Problem 8 - College Physics

Question

\(\bullet\) A sinusoidal electromagnetic wave having a magnetic field of amplitude 1.25 \(\mu\) and a wavelength of 432 nm is traveling in the \(+x\) direction through empty space. (a) What is the frequency of this wave? (b) What is the amplitude of the associated electric field? (c) Write the equations for the electric and magnetic fields as functions of \(x\) and \(t\) in the form of Equations \((23.3) .\)

Step-by-Step Solution

Answer

(a) Frequency: \(6.94 \times 10^{14}\) Hz. (b) Electric field amplitude: 375 V/m. (c) See steps for field equations.

Step 1: Calculate the Frequency of the Wave

To find the frequency \(f\) of the wave, we use the wave equation relating speed \(c\), frequency \(f\), and wavelength \(\lambda\): \[ c = f \cdot \lambda \]Given that the speed of light is \(c = 3 \times 10^8 \text{ m/s}\), and the wavelength \(\lambda = 432 \text{ nm} = 432 \times 10^{-9} \text{ m}\), we can solve for \(f\):\[ f = \frac{c}{\lambda} = \frac{3 \times 10^8 \text{ m/s}}{432 \times 10^{-9} \text{ m}} \]\( f \approx 6.94 \times 10^{14} \text{ Hz} \).

Step 2: Determine the Amplitude of the Electric Field

Use the relationship between the magnitudes of the electric field \(E\) and the magnetic field \(B\) amplitude for electromagnetic waves:\[ E = cB \]The amplitude of the magnetic field is given as \(B = 1.25 \times 10^{-6} \text{ T}\), so:\[ E = (3 \times 10^8) \times (1.25 \times 10^{-6}) \approx 375 \text{ V/m} \]

Step 3: Write the Equations for Electric and Magnetic Fields

The equations for the electric and magnetic fields of a sinusoidal wave can be written as:For the electric field:\[ E(x,t) = E_0 \sin(kx - \omega t) \]For the magnetic field:\[ B(x,t) = B_0 \sin(kx - \omega t) \]Where:- \(E_0 = 375 \text{ V/m}\) is the amplitude of the electric field.- \(B_0 = 1.25 \times 10^{-6} \text{ T}\) is the amplitude of the magnetic field.- \(k = \frac{2\pi}{\lambda}\) is the wave number: \[ k = \frac{2\pi}{432 \times 10^{-9}} \approx 1.45 \times 10^7 \text{ m}^{-1} \]- \(\omega = 2\pi f\) is the angular frequency:\[ \omega = 2\pi \times 6.94 \times 10^{14} \approx 4.36 \times 10^{15} \text{ rad/s} \]Putting it all together, the equations are:\[ E(x,t) = 375 \sin\left(1.45 \times 10^7 x - 4.36 \times 10^{15} t\right) \]\[ B(x,t) = 1.25 \times 10^{-6} \sin\left(1.45 \times 10^7 x - 4.36 \times 10^{15} t\right) \]

Key Concepts

Frequency Calculation

In the world of electromagnetic waves, frequency is a fundamental characteristic that is tightly linked to wavelength. To determine the frequency of a wave, you can use the wave equation which connects the speed of light, wavelength, and frequency. The equation is:\[ c = f \cdot \lambda \]Where:

• c is the speed of light, approximately \(3 \times 10^8\) m/s
• f is the frequency
• \(\lambda\) is the wavelength

In our exercise, we’re given a wavelength of 432 nm, which is converted into meters by multiplying with \(10^{-9}\). Substitute these values into the equation to solve for frequency:\[ f = \frac{c}{\lambda} = \frac{3 \times 10^8}{432 \times 10^{-9}} \approx 6.94 \times 10^{14} \text{ Hz} \]This calculation shows the frequency of the wave as about 694 THz, illustrating how short wavelengths correspond to high frequencies in electromagnetic waves.

Electric and Magnetic Fields

Electromagnetic waves are fascinating because they consist of oscillating electric (E) and magnetic (B) fields that travel through space together. A crucial relationship in electromagnetic theory is the proportionality between the amplitude of these fields:\[ E = cB \]Where:

• E is the electric field amplitude
• c is the speed of light
• B is the magnetic field amplitude

To find the electric field amplitude from a given magnetic field amplitude, multiply by the speed of light. In this exercise, given that the magnetic field amplitude \(B = 1.25 \times 10^{-6} \text{ T}\):\[ E = (3 \times 10^8) \cdot (1.25 \times 10^{-6}) \approx 375 \text{ V/m} \]This relationship showcases how both fields are intrinsically connected in electromagnetic radiation, indicating that a wave’s properties can be fully described by either field.

Wave Equation

The wave equation is vital for describing electromagnetism in mathematical form. It characterizes how the electric and magnetic fields vary with space \(x\) and time \(t\). For sinusoidal waves, the general form of the equations are:

• For the electric field: \[ E(x,t) = E_0 \sin(kx - \omega t) \]
• For the magnetic field: \[ B(x,t) = B_0 \sin(kx - \omega t) \]

Where:

• E₀ and B₀ are the amplitudes of the electric and magnetic fields, respectively
• k is the wave number, \( \frac{2\pi}{\lambda} \)
• \(\omega\) is the angular frequency, \(2\pi f\)

In our specific case:

• k was calculated as: \[ k = \frac{2\pi}{432 \times 10^{-9}} \approx 1.45 \times 10^7 \text{ m}^{-1} \]
• \(\omega\) was found to be: \[ \omega = 2\pi \times 6.94 \times 10^{14} \approx 4.36 \times 10^{15} \text{ rad/s} \]

This results in wave equations describing their propagation as:\[ E(x,t) = 375 \sin(1.45 \times 10^7 x - 4.36 \times 10^{15} t) \]\[ B(x,t) = 1.25 \times 10^{-6} \sin(1.45 \times 10^7 x - 4.36 \times 10^{15} t) \] These equations illustrate the oscillatory nature of electromagnetic fields, weaving in space and time as they propagate through a vacuum.