The microwaves in a certain microwave oven have a wavelength of 12.2 cm. Part 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? ΫΠΙ ΑΣΦ ? L= Submit Request Answer Part B What is the frequency of these microwaves? IVE ΑΣΦ f= Submit Request Answer ? cm Hz
The microwaves in a certain microwave oven have a wavelength of 12.2 cm. Part 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? ΫΠΙ ΑΣΦ ? L= Submit Request Answer Part B What is the frequency of these microwaves? IVE ΑΣΦ f= Submit Request Answer ? cm Hz
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
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![### Problem: Microwave Properties in an Oven
#### Background Information:
The microwaves in a certain microwave oven have a wavelength of 12.2 cm.
#### Question Part 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?**
**Answer Input:**
\[ L = \_\_\_\_ \] cm
<div style="margin-top:10px;">
<button>Submit</button> <a href="#">Request Answer</a>
</div>
#### Question Part B:
**What is the frequency of these microwaves?**
**Answer Input:**
\[ f = \_\_\_\_ \] Hz
<div style="margin-top:10px;">
<button>Submit</button> <a href="#">Request Answer</a>
</div>
#### Explanatory Notes:
- An antinodal plane in a standing wave is a position where the electric field amplitude is at a maximum.
- Since there are five antinodal planes, there will be four complete wavelengths fitting within the width of the oven. Therefore, the width \(L\) can be found using the equation for standing waves.
- The frequency \(f\) of the microwaves can be calculated using the relationship \( v = \lambda f \), where \(v\) is the speed of the wave (which for microwaves is the speed of light \( c \approx 3.00 \times 10^8 \; \text{m/s} \)).
Make sure to calculate accurately and review wave properties to answer these questions correctly!](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8d4f1709-330a-4d41-ae4f-aad2db91bb03%2F2013943a-4966-4995-b12b-015d18376055%2Fgqtj5d8_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem: Microwave Properties in an Oven
#### Background Information:
The microwaves in a certain microwave oven have a wavelength of 12.2 cm.
#### Question Part 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?**
**Answer Input:**
\[ L = \_\_\_\_ \] cm
<div style="margin-top:10px;">
<button>Submit</button> <a href="#">Request Answer</a>
</div>
#### Question Part B:
**What is the frequency of these microwaves?**
**Answer Input:**
\[ f = \_\_\_\_ \] Hz
<div style="margin-top:10px;">
<button>Submit</button> <a href="#">Request Answer</a>
</div>
#### Explanatory Notes:
- An antinodal plane in a standing wave is a position where the electric field amplitude is at a maximum.
- Since there are five antinodal planes, there will be four complete wavelengths fitting within the width of the oven. Therefore, the width \(L\) can be found using the equation for standing waves.
- The frequency \(f\) of the microwaves can be calculated using the relationship \( v = \lambda f \), where \(v\) is the speed of the wave (which for microwaves is the speed of light \( c \approx 3.00 \times 10^8 \; \text{m/s} \)).
Make sure to calculate accurately and review wave properties to answer these questions correctly!
![### Educational Content on Microwave Frequency Calculations in Ovens
---
**Problem Statement:**
Suppose a manufacturing error occurred and the oven was made 6.0 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?
---
**Solution Explanation:**
When dealing with microwave ovens, the resonant frequencies correspond to the standing wave patterns that form inside the cavity. To maintain the same number of antinodal planes (in this case, five), we need to adjust the frequency based on the new dimensions of the cavity.
### Key Steps to Solve:
1. **Identify the original width of the oven (W_old)**: Not given in the problem statement, but assumed to be known.
2. **New width of the oven (W_new)**: This is the original width plus 6.0 cm.
\[
W_{\text{new}} = W_{\text{old}} + 6.0 \, \text{cm}
\]
3. **Determine the relationship between the width and the wavelength (λ)**: For five antinodal planes, the width in multiples of half-wavelengths is:
\[
W_{\text{new}} = \frac{5}{2}\lambda
\]
4. **Calculate the wavelength (λ)**:
\[
\lambda = \frac{2}{5} W_{\text{new}}
\]
5. **Using the speed of light (c = 3 x 10^8 m/s) and the relationship c = fλ, solve for the new frequency (f_new)**:
\[
f_{\text{new}} = \frac{c}{\lambda}
\]
By following these steps, one can determine the adjusted frequency required to maintain the five antinodal planes in the modified oven dimension.
### Example Calculation:
- Assume original width \( W_{\text{old}} = 30 \, \text{cm} \).
- New width \( W_{\text{new}} = 30 \, \text{cm} + 6 \, \text{cm} = 36 \, \text{cm} \).
- Calculate the wavelength:
\[
\lambda = \frac{2}{5} \times 36 \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8d4f1709-330a-4d41-ae4f-aad2db91bb03%2F2013943a-4966-4995-b12b-015d18376055%2Fazs52j_processed.png&w=3840&q=75)
Transcribed Image Text:### Educational Content on Microwave Frequency Calculations in Ovens
---
**Problem Statement:**
Suppose a manufacturing error occurred and the oven was made 6.0 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?
---
**Solution Explanation:**
When dealing with microwave ovens, the resonant frequencies correspond to the standing wave patterns that form inside the cavity. To maintain the same number of antinodal planes (in this case, five), we need to adjust the frequency based on the new dimensions of the cavity.
### Key Steps to Solve:
1. **Identify the original width of the oven (W_old)**: Not given in the problem statement, but assumed to be known.
2. **New width of the oven (W_new)**: This is the original width plus 6.0 cm.
\[
W_{\text{new}} = W_{\text{old}} + 6.0 \, \text{cm}
\]
3. **Determine the relationship between the width and the wavelength (λ)**: For five antinodal planes, the width in multiples of half-wavelengths is:
\[
W_{\text{new}} = \frac{5}{2}\lambda
\]
4. **Calculate the wavelength (λ)**:
\[
\lambda = \frac{2}{5} W_{\text{new}}
\]
5. **Using the speed of light (c = 3 x 10^8 m/s) and the relationship c = fλ, solve for the new frequency (f_new)**:
\[
f_{\text{new}} = \frac{c}{\lambda}
\]
By following these steps, one can determine the adjusted frequency required to maintain the five antinodal planes in the modified oven dimension.
### Example Calculation:
- Assume original width \( W_{\text{old}} = 30 \, \text{cm} \).
- New width \( W_{\text{new}} = 30 \, \text{cm} + 6 \, \text{cm} = 36 \, \text{cm} \).
- Calculate the wavelength:
\[
\lambda = \frac{2}{5} \times 36 \
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