Question

A sinusoidal electromagnetic wave having a magnetic field of amplitude 1.25 \(\mu\)T 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 Eqs. (32.17).

Short answer

(a) Frequency is approximately 6.94 × 10^14 Hz. (b) The electric field amplitude is 375 V/m. (c) See steps 3 and 4 for field equations.

Step-by-step solution

Calculate the Frequency of the Wave

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

Determine the Amplitude of the Electric Field

The relationship between the magnitude of electric field \( E_0 \) and the magnetic field \( B_0 \) in an electromagnetic wave is given by \( E_0 = cB_0 \). Substituting the given values \( c = 3 \times 10^8 \, \text{m/s} \) and \( B_0 = 1.25 \times 10^{-6} \, \text{T} \): \[ E_0 = 3 \times 10^8 \times 1.25 \times 10^{-6} = 375 \, \text{V/m} \]

Write the Equation for the Electric Field

The electric field equation of a wave traveling in the +\(x\)-direction is given by: \[ E(x,t) = E_0 \cos(kx - \omega t) \]where \( k = \frac{2\pi}{\lambda} \) is the wave number and \( \omega = 2\pi f \) is the angular frequency.First, calculate \( k \) and \( \omega \):\[ k = \frac{2\pi}{432 \times 10^{-9}} = 1.45 \times 10^7 \, \text{rad/m} \]\[ \omega = 2 \pi \times 6.94 \times 10^{14} = 4.36 \times 10^{15} \, \text{rad/s} \]Thus, the electric field equation is:\[ E(x,t) = 375 \cos (1.45 \times 10^7 x - 4.36 \times 10^{15} t) \text{ V/m} \]

Write the Equation for the Magnetic Field

The magnetic field has a similar form as the electric field but uses the amplitude \( B_0 \):\[ B(x,t) = B_0 \cos(kx - \omega t) \]Using the previously calculated values of \( k \) and \( \omega \), the magnetic field equation is:\[ B(x,t) = 1.25 \times 10^{-6} \cos(1.45 \times 10^7 x - 4.36 \times 10^{15} t) \text{ T} \]

Key concepts

Frequency Calculation

Understanding the frequency of an electromagnetic wave is essential in many fields, including optics and telecommunications. Frequency (\( f \) ) determines how many cycles of the wave pass a point in one second and is closely tied to the wave's energy. The equation connecting frequency to wavelength (\( \lambda \) ) is: \[ c = \lambda f \]where \( c \) is the speed of light, approximately \( 3 \times 10^8 \text{ m/s} \). To find frequency, rearrange the equation: \[ f = \frac{c}{\lambda}\]Substitute the given wavelength of 432 nm (which is \( 432 \times 10^{-9} \text{ m} \) in meters) into the formula. Calculating gives us a frequency of approximately \( 6.94 \times 10^{14} \text{ Hz} \). This is a high frequency typically associated with visible light.

Electric Field Amplitude

In an electromagnetic wave, the electric field amplitude \( E_0 \) is directly related to the wave's magnetic field amplitude \( B_0 \). The relationship is given by:\[ E_0 = cB_0\]where \( c \) is again the speed of light. This equation shows that the electric field's strength increases with a stronger magnetic field. Given that the magnetic field amplitude is 1.25 \( \mu \text{T} \) (or \( 1.25 \times 10^{-6} \text{ T} \) ), we substitute into the formula:\[ E_0 = 3 \times 10^8 \times 1.25 \times 10^{-6} = 375 \text{ V/m}\]This result tells us that in this wave, the peak electric field is 375 volts per meter, which means it has a significant electric component.

Magnetic Field Equation

The magnetic field in an electromagnetic wave can be described by a sinusoidal equation, similar to the electric field. The form of this equation is:\[ B(x,t) = B_0 \cos(kx - \omega t)\]In this equation, \( B_0 \) is the amplitude of the magnetic field, \( k \) is the wave number, and \( \omega \) is the angular frequency. The wave number \( k \) is determined by:\[ k = \frac{2\pi}{\lambda}\]where \( \lambda \) is the wavelength. Given the wavelength, the calculated wave number is approximately \( 1.45 \times 10^7 \text{ rad/m} \).The angular frequency \( \omega \) is found using:\[ \omega = 2\pi f\]With the earlier calculated frequency, this results in about \( 4.36 \times 10^{15} \text{ rad/s} \). Consequently, the magnetic field equation stands as:\[ B(x,t) = 1.25 \times 10^{-6} \cos(1.45 \times 10^7 x - 4.36 \times 10^{15} t) \text{ T}\]

Wave Equation Formulation

Formulating wave equations helps us understand how electromagnetic waves propagate through space. Both electric and magnetic fields in these waves are described by sinusoidal functions of spatial and temporal variables.For the electric field, use the generic form:\[ E(x,t) = E_0 \cos(kx - \omega t)\]Substituting in the calculated amplitude \( E_0 = 375 \, \text{V/m} \), wave number \( k = 1.45 \times 10^7 \, \text{rad/m} \), and angular frequency \( \omega = 4.36 \times 10^{15} \, \text{rad/s} \), the equation becomes:\[ E(x,t) = 375 \cos (1.45 \times 10^7 x - 4.36 \times 10^{15} t) \, \text{V/m}\]This equation represents how the electric field varies with both position and time, showing its sinusoidal nature as it moves through space. Both the electric and magnetic wave equations show that electromagnetic waves are a complex balance of their field components, propagating with constant phase and frequency.