Question

Show from Maxwell's equations in a vacuum that the magnetic field \(B\) obeys the wave equation.

Short answer

Using Maxwell's equations in a vacuum, the magnetic field \(B\) is found to satisfy the wave equation \(abla^2 B = \mu_0 \epsilon_0\frac{\partial^2 B}{\partial t^2}\).

Step-by-step solution

Write down Maxwell's equations in a vacuum

First, list the relevant Maxwell's equations for electromagnetic fields in a vacuum, which do not involve charges or currents: Faraday's law of induction and the Maxwell-Ampère law (without currents). - Faraday's law of induction in differential form: \(abla \times E = -\frac{\partial B}{\partial t}\) - Maxwell-Ampère law in differential form (in a vacuum with no currents): \(abla \times B = \mu_0 \epsilon_0 \frac{\partial E}{\partial t}\)

Use Faraday's law to find the curl of the curl of B

Apply the curl operator to both sides of Faraday's law to get an equation involving \(abla \times (abla \times B)\). \(abla \times (abla \times E) = abla \times (-\frac{\partial B}{\partial t})\) Now, use the vector identity \(abla \times (abla \times E) = abla (abla \cdot E) - abla^2 E\), and since in a vacuum \(abla \cdot E = 0\), the first term drops, simplifying to: \(-abla^2 E = abla \times (-\frac{\partial B}{\partial t}) \)

Relate E to B using the Maxwell-Ampère law

We know from Maxwell-Ampère's law that \(abla \times B = \mu_0 \epsilon_0 \frac{\partial E}{\partial t}\). Apply the partial derivative with respect to time to both sides, and substitute that in the wave equation from Step 2: \(-abla^2 E = abla \times (-\frac{\partial B}{\partial t})\)\(= -\frac{\partial}{\partial t}(abla \times B)\)\(= -\mu_0 \epsilon_0\frac{\partial^2 E}{\partial t^2}\).Now, plug in \(-abla^2 E\) from the equation above to get the wave equation for \(E\): \(abla^2 E = \mu_0 \epsilon_0\frac{\partial^2 E}{\partial t^2}\).

Apply the curl operator to Maxwell-Ampère law

Apply the curl operator to both sides of the Maxwell-Ampère law (\(abla \times B =\mu_0 \epsilon_0\frac{\partial E}{\partial t}\)) to find an expression for \(abla \times (abla \times B)\): \(abla \times (abla \times B) = abla \times (\mu_0 \epsilon_0\frac{\partial E}{\partial t})\).

Utilize the vector identity and substitute E with B

Using the vector identity \(abla \times (abla \times B) = abla (abla \cdot B) - abla^2 B\) and knowing that in a vacuum \(abla \cdot B = 0\), the equation simplifies to: \(-abla^2 B = abla \times (\mu_0 \epsilon_0\frac{\partial E}{\partial t})\).By substituting \(E\) with \(B\) using Ampère's law, we obtain: \(-abla^2 B = \mu_0 \epsilon_0\frac{\partial}{\partial t}(abla \times E)\).Recall from Step 2 that \(abla \times E = -\frac{\partial B}{\partial t}\), we can substitute \(abla \times E\) with \(-\frac{\partial B}{\partial t}\) to get: \(-abla^2 B = -\mu_0 \epsilon_0\frac{\partial^2 B}{\partial t^2}\).Divide both sides by -1 to get the wave equation for \(B\): \(abla^2 B = \mu_0 \epsilon_0\frac{\partial^2 B}{\partial t^2}\).

Key concepts

Faraday's Law of Induction

One of the cornerstone principles of electromagnetism is Faraday's law of induction. This law tells us how a changing magnetic field can create an electric field. In simple terms, whenever there is a change in the magnetic environment of a region, an electric field is induced in the surrounding space.

In a mathematical form, this law is expressed as a vector calculus equation involving the curl of the electric field (\( abla \times E \)), which equals the negative rate of change of the magnetic field (\( -\frac{\text{\textbf{d}}B}{\text{\textbf{d}}t} \) ). This equation echoes the idea that the induced electric field forms loops, which is characterized by the curl operation in vector calculus.

Understanding Faraday's law is crucial for grasping the concept of electromagnetic induction, which is the working principle behind many devices, including electric generators and transformers.

Maxwell-Ampère Law

The Maxwell-Ampère law is an extension of Ampère's law that includes displacement current, allowing the law to be consistent with charge conservation. In a vacuum, where there are no free currents, the law links the curl of the magnetic field (\( abla \times B \)) to the rate of change of the electric field (\( \frac{\text{\textbf{d}}E}{\text{\textbf{d}}t} \) ). This reflects how changing electric fields can generate magnetic fields.

When written with the vacuum permittivity (\( \text{ε}_0 \) ) and permeability (\( \text{μ}_0 \) ), the law forms part of a wave equation for electromagnetic fields. This relationship is vital for the development of the concept of electromagnetic waves, which can travel through a vacuum without the need for any medium.

Electromagnetic Wave Theory

Electromagnetic waves are ripples in the electromagnetic field that carry energy across space. The electromagnetic wave theory, which arises from Maxwell's equations, indicates that these waves propagate through the vacuum at the speed of light. This speed (\( c \) ) is related to the permittivity (\( \text{ε}_0 \) ) and the permeability (\( \text{μ}_0 \) ) of free space through the equation \( c = \frac{1}{\text{\textbf{√}}(\text{ε}_0 \text{μ}_0)} \) .

The theory suggests that electric and magnetic fields oscillate perpendicularly to each other and to the direction of wave propagation. This concept helped unify the theories of electricity, magnetism, and optics, and paved the way for the modern understanding of light as an electromagnetic phenomenon.

Curl of a Vector Field

In vector calculus, the curl is an operator that measures the rotation of a vector field. For example, when applied to the velocity field of a fluid, it describes the rotation at any point within the fluid. In terms of electromagnetic fields, the curl of the electric field (\( abla \times E \) ) is related to the change in the magnetic field, as stated by Faraday's law,

and the curl of the magnetic field (\( abla \times B \) ) is associated with both electric currents and changes in the electric field, as per Maxwell-Ampère law. The curl operator is a way to capture the swirling of the field lines around a point and is a fundamental tool in describing the dynamics of electromagnetic fields.

Vector Calculus Identities

Vector calculus identities are crucial for simplifying and manipulating vector equations. Common identities include the divergence (\( abla \text{\textbf{·}} \) ) and curl (\( abla \times \) ) operations, as well as the Laplacian (\( abla^2 \) ), which is the divergence of the gradient of a scalar field.

An important identity used in electromagnetism is \( abla \times (abla \times A) = abla (abla \text{\textbf{·}} A) - abla^2 A \) for a vector field A. This identity helps simplify the step from Faraday's law to the wave equation for the magnetic field as seen in the provided exercise. It is one of the many vector calculus tools that enable physicists and engineers to solve complex problems involving electromagnetic fields.