# Understanding Maxwell’s Equations and Their Relation to Waves

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## Maxwell’s Equations and Their Impact

Maxwell’s equations represent a crucial mathematical framework in physics, revealing the profound truth that light is fundamentally an electromagnetic wave.

### Overview of Maxwell’s Equations

To summarize Maxwell’s equations succinctly: they consist of four key equations that define the behavior of electric and magnetic fields, including their configurations and how they can be generated. I will present the integral forms of these equations initially, followed by their differential forms.

### Gauss’s Law

The version commonly introduced in introductory physics courses is shown here.

In this equation, the left side denotes the electric flux through a closed surface, calculated by integrating the component of the electric field that is perpendicular to the surface. The notation indicates that it is a closed surface integral, with dA representing the area element. The unit vector n-hat is perpendicular to the surface.

On the right side, we have the total charge enclosed by that surface. The constant ?? is typically represented as follows:

Essentially, this law indicates that electric charges produce electric fields, which radiate outwards from positive charges and converge towards negative ones.

### Gauss’s Law for Magnetism

This law is somewhat unconventional—it suggests that, unlike electric charges, magnetic charges do not exist. Thus, the magnetic flux across a closed surface is always zero.

This implies that no net "magnetic charge" exists within any surface, leading to the conclusion that magnetic monopoles have not been observed. However, their existence remains a topic of discussion in theoretical physics.

### Faraday’s Law

Electric fields can be generated by electric charges (as explained by Gauss's Law), but there is another method, as Yoda famously said:

> No, there is another.

A changing magnetic flux can also produce an electric field.

The left side of this equation states that the closed line integral of the electric field is non-zero, contrary to the conservative nature of electric fields, while the right side indicates the magnetic flux over the area bounded by the same curve.

Two categories of electric fields arise: the Coulomb field associated with electric charges (linked to Gauss’s Law) and the "curly" fields produced by changing magnetic flux (as per Faraday’s Law).

A few notes on Faraday’s Law: - The magnetic flux on the right side does not feature a closed integral but refers to the area enclosed by the curve. - The negative sign indicates the direction of the generated field. - There is a time derivative, showing that the curly field is dependent on the rate of change of the magnetic flux.

### Ampere-Maxwell Law

One more equation to explore.

In essence, this law states that two methods can create a curly magnetic field: through an electric current (I_in) and by changing electric flux. It parallels Faraday's Law but with two distinctions: - It involves a path integral of the magnetic field instead of the electric field. - It includes an additional term representing the electric current flowing through the surface area bounded by the curve.

The constant involved is:

Faraday’s Law would similarly incorporate a term related to electric currents if magnetic monopoles were discovered.

### Differential Form of Maxwell’s Equations

We still have a lot to cover! I'll start again with Gauss’s Law. Here, I will introduce the concept of electric charge density (?), representing charge per unit volume. Integrating this density over a space provides the total electric charge present.

This integral signifies a volume integral due to the presence of dV.

Next, we apply the divergence theorem, which establishes a link between surface area integrals and volume integrals.

Here, the left side is a surface integral while the right side represents a volume integral. The operator (?) appears in Cartesian coordinates as follows:

By substituting the charge density and applying the divergence theorem, we can rewrite Gauss’s Law, replacing the charge term with a volume integral.

Now, we can transform the surface integral into a volume integral.

Since both integrals cover the same volume, the expressions within them must be identical. This is Gauss's law expressed in terms of volume integration.

As for Gauss’s Law for magnetism, since no magnetic monopoles exist, the magnetic charge density is zero, leading to:

For Faraday’s law, we will utilize Stoke’s Theorem, which relates surface integrals to line integrals.

The left side involves a path integral, while the right features the curl of the electric field, defined as the cross product of the Del operator and the field vector (F). Following this, we calculate the flux as usual.

To apply this to Faraday’s Law, we must address the time derivative. Since magnetic flux is spatially dependent, we can move the derivative inside the integral, converting it into a partial derivative:

Now, applying Stoke’s theorem changes the line integral into a surface integral.

Like before, since we have two area integrals, the expressions must match.

For the Ampere-Maxwell law, we introduce current density (j), indicating current per square meter, allowing us to derive the total current through a given area:

We will use Stoke’s theorem again to convert the line integral of the magnetic field into an area integral.

Next, we will replace the current (I_in) with an area integral of the current density on the right side of the Ampere-Maxwell equation, also moving the derivative inside the integral.

As both area integrals cover the same space, we can combine them. Incorporating Stoke’s theorem leads us to:

Again, the content within the integrals must align. Let’s compile all four equations together:

It's reminiscent of the Avengers, albeit with just four members and no superhero outfits.

## The Wave Equation

Imagine shaking one end of a string perpendicular to its length. This action displaces the string, and the resulting wave travels down its length.

This illustrates a graph of displacement (y) over position (x), which also varies with time. The relationship among y, x, and time (t) is known as the wave equation:

Here, v indicates the wave speed—how quickly a disturbance travels along the string. For a derivation of this wave equation, you can refer to a video.

This equation signifies that the second partial derivative of a function (y) with respect to time equals the wave speed multiplied by the second partial derivative with respect to position. Importantly, this equation is not limited to physical waves; it can apply to any function, denoted as f.

Moreover, this wave equation extends into three dimensions for vector fields (vector F), represented as follows:

The operator ?², referred to as the Laplacian, remains manageable in Cartesian coordinates.

Now, consider a region devoid of free charges (where ? equals zero) and electric currents (where j equals zero). In such a case, we can express Maxwell's equations as follows:

We begin with Faraday’s Law and take the curl of both sides:

Focusing on the right side, we see the partial derivative of B concerning time, which only involves time, while the curl pertains solely to spatial variables. This allows us to interchange their order:

The curl of B can be substituted using the Ampere-Maxwell law:

We require a vector identity, which states that for a vector field (F), the curl of the curl of F is expressed as:

Applying this identity to the curl of E yields:

Using Gauss's Law, we know that with zero charge density, ?·E equals zero. Multiplying both sides by -1 gives:

This represents the wave equation for the electric field. Notably, we can determine the wave speed.

By substituting the values for ?? and ??, we find the wave speed to be approximately 2.99 x 10? meters per second, matching the speed of light—a remarkable correlation!

Now, to verify if the magnetic field also behaves as a wave, we will adopt a similar approach. Starting with the Ampere-Maxwell equation (assuming zero current density), we take the curl of both sides.

Using Faraday’s Law, we can relate E to B on the right side:

Now, we substitute the curl of the curl on the left side:

In the absence of magnetic monopoles (and in empty space), we find that ?·B equals zero. Multiplying both sides by -1 leads to:

This also results in a wave equation, with a wave speed of c = 2.99 x 10? m/s, confirming that the magnetic field exhibits wave-like behavior as well.