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Chapter 25: Problem 50

Electromagnetic wave 1 has a maximum electric field of \(E_{0}=52 \mathrm{V} / \mathrm{m},\) and electromagnetic wave 2 has a maximum magnetic field of \(B_{0}=1.5 \mu T .\) (a) Which wave has the greater intensity? (b) Calculate the intensity of each wave.

Short Answer

Expert verified
Wave 2 has the greater intensity. Intensities are 4.63 W/m虏 for wave 1 and 240.2 W/m虏 for wave 2.

Step by step solution

01

Maximum Electric Field Conversion

For electromagnetic wave 1, we have the maximum electric field \(E_0 = 52 \text{ V/m}\). To find the intensity, we first need the maximum magnetic field, which is related to the electric field by the equation \( E_0 = c B_0 \), where \(c\) is the speed of light \( (3 \times 10^8 \text{ m/s}) \). Rearrange this to find \(B_0\): \[ B_0 = \frac{E_0}{c} \] \[ B_0 = \frac{52}{3 \times 10^8} \approx 1.73 \times 10^{-7} \text{ T} \].
02

Calculate Intensity of Wave 1

The intensity \(I\) of an electromagnetic wave can be calculated using the formula: \[ I = \frac{1}{2}c\epsilon_0E_0^2 \] where \(\epsilon_{0} = 8.85 \times 10^{-12} \text{ F/m} \) is the permittivity of free space. Calculate the intensity of wave 1: \[ I_1 = \frac{1}{2} \times 3 \times 10^8 \times 8.85 \times 10^{-12} \times (52)^2 \] \[ I_1 \approx \frac{1}{2} \times 3 \times 10^8 \times 8.85 \times 10^{-12} \times 2704 \] \[ I_1 \approx 4.63 \text{ W/m}^2 \].
03

Magnetic Field Conversion for Wave 2

Wave 2's maximum magnetic field \(B_0 = 1.5 \mu\text{T} = 1.5 \times 10^{-6} \text{ T}\). Use \( E_0 = c B_0 \) to find the maximum electric field: \[ E_0 = 3 \times 10^8 \times 1.5 \times 10^{-6} \] \[ E_0 \approx 450 \text{ V/m} \].
04

Calculate Intensity of Wave 2

Now calculate the intensity of wave 2 using the same formula for intensity: \[ I_2 = \frac{1}{2}c\epsilon_0E_0^2 \] \[ I_2 = \frac{1}{2} \times 3 \times 10^8 \times 8.85 \times 10^{-12} \times (450)^2 \] \[ I_2 \approx \frac{1}{2} \times 3 \times 10^8 \times 8.85 \times 10^{-12} \times 202500 \] \[ I_2 \approx 240.2 \text{ W/m}^2 \].
05

Compare Intensities

With \(I_1 \approx 4.63 \text{ W/m}^2\) and \(I_2 \approx 240.2 \text{ W/m}^2\), we can clearly see that the intensity of wave 2 is much greater than that of wave 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electric Field
The electric field, sometimes denoted as \(E_0\), is a vital component in understanding electromagnetic waves. It represents the force experienced by a charged particle in the presence of a wave. In our context, the electric field of a wave contributes significantly to the wave's intensity.

  • An electric field is measured in volts per meter (V/m).
  • It indicates how much force a unit charge would experience at any given point in the field's reach.
In vacuum, the electric field \(E_0\) relates to the magnetic field \(B_0\) using the speed of light \(c\) with the equation: \(E_0 = c B_0\). This relationship is crucial because it helps us deduce one field if the other is known.

Understanding the electric field's role is essential because it directly influences other properties like intensity. Hence, accurate calculation of \(E_0\) forms the foundational step in determining the energy carried by electromagnetic waves.
Magnetic Field
In electromagnetic waves, the magnetic field is a crucial counterpart to the electric field. Represented by \(B_0\), the magnetic field oscillates in a direction perpendicular to the electric field, and is measured in teslas (T). Here are some key points to understand better:

  • Magnetic fields arise from moving charges or dynamic changes in electric fields.
  • They interrelate with electric fields to form the complete picture of electromagnetic waves.
When given the maximum magnetic field as in our exercise \(B_0 = 1.5 \mu \text{T}\), it can be transformed into its corresponding electric field value using the speed of light \(c\): \(E_0 = c B_0\). This relationship underscores the magnetic field's significance in calculations of wave properties.

By accurately determining the magnetic field, we lay the ground for further calculations, such as the determination of wave intensity, which gives insight into the wave's power and effectiveness in transmitting energy.
Wave Intensity
The intensity of an electromagnetic wave is a measure of its power per unit area. This concept quantifies how much energy is carried by the wave through space. The intensity \(I\) depends on both electric and magnetic fields and is given by the formula: \[I = \frac{1}{2} c \epsilon_0 E_0^2\].

Here, \(\epsilon_0\) is the permittivity of free space, which is about \(8.85 \times 10^{-12} \text{ F/m}\). This formula reflects:
  • The role of the speed of light \(c\) in propagating wave energy.
  • How the strength of the electric field \(E_0\) directly affects the wave's intensity.
In the provided exercise, we calculated two different wave intensities to determine which wave was stronger. Wave 2, with a calculated intensity of approximately \(240.2 \text{ W/m}^2\), proved more intense than Wave 1's \(4.63 \text{ W/m}^2\). This understanding of intensity helps compare and contrast waves, shedding light on how energetic or penetrating they might be in practical applications.

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Most popular questions from this chapter

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