Question

In four dimensions define second-order antisymmetric tensors \(F_{i j}\) and \(Q_{i j}\) and a first-order tensor \(S_{i}\) as follows: (a) \(F_{23}=H_{1}, Q_{23}=B_{1}\) and their cyclic permutations; (b) \(F_{i 4}=-D_{i}, Q_{i 4}=E_{i}\) for \(i=1,2,3 ;\) (c) \(S_{4}=\rho, S_{i}=J_{i}\) for \(i=1,2,3\). Then, taking \(x_{4}\) as \(t\) and the other symbols to have their usual meanings in electromagnetic theory, show that the equations \(\sum_{j} \partial F_{i j} / \partial x_{j}=S_{i}\) and \(\partial Q_{j k} / \partial x_{i}+\) \(\partial Q_{k i} / \partial x_{j}+\partial Q_{i j} / \partial x_{k}=0\) reproduce Maxwell's equations. Here \(i, j, k\) is any set of three subscripts selected from \(1,2,3,4\), but chosen in such a way that they are all different.

Short answer

The tensor equations reproduce Maxwell's equations by interpreting the tensor components and relating them to the electromagnetic field components.

Step-by-step solution

- Identify given tensors and their components

The given tensors are second-order antisymmetric tensors: \(F_{ij}\) and \(Q_{ij}\), and a first-order tensor: \(S_i\). The components are defined as follows:1. \(F_{23} = H_1, Q_{23} = B_1\) and other cyclic permutations.2. \(F_{i4} = -D_i, Q_{i4} = E_i\) for \(i=1,2,3\).3. \(S_4 = \rho, S_i = J_i\) for \(i=1,2,3\).Here, \(x_4\) is time (\(t\)) and the other symbols have their usual meanings in electromagnetic theory.

- Write out the components of the tensors

The explicit components for the tensors are:1. \(F_{12} = H_3, F_{13} = -H_2, F_{23} = H_1\) and their permutations.2. \(F_{14} = -D_1, F_{24} = -D_2, F_{34} = -D_3\).3. \(Q_{12} = B_3, Q_{13} = -B_2, Q_{23} = B_1\) and their permutations.4. \(Q_{14} = E_1, Q_{24} = E_2, Q_{34} = E_3\).5. \(S_4 = \rho, S_1 = J_1, S_2 = J_2, S_3 = J_3\).

- Relate Maxwell's equations to the tensor equations

Use the provided equations:1. \(\sum_j \frac{\partial F_{ij}}{\partial x_j} = S_i\)2. \(\frac{\partial Q_{jk}}{\partial x_i} + \frac{\partial Q_{ki}}{\partial x_j} + \frac{\partial Q_{ij}}{\partial x_k} = 0\)Substitute different values of \(i, j, k\) to relate these equations to Maxwell's equations.

- Expand the first tensor equation and interpret

Expanding the first equation for \(i=1\):\(\frac{\partial F_{12}}{\partial x_2} + \frac{\partial F_{13}}{\partial x_3} + \frac{\partial F_{14}}{\partial x_4} = \frac{\partial H_3}{\partial y} - \frac{\partial H_2}{\partial z} - \frac{\partial D_1}{\partial t} = J_1\).This matches one of Maxwell's equations: \(abla \times \mathbf{H} - \frac{\partial \mathbf{D}}{\partial t} = \mathbf{J}\).

- Expand the second tensor equation and interpret

Expanding for \(i=1, j=2, k=3\):\(\frac{\partial Q_{23}}{\partial x_1} + \frac{\partial Q_{31}}{\partial x_2} + \frac{\partial Q_{12}}{\partial x_3} = \frac{\partial B_1}{\partial x} + \frac{\partial B_2}{\partial y} + \frac{\partial B_3}{\partial z} = 0\).This corresponds to Gauss's law for magnetism: \(abla \cdot \mathbf{B} = 0\).

Key concepts

Maxwell's equations

Maxwell's equations are a set of four fundamental laws that describe how electric and magnetic fields interact with matter. These equations are:

• Gauss's law for electricity: \(abla \cdot \, \mathbf{E} \, = \, \frac {\rho} {\epsilon_0}\)
• Gauss's law for magnetism: \(abla \cdot \, \mathbf{B} \, = \, 0\)
• Faraday's law of induction: \(abla \times \, \mathbf{E} \, = \, -\frac {\, \partial \, \mathbf{B}} {\partial t}\)
• Ampère's law (with Maxwell's correction): \(abla \times \, \mathbf{H} \, = \, \mathbf{J} \, + \frac {\, \partial \, \mathbf{D}} {\partial t}\)

These equations form the foundation of classical electrodynamics, describing how electric charges produce electric fields (Gauss's law for electricity), how magnetic monopoles do not exist (Gauss's law for magnetism), how changing magnetic fields induce electric fields (Faraday's law), and how electric currents and changing electric fields produce magnetic fields (Ampère's law). Together, these laws explain the behavior of electromagnetic fields and their interactions with matter.

Tensor calculus

Tensor calculus is an extension of vector calculus to tensors, which are mathematical objects that generalize scalars, vectors, and matrices. It is particularly useful in physics and engineering because it provides a framework to describe physical laws in a way that is invariant under coordinate transformations.

• Tensors can be thought of as multi-dimensional arrays of numbers.
• They have specific transformation properties under changes of coordinates.
• Important operations in tensor calculus include contraction, raising and lowering indices, and differentiation.

In the context of electromagnetism, tensors help encode information about electric and magnetic fields in a compact form, allowing for easier manipulation and application of Maxwell's equations in various coordinate systems. Tensors like the electromagnetic field tensor \(F_{ij}\) and the dual tensor \(Q_{ij}\) represent electric and magnetic field components in a unified framework.

Antisymmetric tensors

Antisymmetric tensors are a special class of tensors for which swapping any two indices changes the sign of the tensor component. For example, if \(A_{ij}\) is an antisymmetric tensor, then \(A_{ij} = -A_{ji}\).
This property implies that \(A_{ii} = 0\), making the diagonal elements zero.

• Antisymmetric tensors naturally arise in the study of electromagnetic fields.
• The electromagnetic field tensor \(F_{ij}\) is an antisymmetric tensor that combines the electric and magnetic fields.
• The antisymmetry property simplifies many physical equations and provides insights into the nature of the fields.

In our specific problem, we define second-order antisymmetric tensors \(F_{ij}\) and \(Q_{ij}\), which incorporate electric and magnetic field components in a way that reflects the underlying physical symmetries and conservation laws.

Electromagnetic fields

Electromagnetic fields are physical fields produced by electrically charged objects. They affect the behavior of charged objects in the vicinity of the field. An electromagnetic field is made up of:

• Electric field (\(\mathbf{E}\)), which exerts forces on charges.
• Magnetic field (\(\mathbf{B}\)), which affects moving charges and magnetic dipoles.

These fields are interrelated and propagate through space as electromagnetic waves, such as light. In the context of Maxwell's equations and their tensorial representations:

• Electric field components (\(\mathbf{E}\)) appear in the tensor \(Q_{ij}\).
• Magnetic field components (\(\mathbf{H}\), related to \(\mathbf{B}\)) are incorporated in the tensor \(F_{ij}\).

Understanding the interplay between \(\mathbf{E}\) and \(\mathbf{B}\) is crucial for solving many problems in electromagnetism, from the propagation of light to the design of electrical circuits.

Partial differential equations

Partial differential equations (PDEs) involve functions and their partial derivatives with respect to multiple variables. PDEs are essential in describing many physical systems, including electromagnetism, fluid dynamics, heat transfer, and more.

• PDEs like those found in Maxwell's equations describe how fields and potentials evolve over time and space.
• Solving PDEs often requires advanced mathematical techniques, including separation of variables, Fourier transforms, and numerical methods.

In our problem, we deal with two key PDEs:

• \(abla \times \, \mathbf{H} \, - \, \frac {\partial \, \mathbf{D}} {\partial t} \, = \, \mathbf{J}\)
• \(abla \cdot \, \mathbf{B} \, = 0\)

These equations are derived from the tensor equations given in the exercise, demonstrating how PDEs bridge the connection between abstract mathematical formulations and physical phenomena observed in nature.