Use Gauss' and Stokes' theorems (Appendix A) to convert Maxwell's differential equations for vacuum, \((82.1)\) to \((8.2 .4)\), to their integral form $$ \begin{aligned} &\oint_{S} \mathbf{E} \cdot d \mathbf{S}=\frac{q}{\epsilon_{0}} \\ &\oint_{L} \mathbf{E} \cdot d \mathbf{l}=-\frac{d \Phi_{m}}{d t} \\ &\oint_{S} \mathbf{B} \cdot d \mathbf{S}=0 \\ &\oint_{L} \mathbf{B} \cdot d \mathbf{l}=\mu_{0} I+\mu_{0} \frac{d \Phi_{*}}{d t} \end{aligned} $$ † See Sec. \(5.4\) and Prob. 8.2.4. where the closed surface \(S\) contains the net charge \(q\) and the closed line (loop) \(L\) is linked by the net current \(I\), the magnetic flux \(\Phi_{m}=\int \mathbf{B} \cdot d \mathbf{S}\), and the electric flux \(\Phi_{e}=\epsilon_{0} \int \mathbf{E} \cdot d \mathbf{S}\). Note: The corresponding equations for a general electromagnetic medium are developed in Prob. \(8.6 .1 .\)

Gauss' theorem, also known as Gauss' law, is a fundamental principle in electromagnetism that relates the electric field \textbf{E} around a closed surface to the charge enclosed by that surface. Mathematically, the theorem can be expressed as \[ \oint_{S} \mathbf{E} \cdot d \mathbf{S} = \frac{q}{\epsilon_{0}} \], where \(\oint_{S}\) denotes the closed surface integral over surface \(S\), \mathbf{E}\ is the electric field, \(d\mathbf{S}\) is a vector representing an infinitesimal area on surface \(S\) pointing outward, \(q\) is the total electric charge enclosed within the surface, and \(\epsilon_{0}\) is the permittivity of free space.

Gauss' theorem is one of the four Maxwell's equations, which are the foundation of classical electrodynamics, optics, and electric circuits. The theorem has a deep physical significance: it implies that electric field lines originate from positive charges and terminate on negative charges, and the net flux through a closed surface depends only on the charge enclosed, not on the specifics of how the field is configured outside the surface.

Stokes' theorem bridges the gap between the concepts of circulation and curl. It is used to relate the surface integral of the curl of a vector field to the line integral of the vector field around the boundary of the surface. In the context of electromagnetism, Stokes' theorem can be applied to the magnetic vector field \textbf{B}, enabling us to connect the electric field \textbf{E} around a closed loop to the rate of change of the magnetic flux through the enclosed surface. Stokes' theorem is expressed as \[ \oint_{L} \mathbf{B} \cdot d \mathbf{l} = \iint_{S} abla \times \mathbf{B} \cdot d \mathbf{S} \], where \(\oint_{L}\) is the line integral around closed loop \(L\), and the right-hand side involves the surface integral over the surface bounded by \(L\).

Understanding Stokes' theorem is essential when studying electromagnetic waves and the dynamic relationships in electromagnetism, such as how varying magnetic fields give rise to electric fields, and vice versa. This theorem plays a crucial role in the integral form of Maxwell's equations, providing a deeper look into the symmetry of nature's laws.

Electromagnetic waves are a form of energy transfer through space at the speed of light, consisting of oscillating electric and magnetic fields that are perpendicular to each other and to the direction of propagation. Maxwell's equations predict the existence of these waves and describe how electric and magnetic fields are generated and altered by each other and by charges and currents.

Electromagnetic waves are fundamental to our understanding of the electromagnetic spectrum, including radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays. These waves are capable of traveling through a vacuum, unlike sound waves, which require a medium. The ability of electromagnetic waves to carry energy and information across vast distances is what enables technologies such as television, radio communication, and cellular phones. In the integral form of Maxwell's equations, these waves are described by time-varying fields, showing the interrelation between changing electric and magnetic fields in space.

The integral form of Maxwell's equations provides a powerful framework for understanding how electric and magnetic fields behave under various circumstances. These equations summarize the generation and interrelation of these fields and serve as the basis for classical electrodynamics. They appear in two forms: differential and integral. The integral form, derived from the differential form using theorems like Gauss' and Stokes', is often more practical when solving problems with symmetric geometries because it deals with the net effects over an entire region.

The integral form of Maxwell's equations consists of the following four equations:

• Gauss' law for electricity, which relates the electric field to the charge distribution.
• Gauss' law for magnetism, which states that there are no magnetic monopoles.
• Faraday's law of induction, which describes how a changing magnetic field creates an electric field.
• Ampère's law with Maxwell's addition, which relates the magnetic field to the electric current and the change in the electric field.

These equations are pivotal for predicting how fields interact with materials, how they propagate through space, and how they give rise to phenomena such as electromagnetic waves.