Question

Write Maxwell's equations in point form in terms of \(\mathbf{E}\) and \(\mathbf{H}\) as they apply to a sourceless medium, where \(\mathbf{J}\) and \(\rho_{v}\) are both zero. Replace \(\epsilon\) by \(\mu, \mu\) by \(\epsilon, \mathbf{E}\) by \(\mathbf{H}\), and \(\mathbf{H}\) by \(-\mathbf{E}\), and show that the equations are unchanged. This is a more general expression of the duality principle in circuit theory.

Short answer

Question: Show that the duality principle is valid in the sourceless region by replacing the variables in Maxwell's equations and demonstrating that they remain unchanged. Answer: The duality principle in circuit theory was demonstrated by replacing the variables in sourceless Maxwell's equations as follows: ε by μ, μ by ε, E by H, and H by -E. After making these replacements and simplifying the resulting equations, we found that the equations remained unchanged, proving the validity of the duality principle in a sourceless region.

Step-by-step solution

Write Maxwell's Equations in General Form

In the general form, Maxwell's equations are given by: 1. Gauss's Law for Electric field: \(\nabla \cdot \mathbf{E} = \frac{\rho_v}{\epsilon_0}\) 2. Gauss's Law for Magnetic field: \(\nabla \cdot \mathbf{H} = 0\) 3. Faraday's Law: \(\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}\) 4. Ampere's Law (with Maxwell's addition): \(\nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t}\) Where \(\mathbf{E}\) is the electric field, \(\mathbf{H}\) is the magnetic field, \(\mathbf{B}\) is the magnetic flux density, \(\mathbf{D}\) is the electric flux density, \(\rho_v\) is the volume charge density, and \(\mathbf{J}\) is the current density.

Apply Conditions for Sourceless Medium

In a sourceless medium, we have \(\mathbf{J} = 0\) and \(\rho_v = 0\). Applying these conditions, we can simplify Maxwell's equations as: 1. Gauss's Law for Electric field: \(\nabla \cdot \mathbf{E} = 0\) 2. Gauss's Law for Magnetic field: \(\nabla \cdot \mathbf{H} = 0\) 3. Faraday's Law: \(\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}\) 4. Ampere's Law (with Maxwell's addition): \(\nabla \times \mathbf{H} = \frac{\partial \mathbf{D}}{\partial t}\)

Perform the Replacement of Variables

Now, we are asked to replace \(\epsilon\) by \(\mu\), \(\mu\) by \(\epsilon\), \(\mathbf{E}\) by \(\mathbf{H}\), and \(\mathbf{H}\) by \(-\mathbf{E}\). Before doing this, we need to represent \(\mathbf{B}\) and \(\mathbf{D}\) in terms of \(\mathbf{E}\) and \(\mathbf{H}\). We know that: \(\mathbf{B} = \mu \mathbf{H}\) \(\mathbf{D} = \epsilon \mathbf{E}\) Now, we can replace the variables in the sourceless Maxwell's equations: 1. Gauss's Law for Electric field (with replacements): \(\nabla \cdot \mathbf{H} = 0\) 2. Gauss's Law for Magnetic field (with replacements): \(\nabla \cdot (-\mathbf{E}) = 0\) 3. Faraday's Law (with replacements): \(\nabla \times \mathbf{H} = -\frac{\partial (\epsilon(-\mathbf{E}))}{\partial t}\) 4. Ampere's Law (with replacements): \(\nabla \times (-\mathbf{E}) = \frac{\partial (\mu \mathbf{H})}{\partial t}\)

Show that Equations are Unchanged

Now we compare these equations after the replacements with the sourceless Maxwell's equations that we derived in step 2. 1. Gauss's Law for Electric field is unchanged. 2. Gauss's Law for Magnetic field is also unchanged (the negative sign can be absorbed by the curl's cross product). 3. Faraday's Law, after simplifying the right-hand side, becomes: \(\nabla \times \mathbf{H} = \frac{\partial (\epsilon \mathbf{E})}{\partial t}\) Which is the same as Ampere's Law in sourceless medium (step 2.4). 4. Ampere's Law, after simplifying the right-hand side, becomes: \(\nabla \times \mathbf{E} = -\frac{\partial (\mu \mathbf{H})}{\partial t}\) Which is the same as Faraday's Law in sourceless medium (step 2.3). As we can see, after the replacement of variables, the resulting equations are the same as the original sourceless Maxwell's equations. This demonstrates the duality principle in circuit theory.

Key concepts

Electromagnetic Field Theory

Electromagnetic field theory is a fundamental pillar of modern physics and electrical engineering, encompassing the study of electric fields, magnetic fields, and their interrelation. At the core of this theory lie Maxwell's equations, which describe how electric charges and currents produce electric and magnetic fields, and how those fields interact with each other.

Maxwell's equations are elegantly simple yet powerful in their ability to predict and explain a wide range of phenomena, from visible light to radio waves. In essence, these equations tell us that changing magnetic fields induce electric fields (Faraday's Law), electric charges produce electric fields (Gauss's Law for the electric field), electric currents and changing electric fields create magnetic fields (Ampere's Law with Maxwell's addition), and that magnetic fields do not arise from 'magnetic charges' but rather from electric currents or changing electric fields (Gauss's Law for the magnetic field).

In the context of the exercise, these equations were applied in a sourceless medium—a scenario where there are no free charges or currents to generate fields.

Sourceless Medium

In the scenario presented, we discuss a 'sourceless' medium. This concept simply means that the medium through which electromagnetic fields are propagating is devoid of free charges (\(\rho_v = 0\) – no volume charge density) and free currents (\(\textbf{J} = 0\) – no current density). In such a medium, the generation of fields must be self-sustaining, without being produced or altered directly by charge or current sources.

When we look at Maxwell's equations in a sourceless medium, we find that the presence of charges (\( \rho_v \) and currents (\( \textbf{J} \) naturally simplify the equations, allowing us to focus on the characteristics of free-floating electromagnetic fields as they evolve in space and time. Because the medium lacks sources, the electric field divergence (\( abla \bullet \textbf{E} \) and magnetic field divergence (\( abla \bullet \textbf{H} \) both become zero, indicating field continuity without external influence. This condition is quintessential for understanding how fields behave under vacuum-like conditions.

Duality Principle

The duality principle is an intriguing aspect of electromagnetic field theory which states that there is a symmetry between electric and magnetic fields. Here, it's being shown in terms of the mathematical form of Maxwell's equations—specifically, how they can be transformed into each other through a set of variable substitutions.

In the context of the problem, we see that by switching electric permittivity (\( \textbf{E} \) with magnetic permeability (\( \textbf{H} \) and vice versa, and applying the appropriate sign changes, we end up with a set of equations that are identical in form to the original equations. From a physical perspective, this underpins the concept that electric and magnetic phenomena are two facets of the same fundamental entity - the electromagnetic field.

The exercise demonstrates this principle by mandating replacements in the sourceless Maxwell's equations and showing that, even after swapping parameters associated with electric fields to those associated with magnetic fields, the form of the equations remains unchanged. This intrinsic symmetry is what makes the study of electromagnetic fields so elegant and conceptually rich.