A plane electromagnetic wave having a frequency v = 23.9 GHz propagates along the positive z-directi

Q: Why is the phase $(kz - \omega t)$ and not $(kz + \omega t)$?

A phase of $(kz - \omega t)$ represents a wave traveling in the positive z-direction. If it were $(kz + \omega t)$, the wave would be traveling in the negative z-direction.

Q: How do we know the magnetic field is along the x or y axis?

In an electromagnetic wave, the electric field, magnetic field, and direction of propagation are all mutually perpendicular. Since the wave travels along the z-axis, the magnetic field must lie in the xy-plane.

Q: Why do we use angular frequency ($\omega$) instead of just frequency ($\nu$) in the wave equation?

The sine function requires an angle in radians. Multiplying the frequency (cycles per second) by $2\pi$ gives the angular frequency in radians per second, making the units consistent for the phase term.

Q: What if the electric field was given as an RMS value instead of a peak value?

You would first need to convert the RMS value to the peak value by multiplying it by $\sqrt{2}$ before using the formula $E_0 = c B_0$.

Q: Can the magnetic field be in the z-direction?

No, electromagnetic waves are transverse waves. The oscillating fields must be perpendicular to the direction of energy transfer (propagation).

Question

A plane electromagnetic wave having a frequency $
u = 23.9 	ext{ GHz}$ propagates along the positive z-direction in free space. The peak value of the electric field is $60 	ext{ V/m}$. Which among the following is the acceptable magnetic field component in the electromagnetic wave?

Options:

(A) $\mathbf{B} = 2 	imes 10^{-7} \sin(0.5 	imes 10^3 z + 1.5 	imes 10^{11} t) \hat{i}$

(B) $\mathbf{B} = 2 	imes 10^{-7} \sin(0.5 	imes 10^3 z - 1.5 	imes 10^{11} t) \hat{i}$

(C) $\mathbf{B} = 60 \sin(0.5 	imes 10^3 x + 1.5 	imes 10^{11} t) \hat{k}$

(D) $\mathbf{B} = 2 	imes 10^{-7} \sin(1.5 	imes 10^2 x + 0.5 	imes 10^{11} t) \hat{j}$

Sigma Percentile · Accepted Answer

The universe is constantly buzzing with invisible energy, and electromagnetic waves are the ultimate messengers. From the light we see to the Wi-Fi signals connecting our devices, these waves are everywhere.

In this problem, we are tasked with decoding the exact mathematical signature of one such wave. Let's break down the physics and construct the magnetic field equation step by step.

### Analyzing the Setup

We are given a plane electromagnetic wave traveling through free space. The wave is propagating along the positive z-direction.

This is a crucial piece of information because the direction of travel dictates the structure of our wave equation. We are also given the frequency of the wave, $
u = 23.9 	ext{ GHz}$, and the peak electric field, $E_0 = 60 	ext{ V/m}$.

In any electromagnetic wave, the electric field ($\mathbf{E}$) and the magnetic field ($\mathbf{B}$) are intimately connected. They oscillate perpendicular to each other and perpendicular to the direction of propagation.

### The Master Equation

To write the equation for the magnetic field, we need three key ingredients: the peak amplitude ($B_0$), the angular frequency ($\omega$), and the wave number ($k$).

Let's start with the amplitude. The ratio of the peak electric field to the peak magnetic field is always equal to the speed of light ($c$).

$$B_0 = \frac{E_0}{c}$$

Substituting our known values, we get:

$$B_0 = \frac{60}{3 	imes 10^8} = 2 	imes 10^{-7} 	ext{ T}$$

Next, we calculate the angular frequency, which tells us how fast the fields are oscillating in time.

$$\omega = 2\pi
u$$

Plugging in our frequency, we find:

$$\omega = 2\pi 	imes 23.9 	imes 10^9 \approx 1.5 	imes 10^{11} 	ext{ rad/s}$$

Finally, we need the wave number, which describes how the wave varies in space.

$$k = \frac{\omega}{c}$$

Using our previously calculated $\omega$, we get:

$$k = \frac{1.5 	imes 10^{11}}{3 	imes 10^8} = 0.5 	imes 10^3 	ext{ m}^{-1}$$

### Constructing the Final Wave

Now comes the most critical part: assembling the phase of the wave. Because the wave is traveling in the **positive z-direction**, the spatial and temporal parts of the phase must have opposite signs.

This means our phase term will look like $(kz - \omega t)$. If the wave were traveling in the negative z-direction, it would be $(kz + \omega t)$.

Putting it all together, the general form of our magnetic field is:

$$\mathbf{B} = B_0 \sin(kz - \omega t) \hat{i}$$

Substituting all our calculated values into this framework, we arrive at the final equation:

$$\mathbf{B} = 2 	imes 10^{-7} \sin(0.5 	imes 10^3 z - 1.5 	imes 10^{11} t) \hat{i}$$

This perfectly matches option (b). By carefully analyzing the direction of propagation and the fundamental relationships between the fields, we've successfully decoded the wave's signature!

Select Answer:

Visualized Solution

$Visualizing the EM Wave$

$Given Parameters$

$Peak Magnetic Field (B_{0})$

$Angular Frequency (ω)$

$Wave Number (k)$

$Phase of the Wave$

$Final Equation$

The Sigma Insight: Characteristics of Electromagnetic Waves

Analyzing the Setup

The Master Equation

Constructing the Final Wave

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Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Visualizing the EM Wave

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Given Parameters

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Peak Magnetic Field (B0​)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Angular Frequency (ω)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Wave Number (k)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Phase of the Wave

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Final Equation