Question

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

Short answer

Answer: The distance between successive heating antinodes is approximately 6.25 cm.

Step-by-step solution

Find the speed of light in a vacuum

In order to find the wavelength, we first need to know the speed of light in a vacuum. The speed of light is a constant, denoted by "\(c\)", and is approximately \(3.00 \times 10^8 \mathrm{m/s}\).

Calculate the wavelength of microwaves

Now we can find the wavelength of the microwaves produced by the oven. The relationship between frequency (\(f\)), speed of light (\(c\)), and wavelength (\(\lambda\)) is given by the equation: \[c = f \cdot \lambda\] We are given the frequency as \(2.4\, \mathrm{GHz}\), which can be converted to Hz by multiplying by \(10^9\): \[2.4\,\mathrm{GHz} = 2.4 \times 10^9 \,\mathrm{Hz}\] Now, we can use the relationship to find the wavelength: \[\lambda = \frac{c}{f}\] \[\lambda = \frac{3.00 \times 10^8\,\mathrm{m/s}}{2.4 \times 10^9\,\mathrm{Hz}}\]

Calculate the distance between successive heating antinodes

We now have the wavelength of the microwaves. Since antinodes are half a wavelength apart, we can find the distance between them by dividing the wavelength by 2: \[d = \frac{\lambda}{2}\] Substituting the wavelength found in step 2, \[d = \frac{(3.00 \times 10^8\,\mathrm{m/s})/(2.4 \times 10^9\,\mathrm{Hz})}{2}\]

Calculate the final result

Finally, we can calculate the distance between successive heating antinodes by doing the arithmetic operations: \[d = \frac{3.00 \times 10^8\,\mathrm{m/s}}{2.4 \times 10^9\,\mathrm{Hz} \cdot 2}\] \[d = \frac{3.00 \times 10^8\,\mathrm{m}}{4.8 \times 10^9}\] \[d \approx 6.25 \times 10^{-2}\,\mathrm{m}\] So, the distance between successive heating antinodes in a microwave oven's cavity is approximately \(6.25 \times 10^{-2}\,\mathrm{m}\) or \(6.25\,\mathrm{cm}\).