Question

A microwave oven is designed to operate at a frequency of \(2.2 \times 10^{9} \mathrm{~Hz}\). Determine the wavelength of these microwaves and the energy of each microwave.

Short answer

Answer: The wavelength of the microwaves is \(1.36 \times 10^{-1} \mathrm{m}\), and the energy of each microwave is \(1.46 \times 10^{-24} \mathrm{J}\).

Step-by-step solution

Recall the constants involved

In order to proceed, we will need the values for the speed of light (c) and Planck's constant (h). The speed of light is \(c=3 \times 10^8 \mathrm{m/s}\), and Planck's constant is \(h=6.63 \times 10^{-34} \mathrm{Js}\).

Calculate the wavelength using the frequency

Use the wavelength-frequency relationship \(c = fλ\) to determine the wavelength. We have the values for the speed of light (c) and frequency (f), and we need to find the wavelength (λ). By rearranging the formula as \(λ=\frac{c}{f}\) and substituting the given values, we obtain: \[ λ = \frac{3 \times 10^8 \mathrm{m/s}}{2.2 \times 10^9 \mathrm{Hz}}= 1.36 \times 10^{-1} \mathrm{m} \] So, the wavelength of these microwaves is \(1.36 \times 10^{-1} \mathrm{m}\).

Calculate the energy of each microwave

Use the energy-frequency relationship \(E = hf\) to determine the energy of each microwave. We have the values for Planck's constant (h) and frequency (f), so we can directly calculate the energy (E) as follows: \[ E=(6.63 \times 10^{-34} \mathrm{Js})(2.2 \times 10^9 \mathrm{Hz}) = 1.46 \times 10^{-24} \mathrm{J} \] Therefore, the energy of each microwave is \(1.46 \times 10^{-24} \mathrm{J}\).

Key concepts

Wavelength Calculation

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.

Energy Calculation

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

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.