Problem 2 - University Physics with Modern Physics

Question

Consider each of the electric- and magnetic-field orientations given next. In each case, what is the direction of propagation of the wave? (a) \(\vec{E}\) in the +\(x\)-direction, \(\vec{B}\) in the +\(y\)-direction; (b) \(\vec{E}\) in the -\(y\)-direction, \(\vec{B}\) in the +\(x\)-direction; (c) \(\vec{E}\) in the +\(z\)-direction, \(\vec{B}\) in the -\(x\)-direction; (d) \(\vec{E}\) in the +\(y\)-direction, \(\vec{B}\) in the -\(z\)-direction.

Step-by-Step Solution

Answer

(a) +z; (b) -z; (c) +y; (d) +x.

Step 1: Understanding Electromagnetic Wave Propagation

In electromagnetic wave propagation, the direction of propagation of the wave is given by the cross product of the electric field \(\vec{E}\) and the magnetic field \(\vec{B}\). The direction of this propagation is given by the right-hand rule, meaning that if you point your index finger in the direction of \(\vec{E}\) and your middle finger in the direction of \(\vec{B}\), then your thumb will point in the direction of wave propagation.

Step 2: Case (a): Fields in +x and +y Directions

Given \(\vec{E}\) in the +\(x\)-direction and \(\vec{B}\) in the +\(y\)-direction. Using the right-hand rule, point your index finger in the +\(x\)-direction and your middle finger in the +\(y\)-direction. Your thumb points in the +\(z\)-direction, indicating the wave propagates in the +\(z\)-direction.

Step 3: Case (b): Fields in -y and +x Directions

Here \(\vec{E}\) is in the -\(y\)-direction and \(\vec{B}\) in the +\(x\)-direction. Point your index finger in the -\(y\)-direction and your middle finger in the +\(x\)-direction. Your thumb points in the -\(z\)-direction, so the wave propagates in the -\(z\)-direction.

Step 4: Case (c): Fields in +z and -x Directions

With \(\vec{E}\) in the +\(z\)-direction and \(\vec{B}\) in the -\(x\)-direction, point your index finger in the +\(z\)-direction and your middle finger in the -\(x\)-direction. Your thumb points in the +\(y\)-direction, indicating that the wave propagates in the +\(y\)-direction.

Step 5: Case (d): Fields in +y and -z Directions

Here \(\vec{E}\) is in the +\(y\)-direction and \(\vec{B}\) is in the -\(z\)-direction. Using the right-hand rule, point your index finger in the +\(y\)-direction and your middle finger in the -\(z\)-direction. Your thumb will point in the +\(x\)-direction, meaning the wave propagates in the +\(x\)-direction.

Key Concepts

Right-Hand Rule

The right-hand rule is a handy mnemonic that helps you determine the direction of electromagnetic wave propagation. It uses the orientation of your fingers to describe vectors in relation to one another. Imagine you're holding your right hand out:

• Extend your index finger in the direction of the electric field \(\vec{E}\).
• Extend your middle finger perpendicular to your index finger, pointing in the direction of the magnetic field \(\vec{B}\).
• Your thumb, extended at a right angle to both fingers, will then indicate the direction of wave propagation.

This is a quick and effective way to visualize and calculate the direction of an electromagnetic wave using vector quantities. It helps you comprehend how these two orthogonal components, the electric and magnetic fields, interact to produce a wave in space.

Electric Field Orientation

An electric field is a vector field that represents the force exerted on a charged particle. Its direction can be described as the path along which a positive charge would accelerate.
In the context of electromagnetic waves, this field determines the orientation of one component of the wave. It oscillates perpendicular to the magnetic field component.

• In Case (a), the electric field \(\vec{E}\) is oriented in the +\(x\)-direction.
• In Case (b), it points in the -\(y\)-direction.
• In Case (c), \(\vec{E}\) is aligned in the +\(z\)-direction.
• In Case (d), it's directed in the +\(y\)-direction.

These orientations are fundamental in evaluating the electromagnetic wave's overall direction using the right-hand rule.

Magnetic Field Orientation

Magnetic field orientation is another critical component in electromagnetic wave propagation. The magnetic field, much like its electric counterpart, is a vector field that affects charged particles. It oscillates perpendicular to both the electric field and the wave propagation direction.

• In Case (a), the magnetic field \(\vec{B}\) points in the +\(y\)-direction.
• In Case (b), it is oriented in the +\(x\)-direction.
• In Case (c), \(\vec{B}\) emerges in the -\(x\)-direction.
• In Case (d), it is arranged in the -\(z\)-direction.

These orientations guide the formation of electromagnetic waves when used in conjunction with the electric field direction and the right-hand rule. Understanding the relationship between these vector positions is crucial for identifying the wave's path.