Question

An electromagnetic field \(\Gamma_{\mu 1}\) is sard to be of 'electric type' at an event \(p\) if therc exists a unut tanelike 4-vector \(U_{n}\) at \(p, U_{\alpha} U^{e}=-1\), and a spacelike 4 -vecter field \(E_{\mu}\) ot thogonal to \(U^{\prime \prime}\) such that $$ F_{u}=U_{\mu} E_{v}-U_{v} E_{\mu}, \quad E_{e} U^{k}=0 $$ (a) Show that any purely electne ficld, ie one havmg \(\mathrm{B}=0, \mathrm{ts}\) of electric type. (b) If \(F_{\mu,}\) is of clectric type at \(\mu\), show that there is a vclocity \(\mathrm{v}\) such that $$ \mathbf{B}=\frac{\mathbf{v}}{c} \times \mathbf{E}, \quad(|\mathbf{v}|

Short answer

To summarize, the concepts of spacetime vectors, Lorentz transformations and Maxwell equations have been used to study the characteristics of electric and magnetic fields. It's proved the existence of a reference frame where the electromagnetic field is purely electric and the concept of divergence of the electric field in a vacuum has been discussed.

Step-by-step solution

Analyzing a purely electric field

A purely electric field implies that the magnetic field \(B = 0\). Using the equation \(F_{u}=U_{\mu} E_{v}-U_{v} E_{\mu}, \quad E_{e} U^{k}=0\), we can observe that if \(B = 0\), then all components of the field tensor \(F_{\mu,}\) that relate to \(B\) are zero. Thus, the field tensor \(F_{\mu,}\) corresponds to an electric field.

Defining velocity

We can prove the existence of a velocity \(v\) such that \(\mathbf{B}=\frac{\mathbf{v}}{c} \times \mathbf{E}, \quad(|\mathbf{v}|<c)\) through the Lorentz transformations. By choosing a suitable Lorentz transformation, we can always have relative motion with velocity \(v\) such that a field which was originally purely electric, can have a magnetic component as well. This condition can be expressed as \(\mathbf{B}=\frac{\mathbf{v}}{c} \times \mathbf{E}\).

Working with Lorenz transformation

Given that there exists a Lorentz transformation that transforms the electromagnetic field to one that is purely electric at \(p\), we can state that for any electromagnetic field there always exists a frame of reference where the field is purely electric.

The divergence of the electric field

Given that the vector field \(E^{n}\) is divergence-free, we can write the Maxwell's equations in vacuo as \(E_{*}^{v}=0\). Since these equations are satisfied everywhere, with \(U^{\mu}\) a constant vector field, this implies that the electric field has zero divergence.

Key concepts

Lorentz Transformation

Understanding the Lorentz Transformation is crucial in the realm of electromagnetism and relativity. This mathematical framework helps us to connect different events as observed in various inertial frames. Lorentz transformations are named after Dutch physicist Hendrik Lorentz. They adjust the time and spatial coordinates between two observers moving relative to each other at a constant velocity.
A key feature is that the speed of light remains constant for all observers, regardless of their relative motion. This leads to fascinating outcomes like time dilation and length contraction.
When applying Lorentz transformations to electromagnetic fields, it helps us analyze how the electric and magnetic components of a field interconvert under motion. For example, a purely electric field in one frame can appear as a mixed electric and magnetic field in another moving frame. Understanding this principle is vital to grasp the full depth of electromagnetism.

Electric and Magnetic Fields

Electric and magnetic fields are fundamental concepts in physics describing the electromagnetic force fields produced by electric charges and currents. They are intertwined, manifesting as two aspects of a single electromagnetic field.
The electric field is produced by stationary charges and is represented by the vector field \( \mathbf{E} \). It defines how charged particles such as electrons will move through space. On the other hand, the magnetic field \( \mathbf{B} \) arises from moving charges (currents) and affects other moving charges within its influence.

• Electric fields exert force on charges moving towards or away from it depending on the charge type.
• Magnetic fields induce circular motion among charged particles.

Through specific conditions, we understand how, in one frame of reference, a field might appear purely electric or magnetic, while in another, both these components might exist.

Maxwell's Equations

Maxwell's Equations form the bedrock of classical electromagnetism, optics, and electric circuits. They consist of four equations that describe how electric and magnetic fields propagate, interact, and how they are affected by matter.
These equations are:

• Gauss's Law for Electricity: \( abla \cdot \mathbf{E} = \rho / \varepsilon_0 \)
• Gauss's Law for Magnetism: \( abla \cdot \mathbf{B} = 0 \)
• Faraday's Law of Induction: \( abla \times \mathbf{E} = -\partial \mathbf{B} / \partial t \)
• Ampère-Maxwell Law: \( abla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \partial \mathbf{E} / \partial t \)

These equations collectively imply the nature and directions of electric and magnetic fields within a vacuum, and the massless, divergence-free characteristics of \( \mathbf{B} \) and \( \mathbf{E} \) are encapsulated by these checks, paving the way for understanding the relationship between charge, current, and field.

Divergence-Free Field

A divergence-free field refers to a vector field whose divergence is zero. In simpler terms, it means there are no 'sources' or 'sinks' in the field—what flows in also flows out. In electromagnetic theory, such a condition is vital.
A classic example is the magnetic field \( \mathbf{B} \). According to Gauss's Law for Magnetism, its divergence being zero means that magnetic monopoles do not exist, or in essence, magnetic field lines are closed loops.
For electric fields, in cases where we talk about divergence-free characteristics, it usually means the region is charge-free, reinforcing that the field lines are consistent and uniform across that space. This fundamental idea helps ensure the continued perpetuation of field lines while maintaining the conservation of electromagnetic energy.