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Calculating the wavelength of microwaves involves understanding the relationship between the speed of light, frequency, and wavelength. The speed of light, denoted as \(c\), is a fundamental constant with a value of \(3 \times 10^8 \text{ m/s}\). Microwaves in a typical oven operate at a frequency \(f\), which is given as \(2.2 \times 10^9 \text{ Hz}\).

Using the formula \(c = fλ\) (where \(λ\) is the wavelength), we can rearrange it to find \(λ\) by \(λ = \frac{c}{f}\). Substituting in the values:

• Speed of light, \(c = 3 \times 10^8 \text{ m/s}\)
• Frequency, \(f = 2.2 \times 10^9 \text{ Hz}\)

we get:

\[λ = \frac{3 \times 10^8}{2.2 \times 10^9} = 1.36 \times 10^{-1} \text{ meters}\]

This means the wavelength of the microwaves used in the oven is about 13.6 cm, which illustrates how microwaves fall within the millimeter to meter range.

The energy of a photon is related to its frequency through Planck's equation \(E = hf\). Here, \(E\) is the energy of the photon, \(h\) is Planck's constant \(6.63 \times 10^{-34} \text{ Js}\), and \(f\) is the frequency \(2.2 \times 10^9 \text{ Hz}\).

To find the energy of a microwave photon, use the given values:

• Planck's constant, \(h = 6.63 \times 10^{-34} \text{ Js}\)
• Frequency, \(f = 2.2 \times 10^9 \text{ Hz}\)

Substituting these into the formula \(E = hf\):

\[E = (6.63 \times 10^{-34})(2.2 \times 10^9) = 1.46 \times 10^{-24} \text{ Joules}\]

This shows that the energy of a single microwave photon is very small, about \(1.46 \times 10^{-24} \text{ J}\). This is why many photons are needed to heat food effectively, despite each having such a tiny amount of energy.

Physics constants are fundamental values used in a multitude of calculations. In this exercise, two key constants are essential: the speed of light \(c\) and Planck's constant \(h\).

• The speed of light \(c = 3 \times 10^8 \text{ m/s}\) is crucial in wave calculations and relates the wavelength and frequency of electromagnetic waves.
• Planck's constant \(h = 6.63 \times 10^{-34} \text{ Js}\) connects the energy of photons with their frequency, a fundamental principle of quantum mechanics.

Understanding these constants helps simplify solving problems involving waves and energy. For example, the speed of light connects the concepts of wavelength and frequency, allowing us to determine one if we have the other. Meanwhile, Planck's constant helps us find the energy of a photon once its frequency is known.

Both constants are used extensively in various areas of physics, reflecting their significance in explaining natural phenomena and enabling technological advancements, such as the development of microwave ovens to efficiently heat food using electromagnetic waves.