Question

Prove that the magnetic flux density, B, satisfies the wave equation.

Short answer

Both the electric and magnetic flux densities, D and B, satisfy the wave equation by the application of vector calculus and Maxwell's equations, and therefore, the magnetic flux density B does as well.

Step-by-step solution

Start with Maxwell's equations

The useful Maxwell's equations here are \(\nabla × E = - \frac{\partial B}{\partial t}\)and \(\nabla × H = J + \frac{\partial D}{\partial t}\) . Here, D is electric flux density, E and H are electric and magnetic field strength respectively, J is current density and \( \frac{\partial}{\partial t} \) defines a partial derivative with respect to time.

Apply Vector Identity

Use the relation \(\nabla × (\nabla × A) = \nabla (\nabla \cdot A) - \nabla^2 A \) with A being E and B in respective Maxwell's equations.

Apply to Maxwell's equations

For \(\nabla × E\) (let's call this equation 1) and \(\nabla × H\) (equation 2), use the vector Identity. For equation 1, this gives \(\nabla × (\nabla × E) = -\nabla × (\frac{\partial B}{\partial t})\). And for equation 2, we get \(\nabla × (\nabla × H) = \nabla × J + \nabla × (\frac{\partial D}{\partial t})\)

Simplify equations

\(\nabla × (\nabla × E) = \nabla (\nabla \cdot E) - \nabla^2 E\). Since \(\nabla \cdot E = 0\) (another Maxwell's equation), drop off that term. Also, \(\nabla × (\nabla × H) = \nabla (\nabla \cdot H) - \nabla^2 H\). Since \(\nabla \cdot H = 0\), drop off that term here also. Substituting B for E and D for H in respective equations, we get \(-\nabla^2B = -\frac{\partial^2 B}{\partial t^2}\) and \(-\nabla^2D = \nabla ×J + \frac{\partial^2 D}{\partial t^2}\). Compared with general wave equation \(\nabla^2 \phi = \frac{\partial^2 \phi}{\partial t^2}\), both B and D are solutions of the wave equation.

Conclusion

It's proven that both magnetic and electric flux densities satisfy the wave equation through application of vector manipulation and Maxwell's equations.

Key concepts

Maxwell's equations

Maxwell's equations are a set of four fundamental laws that describe how electric and magnetic fields interact. These equations are critical in the study of electromagnetism and form the foundation of classical electromagnetism, optics, and electric circuits.

• Gauss's Law for Electricity: This law highlights how electric fields originate from electric charges and is expressed as \( abla \cdot D = \rho \), where \( D \) is the electric flux density and \( \rho \) is the charge density.
• Gauss's Law for Magnetism: This indicates that there are no magnetic "charges" or monopoles, characterized by \( abla \cdot B = 0 \) where \( B \) represents magnetic flux density.
• Faraday's Law of Induction: Shows the relationship between a changing magnetic field and the electric field it creates, expressed as \( abla \times E = -\frac{\partial B}{\partial t} \).
• Ampere-Maxwell Law: Integrates current and the changing electric field, described by \( abla \times H = J + \frac{\partial D}{\partial t} \), where \( H \) is magnetic field strength, and \( J \) is current density.

These equations are expressed in terms of vector calculus and are essential for proving properties such as the wave-like behavior of electromagnetic fields.

Magnetic flux density

Magnetic flux density, denoted by \( B \), is a vector that represents the strength and direction of a magnetic field. It is critical in determining how a magnetic field influences its surroundings and interacts with different materials.
The units of magnetic flux density are Tesla (T) in the International System of Units (SI), and it describes how dense the magnetic field lines are in a given area. Higher density means stronger magnetic effects.
Magnetic flux density plays a significant role in electromagnetic systems such as motors, generators, and transformers and is crucial for understanding the wave equation. When fields change, they must adhere to Maxwell's equations. For example, the equation \( abla \times E = -\frac{\partial B}{\partial t} \), emphasizes how a time-varying magnetic field generates an electric field, indicating interconnected dynamics in electromagnetism. Wave solutions reveal that \( B \)'s behavior can be defined by a wave equation, leading to predictions about field propagation.

Vector calculus

Vector calculus is a branch of mathematics that deals with differentiation and integration of vector fields. It is essential in physics and engineering, as it provides tools to describe physical phenomena in three-dimensional space.
Key concepts in vector calculus include:

• Gradient: Measures the rate of change of a scalar field, providing a vector field that points in the direction of the greatest rate of increase.
• Divergence: A scalar that shows a vector field's tendency to "spread out" from a point, used in the context of fields like electric and magnetic field intensities.
• Curl: Represents the rotation or "twist" of a vector field, crucial in electromagnetism to describe how fields change over time.

Applying vector calculus to Maxwell's equations, particularly using the identity \( abla \times (abla \times A) = abla (abla \cdot A) - abla^2 A \), allows us to simplify complex interactions into equations like the wave equation. This technique is indispensable for unveiling the nature of electromagnetic fields and how they propagate.

Partial derivatives

Partial derivatives are mathematical tools for finding how a function changes as its variables change, holding others constant. In the context of physics, they are crucial for describing how fields vary in space and time.
In Maxwell's equations, partial derivatives express how electric and magnetic fields evolve.

• For instance, \( \frac{\partial B}{\partial t} \) shows the rate of change of the magnetic flux density over time, holding other spatial variables fixed.
• They are vital for defining wave equations, as evidenced by comparing terms from vector identities \( -\frac{\partial^2 B}{\partial t^2} \) with spatial derivatives, like \(-abla^2 B\), found in the wave equation.

In this exercise, using partial derivatives helps in simplifying and manipulating the relationship between time-varying fields in electromagnetic theory, emphasizing their wave-like solutions.