Problem 30 - A Course in Mathematical Methods for Physicists

Question

Use Stokes' Theorem to derive the integral form of Faraday's law, $$ \int_{C} \mathbf{E} \cdot d \mathbf{s}=-\frac{\partial}{\partial t} \iint_{D} \mathbf{B} \cdot d \mathbf{S} $$ from the differential form of Maxwell's equations.

Step-by-Step Solution

Answer

By applying Stokes' theorem to the differential form of Faraday's law drawn from Maxwell's equations, we can successfully transform it and arrive at its integral form which is \( \int_{C} \mathbf{E} \cdot d \mathbf{s}=-\frac{\partial}{\partial t} \iint_{D} \mathbf{B} \cdot d \mathbf{S} \)

Step 1: Recall Maxwell's equations and the differential form of Faraday's law

Applying the differential form of Faraday's law from Maxwell's equation we derive that, \( \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \)

Step 2: Recognizing Stokes' Theorem and its application

The Stokes' theorem states that for any vector field, its curl integrated over some surface is equal to the line integral of the vector field around the boundary of the surface. That is if \( \mathbf{F} \) is a vector field, then \( \int_{C} \mathbf{F} \cdot d \mathbf{s} = \iint_{D} \nabla \times \mathbf{F} \cdot d \mathbf{S} \), where \( D \) is a surface bounded by the closed contour \( C \).

Step 3: Apply Stokes' Theorem

Stokes' theorem allows us to transform the right hand side of the equation. From \( \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \), we substitute this into Stokes’s theorem, obtaining \( \iint_{D} (\nabla \times \mathbf{E}) \cdot d \mathbf{S} = -\frac{\partial}{\partial t} \iint_{D} \mathbf{B} \cdot d \mathbf{S} \)

Step 4: Derive the integral form of Faraday's law

On the left side of the equation, by definition of the surface integral of a curl of a vector field over a surface, we can rewrite this as a line integral along the boundary of that surface. Therefore, we obtain the final form of the Faraday's law in integral form: \( \int_{C} \mathbf{E} \cdot d \mathbf{s}=-\frac{\partial}{\partial t} \iint_{D} \mathbf{B} \cdot d \mathbf{S} \)

Key Concepts

Faraday's Law

Faraday's Law is a fundamental principle in electromagnetism that explains how a magnetic field can produce an electric field, a phenomenon known as electromagnetic induction. At its core, Faraday's Law links the change in magnetic field over time to the electric field generated by that change. This concept is pivotal in understanding how electric generators and transformers work. In its integral form, Faraday's Law states:

• \( \int_{C} \mathbf{E} \cdot d \mathbf{s} = -\frac{\partial}{\partial t} \iint_{D} \mathbf{B} \cdot d \mathbf{S} \)

This equation tells us that the electric field \( \mathbf{E} \) around a closed loop \( C \) is equal to the negative rate of change of the magnetic flux \( \mathbf{B} \) through the surface \( D \).
This integral form of Faraday's Law is especially useful when analyzing systems with changing magnetic fields, such as inductors and transformers.
In practical applications, Faraday's Law is used to devise electric circuits that can harness energy from varying magnetic fields, enabling technologies like wireless charging and electric generators.

Maxwell's Equations

Maxwell's Equations are the set of four equations that form the foundation of classical electromagnetism, classical optics, and electric circuits. They beautifully unify all previous electric and magnetic theories into a single, consistent framework. Maxwell's Equations consist of:

• Gauss's law for electricity
• Gauss's law for magnetism
• Faraday's law of induction
• Ampère's law with Maxwell's addition

These equations describe how electric charges produce electric fields (Gauss's law for electricity), how magnetic poles (if they existed) do not exist (Gauss’s law for magnetism), how time-varying magnetic fields induce electric fields (Faraday's law, which we derived the integral form of using Stokes' theorem), and how currents and changing electric fields produce magnetic fields (Ampère's law with Maxwell's addition).
In the context of the exercise, we specifically used the differential form of Faraday's Law:

• \( abla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \)

This equation expresses the relationship between time-varying magnetic fields and the curls (or loops) in the electric field.

Vector Calculus

Vector calculus is an essential mathematical tool used in the study of fields like physics and engineering. It involves vector fields and operations on these fields, such as divergence, gradient, and curl. In understanding electromagnetism, vector calculus is crucial for describing how fields vary in space and time.
One key concept used in the original exercise is the curl of a vector field, denoted as \( abla \times \mathbf{E} \). The curl measures the rotation at a point in the field, which in this case, translates to how a changing magnetic field creates an electric field.

• Divergence measures how much a point acts as a source or sink of the field.
• Gradient provides the direction and rate of fastest increase of a scalar field.
• Curl indicates how much a field tends to swirl around a point.

For the problem at hand, Stokes' Theorem plays a pivotal role in transitioning from differential to integral forms of equations. Stokes' Theorem relates a surface integral of a curl over a surface to a line integral around the boundary of that surface. This interpretation allows us to translate the local properties described by differential equations into global properties described by integrals.
By applying Stokes' Theorem, we moved from the differential expression \( abla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \) to the integral result \( \int_{C} \mathbf{E} \cdot d \mathbf{s} = -\frac{\partial}{\partial t} \iint_{D} \mathbf{B} \cdot d \mathbf{S} \), bridging the gap between local and global understanding.