Question

If \(\mathbf{E}=\) electric field and \(\mathbf{B}=\) magnetic field, is \(\mathbf{E} \times \mathbf{B}\) a vector or a pseudovector? Comment: \(\mathbf{E} \times \mathbf{B} / \mu_{0}\) is called the Poynting vector; it points in the direction of transfer of energy. Does that tell you from the physics whether it is a vector or a pseudovector?

Short answer

\( \mathbf{E} \times \mathbf{B} \) is a pseudovector because it is the cross product of two vectors.

Step-by-step solution

Define Vector and Pseudovector

A vector is a quantity that has both magnitude and direction and transforms like a coordinate under rotation. A pseudovector, also known as an axial vector, behaves like a vector when rotated but gains an additional sign change under improper transformations like reflections.

Understand Cross Product

The cross product of two vectors \( \mathbf{A} \times \mathbf{B} \) results in a vector that is perpendicular to both \( \mathbf{A} \) and \( \mathbf{B} \.\) Importantly, the cross product also follows the right-hand rule in determining its direction.

Properties of E and B Fields

The electric field \( \mathbf{E} \) and the magnetic field \( \mathbf{B} \) are both true vectors (polar vectors). They are quantities derived from actual physical phenomena and they transform as vectors under coordinate transformations.

Analyze The Product \( \mathbf{E} \times \mathbf{B} \)

Given that both \( \mathbf{E} \) and \( \mathbf{B} \) are vectors, their cross product \( \mathbf{E} \times \mathbf{B} \) will behave like a pseudovector (axial vector) because the cross product of two polar vectors yields an axial vector.

Evaluate Physical Interpretation

The Poynting vector \( \mathbf{S} = \frac{1}{\mu_{0}}( \mathbf{E} \times \mathbf{B} ) \) represents the direction of energy transfer in an electromagnetic field. This physical interpretation does not change the mathematical nature of \( \mathbf{E} \times \mathbf{B} \,\) which we have established as an axial vector.

Key concepts

Electric Field

The electric field, denoted as \(\mathbf{E}\), is a vector field that describes the electric force per unit charge at each point in space. It arises from electric charges and varies with position.
The direction of \(\mathbf{E}\) is the direction of the force it exerts on a positive test charge.

• The electric field can be derived from the electric potential, \(V\).
• \[ \mathbf{E} = -\abla V \]
• Units: Volts per meter (V/m) or Newtons per Coulomb (N/C).

Magnetic Field

The magnetic field, denoted as \(\mathbf{B}\), is another vector field that represents the magnetic influence on moving electric charges, currents, and magnetic materials. It is a fundamental element of electromagnetism.
The direction of \(\mathbf{B}\) is found using the right-hand rule around current-carrying wires.

• The magnetic field can also be described via the vector potential \(\mathbf{A}\).
• \[ \mathbf{B} = \abla \times \mathbf{A} \]
• Units: Tesla (T) or Gauss (G).

Cross Product

The cross product of two vectors \(\mathbf{A}\) and \(\mathbf{B}\) results in a third vector that is perpendicular to both \(\mathbf{A}\) and \(\mathbf{B}\). It is also called the vector product.
The magnitude of \(\mathbf{A} \times \mathbf{B}\) is given by \|\mathbf{A}\|\|\mathbf{B}\|\sin \theta\, where \theta\ is the angle between the two vectors.

• It follows the right-hand rule to determine its direction.
• Formula: \[\mathbf{A} \times \mathbf{B} = (a_2b_3 - a_3b_2)\mathbf{i} - (a_1b_3 - a_3b_1)\mathbf{j} + (a_1b_2 - a_2b_1)\mathbf{k} \]

Vector vs Pseudovector

Vectors and pseudovectors are both vector quantities but differ in their transformation properties. Vectors (or true vectors) transform linearly under rotations: they change direction but keep the same length. In contrast, pseudovectors (or axial vectors) behave similarly to true vectors under rotation but gain an extra sign change under improper transformations like reflections.
Examples:

• True vector: Displacement, velocity.
• Pseudovector: Angular momentum, magnetic field.

Right-Hand Rule

The right-hand rule is a mnemonic for understanding orientation conventions for vector cross products and the direction of magnetic forces. When performing a cross product \(\mathbf{A} \times \mathbf{B}\), point your right hand's fingers in the direction of \(\mathbf{A}\) and curl them towards \(\mathbf{B}\). Your thumb then points in the direction of the resultant vector.
Applications:

• Determining the direction of magnetic force.
• Determining the direction of induced current.

Coordinate Transformations

Coordinate transformations are operations that change the coordinates of a point or vector in a physical space to a new coordinate system. These transformations can be rotations, translations, or reflections. They are crucial in understanding how vectors and pseudovectors transform.
Main types:

• Rotation: Changes the orientation of vectors.
• Translation: Moves vectors without changing their orientation.
• Reflection: Changes the sign of pseudovectors.

Electromagnetic Energy Transfer

Electromagnetic energy transfer refers to the movement of energy through the electromagnetic field, described by the Poynting vector \(\mathbf{S} = \frac{1}{\mu_0}(\mathbf{E} \times\mathbf{B})\). The Poynting vector indicates the direction of energy flow and its magnitude represents the energy transfer rate.

• The direction is given by the right-hand rule, perpendicular to both \(\mathbf{E} \) and \(\mathbf{B}\).
• Energy transfer is in the direction of \(\mathbf{S}\).

Understanding this helps in applications like power transmission, antenna theory, and radiation pressure.