Problem 13 - University Physics with Modern Physics

Question

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

Step-by-Step Solution

Answer

(a) 361.45 m, (b) 0.0174 m⁻¹, (c) 5.22 × 10⁶ rad/s, (d) 0.0145 V/m.

Step 1: Calculate Wavelength

The wavelength \( \lambda \) of an electromagnetic wave is related to its frequency \( f \) and the speed of light \( c \). The speed of light is given by \( c = 3 \times 10^8 \text{ m/s} \). Use the formula\[\lambda = \frac{c}{f}\]Given, \( f = 830 \text{ kHz} = 830 \times 10^3 \text{ Hz} \). Substitute the values to find \( \lambda \):\[\lambda = \frac{3 \times 10^8}{830 \times 10^3} = \frac{3 \times 10^8}{8.3 \times 10^5} \approx 361.45 \text{ meters}\]

Step 2: Calculate Wave Number

The wave number \( k \) is given by the formula:\[k = \frac{2\pi}{\lambda}\]Use the previously calculated wavelength \( \lambda = 361.45 \text{ m} \):\[k = \frac{2\pi}{361.45} \approx 0.0174 \text{ m}^{-1}\]

Step 3: Calculate Angular Frequency

Angular frequency \( \omega \) is given by the formula:\[\omega = 2\pi f\]Substitute \( f = 830 \times 10^3 \text{ Hz} \):\[\omega = 2\pi \times 830 \times 10^3 \approx 5.22 \times 10^6 \text{ radians/second}\]

Step 4: Calculate Electric Field Amplitude

The amplitude of the electric field \( E_{max} \) is related to the amplitude of the magnetic field \( B_{max} \) by the formula:\[E_{max} = c \times B_{max}\]Given \( B_{max} = 4.82 \times 10^{-11} \text{ T} \) and \( c = 3 \times 10^8 \text{ m/s} \):\[E_{max} = 3 \times 10^8 \times 4.82 \times 10^{-11} \approx 0.0145 \text{ V/m}\]

Key Concepts

Wavelength Calculation

To determine the wavelength of electromagnetic waves, we need to know the wave's frequency and the speed of light. For any electromagnetic wave traveling in a vacuum, light travels at approximately \( c = 3 \times 10^8 \text{ m/s} \). The wavelength \( \lambda \) can be calculated using the formula:\[ \lambda = \frac{c}{f} \]where \( f \) is the frequency. In the given problem, the radio station broadcasts at a frequency of 830 kHz, which is equivalent to \( 830 \times 10^3 \text{ Hz} \). Plugging the values into the formula:\[ \lambda = \frac{3 \times 10^8}{830 \times 10^3} \approx 361.45 \text{ meters} \]Understanding wavelength is important as it tells us the physical length of one complete wave cycle. This helps in designing antennas and other communication equipment to match the wave's length for optimal transmission and reception.

Wave Number

The wave number \( k \) is a way of describing the number of wave cycles in a unit distance, and is related to the wavelength. It gives us information about how "tightly" the wave cycles are packed in space. The formula for wave number is:\[ k = \frac{2\pi}{\lambda} \]Using the wavelength calculated previously, \( \lambda = 361.45 \text{ m} \), we substitute it into the equation:\[ k = \frac{2\pi}{361.45} \approx 0.0174 \text{ m}^{-1} \]The wave number is useful in many calculations, particularly in physics scenarios involving wave interference and diffraction. It essentially represents how many radians of phase you would move through per meter traveled.

• It's measured in units of inverse meters (\( \text{m}^{-1} \)).
• A higher wave number means shorter wavelengths, which corresponds to higher frequencies.

Angular Frequency

Angular frequency \( \omega \) relates to how fast the wave oscillates at any given point in space. Think of it as the rate of rotation for the wave -- it's measured in radians per second. The formula to calculate angular frequency is:\[ \omega = 2\pi f \]For the frequency \( f = 830 \times 10^3 \text{ Hz} \) from our example, the angular frequency is:\[ \omega = 2\pi \times 830 \times 10^3 \approx 5.22 \times 10^6 \text{ radians/second} \]Angular frequency is crucial in many physical situations:

• It defines how many cycles a wave undergoes per unit of time.
• It's directly proportional to the frequency and is an alternative way of expressing frequency.

When working with mathematical models of waves, using radians is advantageous because of the direct correspondence to the trigonometric functions often used in wave equations.

Electric Field Amplitude

The electric field amplitude \( E_{max} \) represents the maximum strength of the electric field component of an electromagnetic wave. It's directly interrelated with the magnetic field amplitude \( B_{max} \), and for light traveling through a vacuum or air, is given by:\[ E_{max} = c \times B_{max} \]Given that \( B_{max} = 4.82 \times 10^{-11} \text{ T} \) and the speed of light \( c = 3 \times 10^8 \text{ m/s} \), we can find the electric field amplitude as follows:\[ E_{max} = 3 \times 10^8 \times 4.82 \times 10^{-11} \approx 0.0145 \text{ V/m} \]This value tells us how strong the electric field is at its peak within the wave and is useful for understanding the wave's ability to induce currents or voltages in conductors it encounters.

• The electric field amplitude is measured in volts per meter (\( \text{V/m} \)).
• It plays a critical role in determining the intensity of the electromagnetic wave.