Question

Radio station WCCO in Minneapolis broadcasts at a frequency of 830 kHz. Wavelength and angular wave number are

• A. 361 m, 0.0174/m
• B. 381 m, 0.0174 rad/m
• C. 391 m, 0.0174 rad/m
• D. 371 m, 0.0174 rad/m

Answer 1

Answer: (A)

361 m, 0.0174/m

Answer 2

Here's a step-by-step solution to find the wavelength and angular wave number:

1. Identify the given frequency:
   The frequency of the radio station, $f = 830 \text{ kHz}$.
   Convert this to Hertz (Hz):
   $f = 830 \times 10^3 \text{ Hz} = 8.30 \times 10^5 \text{ Hz}$

2. Recall the speed of electromagnetic waves:
   Radio waves are electromagnetic waves, so they travel at the speed of light in a vacuum, $c = 3 \times 10^8 \text{ m/s}$.

3. Calculate the wavelength ($\lambda$):
   The relationship between speed, frequency, and wavelength is given by:
   $c = f \lambda$
   Rearranging to solve for wavelength:
   $\lambda = \frac{c}{f}$
   Substitute the values:
   $\lambda = \frac{3 \times 10^8 \text{ m/s}}{8.30 \times 10^5 \text{ Hz}}$
   $\lambda = \frac{3000}{8.3} \text{ m}$
   $\lambda \approx 361.445 \text{ m}$
   Rounding to the nearest integer, $\lambda \approx 361 \text{ m}$.

4. Calculate the angular wave number ($k$):
   The angular wave number is related to the wavelength by the formula:
   $k = \frac{2\pi}{\lambda}$
   Substitute the calculated wavelength:
   $k = \frac{2 \times 3.14159}{361.445 \text{ m}}$
   $k \approx \frac{6.28318}{361.445} \text{ rad/m}$
   $k \approx 0.01738 \text{ rad/m}$
   Rounding to four decimal places, $k \approx 0.0174 \text{ rad/m}$.

5. Compare with the given options:
   The calculated values are approximately $361 \text{ m}$ for wavelength and $0.0174 \text{ rad/m}$ for angular wave number.
   Option (a) matches these values: $361 \text{ m}, 0.0174/\text{m}$. Note that "/m" is often used as a shorthand for "rad/m" since radians are dimensionless.