Problem 4 - University Physics with Modern Physics

Question

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) \(vec{E} = E\hat{\imath}\), \(\vec{B} = -B\hat{k}\).

Step-by-Step Solution

Answer

(a) -\hat{k}, (b) -\hat{k}, (c) \hat{\jmath}, (d) \hat{\jmath}.

Step 1: Understand the Wave Propagation

For an electromagnetic wave in free space, the electric field \(\vec{E}\), the magnetic field \(\vec{B}\), and the direction of propagation \(\vec{k}\) are all perpendicular to each other. This relationship can be expressed as \(\vec{k} = \vec{E} \times \vec{B}\), where \(\times\) denotes the cross product.

Step 2: Determine Direction for Part (a)

Given \(\vec{E} = E\hat{\imath}\) and \(\vec{B} = -B\hat{\jmath}\), compute \(\vec{k}\) using the cross product. \(\vec{k} = (E\hat{\imath}) \times (-B\hat{\jmath}) = -EB(\hat{\imath} \times \hat{\jmath}) = -EB\hat{k}\). Thus, the wave propagates in the \(-\hat{k}\) direction.

Step 3: Determine Direction for Part (b)

Given \(\vec{E} = E\hat{\jmath}\) and \(\vec{B} = B\hat{\imath}\), compute \(\vec{k}\). \(\vec{k} = (E\hat{\jmath}) \times (B\hat{\imath}) = EB(\hat{\jmath} \times \hat{\imath}) = -EB\hat{k}\). Thus, the wave propagates in the \(-\hat{k}\) direction.

Step 4: Determine Direction for Part (c)

Given \(\vec{E} = -E\hat{k}\) and \(\vec{B} = -B\hat{\imath}\), compute \(\vec{k}\). \(\vec{k} = (-E\hat{k}) \times (-B\hat{\imath}) = EB(\hat{k} \times \hat{\imath}) = EB\hat{\jmath}\). Thus, the wave propagates in the \(\hat{\jmath}\) direction.

Step 5: Determine Direction for Part (d)

Given \(\vec{E} = E\hat{\imath}\) and \(\vec{B} = -B\hat{k}\), compute \(\vec{k}\). \(\vec{k} = (E\hat{\imath}) \times (-B\hat{k}) = -EB(\hat{\imath} \times \hat{k}) = EB\hat{\jmath}\). Thus, the wave propagates in the \(\hat{\jmath}\) direction.

Key Concepts

Wave Propagation Direction

Understanding the direction of wave propagation is crucial in electromagnetism. When dealing with electromagnetic waves, such as light, they propagate through space in a specific direction. This direction is denoted by the vector \( \vec{k} \). In simpler terms, \( \vec{k} \) shows where the wave is headed.

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

The electric field \( \vec{E} \) is a vector field that represents the electric force per unit charge at different points in space. In an electromagnetic wave, the direction of \( \vec{E} \) is one of the key factors influencing the wave's propagation.

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

Just like the electric field, the magnetic field \( \vec{B} \) is a vector. It represents the magnetic influence on moving charges, current loops, and magnetized materials. In electromagnetic waves, \( \vec{B} \) interacts with the electric field to dictate the wave’s direction.

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

The cross product, denoted by \( \times \), is a fundamental operation in vector algebra used in physics to find a vector perpendicular to two given vectors. In the context of electromagnetic waves, it is employed to calculate the direction of wave propagation \( \vec{k} \).

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.