Question

An AM station broadcasts rock music at "950 on your radio dial." Units for AM frequencies are given in kilohertz (kHz). Find the wavelength of the station's radio waves in meters \((\mathrm{m}),\) nanometers (nm), and angstroms (À).

Short answer

The wavelength is 316 m, approximately \(3.16 \times 10^{11}\) nm, and approximately \(3.16 \times 10^{12}\) \text{Å}.

Step-by-step solution

Identify the Given Frequency

The frequency of the radio waves is given as 950 kilohertz (kHz).

Convert Frequency to Hertz

Since 1 kHz is equal to 1000 Hz, multiply the given frequency by 1000 to convert it to hertz (Hz). Thus, the frequency in Hz is \[950 \text{ kHz} \times 1000 = 950,000 \text{ Hz}\]

Use the Speed of Light Formula

Radio waves travel at the speed of light, which is \[c = 3 \times 10^8 \text{ m/s}\]. The wavelength \(\lambda\) can be found using the formula \[\lambda = \frac{c}{f}\], where \(f\) is the frequency in Hz.

Calculate Wavelength in Meters

Substitute the values into the formula to find the wavelength in meters:\[\lambda = \frac{3 \times 10^8 \text{ m/s}}{950,000 \text{ Hz}} = \frac{3 \times 10^8}{9.5 \times 10^5} = 315.79 \text{ m}\]. Thus, the wavelength in meters is approximately 316 m.

Convert Wavelength to Nanometers

1 meter is equal to \[10^9 \text{ nm}\]. To find the wavelength in nanometers, multiply the wavelength in meters by \[10^9\]:\[\lambda = 316 \text{ m} \times 10^9 \text{ nm/m} = 3.16 \times 10^{11} \text{ nm}\]. Thus, the wavelength in nanometers is approximately \(3.16 \times 10^{11}\) nm.

Convert Wavelength to Angstroms

1 meter is equal to \[10^{10} \text{ \text{Å}}\]. To find the wavelength in angstroms, multiply the wavelength in meters by \[10^{10}\]:\[\lambda = 316 \text{ m} \times 10^{10} \text{ \text{Å}/m} = 3.16 \times 10^{12} \text{ \text{Å}}\]. Thus, the wavelength in angstroms is approximately \(3.16 \times 10^{12}\) \text{\text{Å}}.

Key concepts

frequency conversion

Understanding frequency conversion is essential for solving problems related to radio waves and other forms of electromagnetic radiation. Here, the frequency of the AM station is given in kilohertz (kHz). To convert this into a more widely used unit, hertz (Hz), you need to remember that 1 kilohertz equals 1,000 hertz.

In this problem, the radio station broadcasts at 950 kHz. By multiplying 950 by 1,000, you convert the frequency to hertz:
\( 950 \text{kHz} \times 1000 \text{Hz/kHz} = 950,000 \text{Hz} \).

Understanding and performing this conversion allows us to use standard formulas related to wave behavior.

speed of light

The speed of light in a vacuum is a fundamental constant in physics and is denoted by the letter \( c \). The value is approximately \( 3 \times 10^8 \text{ m/s} \).

This constant is crucial when working with any electromagnetic waves, including radio waves. The formula to find the wavelength \( \lambda\ \) of a wave is given by:
\[ \lambda = \frac{c}{f} \]
where \f\ is the frequency.

Using this formula, you can calculate the wavelength of the radio waves emitted by a station if the frequency is known.

wavelength calculation

To find the wavelength using the speed of light and the given frequency, substitute the known values into the wavelength formula:
\ \[ \lambda = \frac{3 \times 10^8 \text{ m/s}}{950,000 \text{ Hz}} \ = \frac{3 \times 10^8}{9.5 \times 10^5} = 315.79 \text{ m} \ \].

This gives us the wavelength of the radio waves in meters. Therefore, the wavelength is approximately 316 meters.

• Remember, performing the division correctly and considering significant figures is essential for an accurate result.

unit conversions in physics

In physics, converting units is a skill you'll use often. Here, we convert the calculated wavelength from meters to nanometers and angstroms, two common units in physics.

• 1 meter \( \text{m} \) is equal to \[ 10^9 \text{ nm} \text{ (nanometers)} \].
  Hence, converting 316 meters to nanometers:
  \[ 316 \text{ m} \times 10^9 \text{ nm/m} = 3.16 \times 10^{11} \text{ nm} \]



• Similarly, 1 meter is equal to \[ 10^{10} \text{ \text{Å} \text{ (angstroms)}} \].
  Converting 316 meters to angstroms:
  \[ 316 \text{ m} \times 10^{10} \text{ \text{Å}/m} = 3.16 \times 10^{12} \text{ \text{Å}} \]

To handle unit conversions effectively:

• Use conversion factors
• Ensure consistency in units throughout your calculations
• Double-check your calculations for accuracy