Problem 63 - University Physics with Modern Physics

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \(2.4 \mathrm{GHz}\)).

Step 2: 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 \(12.5 \mathrm{cm}\).

Step 3: 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 \(6.25 \mathrm{cm}\).

Key Concepts

Microwave Frequency

Microwaves are a type of electromagnetic radiation with frequencies ranging from about 300 MHz to 300 GHz. The frequency is the number of times the wave oscillates in one second. Therefore, a microwave with a frequency of 2.4 GHz oscillates 2.4 billion times per second. This high frequency is perfect for heating food quickly and evenly in microwave ovens because it excites water molecules, causing them to heat up. A microwave oven, therefore, typically operates at a frequency of around 2.4 GHz, making it very efficient for domestic cooking tasks.

Wavelength Calculation

Wavelength refers to the distance between two corresponding points on consecutive waves, like from crest to crest. To determine the wavelength of a wave, you can use the formula:

• \( \lambda = \frac{c}{f} \)

where \( \lambda \) is the wavelength, \( c \) is the speed of light, and \( f \) is the frequency. By substituting the speed of light \( (3.0 \times 10^{8} \mathrm{m/s}) \) and the microwave frequency \( (2.4 \times 10^9 \mathrm{Hz}) \), we can find the wavelength of a microwave to be approximately 0.125 meters or 12.5 centimeters. The wavelength is crucial to understand how the microwave energy is distributed and absorbed by the substance being heated. In this context, shorter wavelengths can also result in more detailed wave patterns.

Speed of Light

The speed of light \((c)\) is a constant that plays a significant role in calculating wave properties. It is approximately \(3.0 \times 10^{8} \mathrm{m/s} \) and represents how fast light travels in a vacuum. This speed allows for the conversion between frequency and wavelength, and it effectively links the two key parameters of wave physics. Since the speed of light is finite, it determines how far and how quickly light and other electromagnetic waves like microwaves travel. This constant is essential when designing devices such as microwave ovens that depend on precise calculations to ensure efficient energy transfer.