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Magazine Chapter 33: Problem 13
Sunlight just outside Earth's atmosphere has an intensity of \(1.40 \mathrm{~kW} / \mathrm{m}^{2} .\) Calculate (a) \(E_{m}\) and (b) \(B_{m}\) for sunlight there, assuming it to be a plane wave.
Short Answer
Expert verified
\(E_{m} \approx 1027 \, \mathrm{N/C}, B_{m} \approx 3.42 \times 10^{-6} \, \mathrm{T}\).
Step by step solution
01
Understand the Problem
The problem provides us with the intensity of sunlight just outside Earth's atmosphere, and we need to find the maximum electric field \(E_{m}\) and magnetic field \(B_{m}\) assuming the sunlight is a plane wave.
02
Use the Formula for Intensity of EM Wave
We use the relationship between intensity \(I\) and the maximum electric field \(E_{m}\) for an electromagnetic wave: \[\[\begin{equation}I = \frac{1}{2} \epsilon_0 c E_{m}^{2}\end{equation}\]\]where \(\epsilon_0\) is the permittivity of free space \(\approx 8.85 \times 10^{-12} \, \mathrm{C}^2/\mathrm{N} \, \mathrm{m}^2\), and \(c\) is the speed of light \(\approx 3 \times 10^8 \, \mathrm{m/s}\).
03
Solve for Maximum Electric Field \(E_{m}\)
Rearrange the intensity formula to solve for \(E_{m}\):\[\[\begin{equation}E_{m} = \sqrt{\frac{2I}{\epsilon_0 c}}\end{equation}\]\]Substitute the given intensity and constants: \(I = 1.40 \, \mathrm{kW/m^2} = 1.40 \times 10^3 \, \mathrm{W/m^2}\), \[E_{m} = \sqrt{\frac{2 \times 1.40 \times 10^3}{8.85 \times 10^{-12} \times 3 \times 10^8}}\approx 1027 \, \mathrm{N/C}\].
04
Use the Relationship Between \(E_{m}\) and \(B_{m}\)
For plane electromagnetic waves, there is a simple relationship between the maximum electric field \(E_{m}\) and the maximum magnetic field \(B_{m}\):\[\[\begin{equation}B_{m} = \frac{E_{m}}{c}\end{equation}\]\]Using the calculated \(E_{m}\) of \(1027 \, \mathrm{N/C}\), calculate \(B_{m}\).
05
Calculate Maximum Magnetic Field \(B_{m}\)
Substitute \(E_{m}\) and \(c\) into the formula:\[B_{m} = \frac{1027}{3 \times 10^8} \approx 3.42 \times 10^{-6} \, \mathrm{T}\]The maximum magnetic field is hence approximately \(3.42 \times 10^{-6} \, \mathrm{T}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Electric Field
An electric field represents the force exerted by an electric charge or a collection of loads in space. The electric field surrounding an object acts directly on other charges in its vicinity, either attracting or repelling them. These forces and interactions are central elements in the study of electromagnetism.
**Understanding the Electric Field in Electromagnetic Waves**
The electric field of an electromagnetic wave, such as light, exhibits a relationship with the wave's intensity. Intensity is the power carried by a wave per unit area, and it's measured in watts per square meter (W/m虏). This relationship is significant in many practical applications, like in solar panels where sunlight needs to be converted into usable energy.
The formula for this relationship is derived from the electromagnetic wave theory, and it's given by:
**Understanding the Electric Field in Electromagnetic Waves**
The electric field of an electromagnetic wave, such as light, exhibits a relationship with the wave's intensity. Intensity is the power carried by a wave per unit area, and it's measured in watts per square meter (W/m虏). This relationship is significant in many practical applications, like in solar panels where sunlight needs to be converted into usable energy.
The formula for this relationship is derived from the electromagnetic wave theory, and it's given by:
- \( I = \frac{1}{2} \epsilon_0 c E_{m}^2 \)
- \(I\) is the intensity.
- \( \epsilon_0 \) is the permittivity of free space, approximately \( 8.85 \times 10^{-12} \, \text{C}^2/\text{N}\, \text{m}^2 \).
- \(c\) is the speed of light in vacuum, approximately \( 3 \times 10^8 \, \text{m/s} \).
- \(E_m\) is the maximum electric field.
Magnetic Field
A magnetic field is a fundamental concept in physics, representing the region around a magnetic material or a moving electric charge within which the force of magnetism acts. Like the electric field, the magnetic field forms a critical part of electromagnetic waves.
**The Role of Magnetic Field in Electromagnetic Waves**
In an electromagnetic wave, the magnetic field fluctuates perpendicular to the electric field and contributes to the overall energy of the wave. The neat trick about electromagnetic waves is that the amplitude of the electric field \(E_m\) directly affects the magnitude of the magnetic field \(B_m\), following the relationship:
\[B_m = \frac{E_m}{c}\]
This equation shows how the maximum magnetic field strength arises from the electric field strength scaled by the speed of light \(c\). It illustrates the inherent balance of electric and magnetic components in light, emphasizing how they propagate together in space.
**The Role of Magnetic Field in Electromagnetic Waves**
In an electromagnetic wave, the magnetic field fluctuates perpendicular to the electric field and contributes to the overall energy of the wave. The neat trick about electromagnetic waves is that the amplitude of the electric field \(E_m\) directly affects the magnitude of the magnetic field \(B_m\), following the relationship:
\[B_m = \frac{E_m}{c}\]
This equation shows how the maximum magnetic field strength arises from the electric field strength scaled by the speed of light \(c\). It illustrates the inherent balance of electric and magnetic components in light, emphasizing how they propagate together in space.
Intensity of Light
The intensity of light refers to the power per unit area received from an electromagnetic wave, often determining how bright or powerful the light appears. In many contexts, intensity relates closely to the wave鈥檚 energy output and application.
**Importance and Calculation of Light Intensity**
Intensity is crucial in various scientific and engineering applications, like designing optical systems and analyzing light emissions. For sunlight or any electromagnetic wave, intensity \(I\) can be connected to the electric field component \(E_m\) by the formula:
\[I = \frac{1}{2} \epsilon_0 c E_m^2\]
This equation allows us to deduce necessary characteristics of an electromagnetic wave if certain variables, like the electric field or the permittivity of free space, are known. It's invaluable in practical conversions of sunlight into electrical energy in solar technologies and gives a solid theoretical foundation for understanding how light interacts with materials.
**Importance and Calculation of Light Intensity**
Intensity is crucial in various scientific and engineering applications, like designing optical systems and analyzing light emissions. For sunlight or any electromagnetic wave, intensity \(I\) can be connected to the electric field component \(E_m\) by the formula:
\[I = \frac{1}{2} \epsilon_0 c E_m^2\]
This equation allows us to deduce necessary characteristics of an electromagnetic wave if certain variables, like the electric field or the permittivity of free space, are known. It's invaluable in practical conversions of sunlight into electrical energy in solar technologies and gives a solid theoretical foundation for understanding how light interacts with materials.
