Problem 11 - University Physics with Modern Physics

Question

Radio station WCCO in Minneapolis broadcasts at a frequency of 830 \(\mathrm{kHz}\) . At a point some distance from the transmitter, the magnetic- field amplitude of the electromagnetic wave from \(\mathrm{WCCO}\) is \(4.82 \times 10^{-11}\) T. Calculate (a) the wavelength; (b) the wave number; (c) the angular frequency; (d) the electric-field amplitude.

Step-by-Step Solution

Answer

(a) 361.45 m; (b) 0.01737 m^-1; (c) 5.216 x 10^6 rad/s; (d) 1.446 V/m.

Step 1: Calculate the Wavelength

The speed of light \( c \) is approximately \( 3 \times 10^8 \) m/s. The wavelength \( \lambda \) is calculated using the formula for wave speed \( v = f \lambda \), where \( f \) is the frequency. Rearranging gives \( \lambda = \frac{c}{f} \).Given the frequency is \( 830 \) \( \mathrm{kHz} = 830 \times 10^3 \) Hz, we have:\[ \lambda = \frac{3 \times 10^8 \text{ m/s}}{830 \times 10^3 \text{ Hz}} \approx 361.45 \text{ m} \]

Step 2: Calculate the Wave Number

The wave number \( k \) is related to the wavelength by the equation \( k = \frac{2\pi}{\lambda} \).Using \( \lambda = 361.45 \text{ m} \):\[ k = \frac{2\pi}{361.45} \approx 0.01737 \text{ m}^{-1} \]

Step 3: Calculate the Angular Frequency

The angular frequency \( \omega \) is calculated using the relation: \( \omega = 2\pi f \).Using \( f = 830 \times 10^3 \text{ Hz} \):\[ \omega = 2\pi \times 830 \times 10^3 \approx 5.216 \times 10^6 \text{ rad/s} \]

Step 4: Calculate the Electric-Field Amplitude

The electric-field amplitude \( E_0 \) can be found using the relationship with the magnetic-field amplitude \( B_0 \) and the speed of light \( c \), given by \( E_0 = cB_0 \).Given \( B_0 = 4.82 \times 10^{-11} \text{ T} \):\[ E_0 = 3 \times 10^8 \times 4.82 \times 10^{-11} \approx 1.446 \text{ V/m} \]

Key Concepts

Wavelength

When we talk about electromagnetic waves, the wavelength is an essential concept to grasp.

A wavelength (\( \lambda \)) is the distance between successive crests of a wave. You can think of it as the length of one complete wave cycle. In the electromagnetic spectrum, different wavelengths correspond to different types of radiation, from radio waves to gamma rays. Radio waves have long wavelengths, whereas gamma rays have very short ones.

• Wavelength is typically measured in meters (m).
• For electromagnetic waves traveling through a vacuum, their speed is the speed of light (\( c \), approximately \( 3 \times 10^8 \) m/s).
• Wavelength is found using the formula: \( \lambda = \frac{c}{f} \), where \( f \) is the frequency.

Understanding wavelength helps us identify what type of electromagnetic wave we're dealing with. In our exercise, the radio station broadcasts with a wavelength of approximately 361.45 meters.

Wave Number

Wave number (\( k \)) is another crucial concept that complements our understanding of electromagnetic waves.

The wave number is the spatial frequency of a wave, which tells us how many wavelengths fit into a unit distance. Essentially, it quantifies the number of wave cycles present over a specific length.

• Mathematically, wave number is defined as \( k = \frac{2\pi}{\lambda} \).
• It is measured in reciprocal meters (\( \text{m}^{-1} \)), indicating the number of wavelengths in a meter.
• Wave number provides insight into the energy and momentum of the wave. Waves with higher wave numbers have shorter wavelengths and typically higher energies.

By calculating the wave number, we obtained approximately \( 0.01737 \, \text{m}^{-1} \) for the radio wave, consistent with its long wavelengths.

Angular Frequency

Angular frequency (\( \omega \)) plays a pivotal role in describing the oscillations of electromagnetic waves.

It conveys how often the wave cycles in terms of radians per second, as opposed to a complete cycle measured in hertz.

• Angular frequency is given by the equation \( \omega = 2\pi f \), where \( f \) is the frequency in Hz.
• Measured in radians per second (\( \text{rad/s} \)), it connects the wave's frequency with its speed and spatial properties.
• This property is beneficial when analyzing wave behaviors in circuits and other systems where sine and cosine functions are used.

For the radio wave in our exercise, the angular frequency is approximately \( 5.216 \times 10^6 \, \text{rad/s} \), indicating its high oscillation rate.

Electric Field Amplitude

The electric field amplitude (\( E_0 \)) is a measure of the strength of the electric field component of an electromagnetic wave.

In electromagnetic waves, both electric and magnetic fields oscillate perpendicular to each other and the direction of wave propagation.

• The electric field amplitude represents the maximum extent of electric field fluctuation from its equilibrium state.
• Mathematically, it is related to the magnetic field amplitude (\( B_0 \)) by the equation \( E_0 = cB_0 \), where \( c \) is the speed of light.
• Measured in volts per meter (\( \text{V/m} \)), it gives insight into the wave's potential to exert force on charges.

From our calculations in the exercise, the electric field amplitude was found to be \( 1.446 \, \text{V/m} \), representing a significant yet typical radio wave intensity.