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Chapter 31: Problem 63

What is the distance between successive heating antinodes in a microwave oven's cavity? A microwave oven typically operates at a frequency of \(2.4 \mathrm{GHz}\).

Short Answer

Expert verified
Answer: The distance between successive heating antinodes in a microwave oven's cavity is approximately 6.25 cm.

Step by step solution

01

Determine the Wavelength

To find the wavelength (λ), we can use the equation: $$ \lambda = \frac{c}{f} $$ where c is the speed of light (\(3.0 \times 10^{8} \mathrm{m/s}\)) and f is the given frequency (\(2.4 \times 10^9 \mathrm{Hz}\) or \(2.4 \mathrm{GHz}\)).
02

Calculate the Wavelength

Now, we can plug the values of c and f into the equation from Step 1 to find the wavelength: $$ \lambda = \frac{3.0 \times 10^8 \mathrm{m/s}}{2.4 \times 10^9 \mathrm{Hz}} = \frac{3.0}{2.4} \times 10^{-1} \mathrm{m} = 1.25 \times 10^{-1} \mathrm{m} $$ So the wavelength of the microwaves is \(0.125 \mathrm{m}\) or \(12.5 \mathrm{cm}\).
03

Find the Distance Between Successive Antinodes

Since there are two antinodes within one complete wavelength and the distance between two successive antinodes is half of the wavelength, we can find the distance between the antinodes as follows: $$ D = \frac{\lambda}{2} = \frac{1.25 \times 10^{-1} \mathrm{m}}{2} = 0.625 \times 10^{-1} \mathrm{m} = 6.25 \mathrm{cm} $$ Therefore, the distance between successive heating antinodes in a microwave oven's cavity is approximately \(6.25 \mathrm{cm}\).

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