a microwave oven to heat efficiently, we must consider how waves of a certain wavelength interact with the appliance's size. For instance, if you want five antinodal planes, you calculate the necessary wavelength to fill the oven correctly:

• Knowing the number of desired antinodal planes and that antinodes occur every half wavelength, we can determine how to fit them within a specific dimension by setting up an equation involving wavelength.
• By rearranging and solving for the wavelength using given or measured dimensions, we ensure optimal wave distribution within the space.

This is how engineers tailor microwaves to maximize energy transfer where needed.

Antinodal Planes

Antinodal planes in a standing wave pattern are crucial because they represent points of maximum amplitude, where the wave achieves its greatest displacement, creating areas of high energy.
In the context of a microwave oven, these planes are vital for efficient heating.

• An antinodal plane occurs half a wavelength apart in a standing wave.
• To ensure adequate heating, microwaves are designed to create a specific number of these planes, arranging wave interference in such a way that these high-energy points are distributed evenly across the food placed inside.
• The ability to calculate and control where these planes form is key to a microwave's cooking efficiency, ensuring that food does not end up with hot and cold spots.

Understanding how to manipulate and calculate these antinodal planes allows appliances to harness wave physics effectively, optimizing the cooking process.