Chapter 32: Problem 4
Consider each of the following electric- and magneticfield orientations. In each case, what is the direction of propagation of the wave? (a) \(\vec{E} = E\hat{\imath}\), \(\vec{B} = -B\hat{\jmath}\); (b) \(\vec{E} = E\hat{\jmath}\), \(\vec{B} = B\hat{\imath}\); (c) \(\vec{E} = -E\hat{k}\) , \(\vec{B} = -B\hat{\imath}\); (d) \(vec{E} = E\hat{\imath}\), \(\vec{B} = -B\hat{k}\).
Short Answer
Step by step solution
Understand the Wave Propagation
Determine Direction for Part (a)
Determine Direction for Part (b)
Determine Direction for Part (c)
Determine Direction for Part (d)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Wave Propagation Direction
In the electromagnetic spectrum, both the electric field \( \vec{E} \) and the magnetic field \( \vec{B} \) are always perpendicular to each other and to the wave propagation direction. This means if you know the orientations of \( \vec{E} \) and \( \vec{B} \), you can determine the direction of \( \vec{k} \).
To find \( \vec{k} \), you use the formula \( \vec{k} = \vec{E} \times \vec{B} \). This equation is a mathematical representation of the direction of the wave by using the cross product. By calculating \( \vec{k} \), you determine the wave's travel path.
Electric Field Orientation
Electric fields are typically oriented along one axis in a three-dimensional coordinate system, such as \( \hat{i} \), \( \hat{j} \), or \( \hat{k} \). The chosen axis depends on the situation or the problem at hand.
Orientation of \( \vec{E} \) is significant because it is directly perpendicular to where the wave will travel and also perpendicular to \( \vec{B} \), the magnetic field. In exercises involving electromagnetic waves, identifying \( \vec{E} \) is the first step in figuring out the direction of \( \vec{k} \), which ultimately determines the propagation direction.
Magnetic Field Orientation
Magnetic fields in these scenarios are also oriented along an axis in the three-dimensional system, such as \( \hat{i} \), \( \hat{j} \), or \( \hat{k} \). A crucial point to remember is \( \vec{B} \) is perpendicular to both the electric field \( \vec{E} \) and the wave propagation direction \( \vec{k} \).
This perpendicular relationship forms the right-handed coordinate system, where \( \vec{E} \), \( \vec{B} \), and \( \vec{k} \) interact according to the right-hand rule. Thus, knowing the orientation of \( \vec{B} \) is vital in determining \( \vec{k} \).
Cross Product in Physics
In mathematical terms, the cross product of two vectors \( \vec{A} \) and \( \vec{B} \) is another vector \( \vec{C} \), which is orthogonal to both \( \vec{A} \) and \( \vec{B} \). The magnitude of \( \vec{C} \) is given by \( |\vec{C}| = |\vec{A}| |\vec{B}| \sin(\theta) \), where \( \theta \) is the angle between \( \vec{A} \) and \( \vec{B} \). The direction of \( \vec{C} \) follows the right-hand rule.
In electromagnetic problems, \( \vec{k} = \vec{E} \times \vec{B} \), helps us find the wave's direction. Using the cross product ensures that \( \vec{k} \) is perpendicular to both the electric and magnetic fields, adhering to the fundamental properties of wave propagation in free space.