Welcome to the fascinating world of Maxwell’s equations, where the laws of electromagnetism come to life! In class 12, we dive into the intricate workings of these four equations that are the cornerstone of modern physics. Developed by the brilliant Scottish physicist James Clerk Maxwell in the 19th century, these equations describe the behavior of electric and magnetic fields and their interdependence.

They are essential to understanding how the world works, from how light travels to the functioning of electronic devices. In this class, you will embark on a journey of discovery, exploring the intricacies of these equations and their practical applications. So, get ready to expand your knowledge and unleash the power of electromagnetism with Maxwell’s equations!

In this article, we will discuss Maxwell’s equations in detail.

# What are Maxwell’s equations?

Maxwell’s equations are a set of four fundamental equations that describe how electric and magnetic fields interact with each other and with charged particles.

They were developed by the brilliant Scottish physicist James Clerk Maxwell in the 19th century and are considered one of the most important contributions to the field of physics.

At their core, these equations reveal the intimate relationship between electric and magnetic fields, showing that they are not independent of each other, but instead intimately linked. They explain how electric charges create electric fields and how moving charges create magnetic fields. Furthermore, they demonstrate how changing magnetic fields produce electric fields, and vice versa, leading to electromagnetic waves, including radio waves, microwaves, and light waves.

Maxwell’s equations have a wide range of practical applications, from the functioning of electronic devices such as radios, televisions, and cell phones, to the behavior of light and other electromagnetic radiation. They have revolutionized our understanding of the physical world and have paved the way for numerous technological advancements.

In essence, Maxwell’s equations are the key to unlocking the secrets of electromagnetism, revealing how the forces that govern the world around us genuinely work. So, if you’re interested in delving into the fascinating world of physics and uncovering the mysteries of electromagnetism, then Maxwell’s equations are a great place to start!

# What are the four Maxwell’s equations?

Maxwell’s equations are a set of four partial differential equations that describe the behavior of electric and magnetic fields and their interaction with charges and currents. They are named after the Scottish physicist James Clerk Maxwell who formulated them in the 19th century.

The four Maxwell’s equations are:

## Gauss’s law for electric fields

Gauss’s law for electric fields is a fundamental law of electromagnetism that relates the flux of an electric field through a closed surface to the charge enclosed within that surface. Mathematically, it can be expressed in integral form as:

$$\begin{equation} \oint_S \vec{E} \cdot \mathrm{d}\vec{A} = \frac{Q_{enc}}{\epsilon_0} \end{equation}$$

where:

- $\oint_S$ is the surface integral operator, which means the integral is taken over the closed surface $S$
- $\vec{E}$ is the electric field at each point on the surface
- $\mathrm{d}\vec{A}$ is the infinitesimal area element of the surface
- $Q_{enc}$ is the total charge enclosed within the surface
- $\epsilon_0$ is the electric constant (also known as the permittivity of free space)

In words, this equation states that the flux of the electric field through any closed surface is proportional to the total charge enclosed within that surface. The constant of proportionality is the electric constant, which has a value of approximately $8.85 \times 10^{-12}$ F/m.

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## Gauss’s law for magnetic fields

Gauss’s Law for Magnetic Fields states that the magnetic flux through any closed surface is always zero. Mathematically, it can be expressed as:

$$\begin{equation} \oint_S \vec{B} \cdot \mathrm{d}\vec{A} = 0 \end{equation}$$

where:

- $\oint_S$ is the surface integral operator, which means the integral is taken over the closed surface $S$
- $\vec{B}$ is the magnetic field at each point on the surface
- $\mathrm{d}\vec{A}$ is the infinitesimal area element of the surface

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## Faraday’s law of electromagnetic induction

Faraday’s Law of Electromagnetic Induction states that whenever the magnetic flux through a closed loop changes over time, an electric field is induced in the loop, which in turn causes an electric current to flow.

In other words, a changing magnetic field through a loop of wire induces an electromotive force (EMF) in the loop, which in turn causes an electric current to flow. Mathematically, it can be expressed as:

$$\begin{equation} \oint_C \vec{E} \cdot \mathrm{d}\vec{l} = -\frac{\mathrm{d}}{\mathrm{d}t}\iint_S \vec{B} \cdot \mathrm{d}\vec{A} \end{equation}$$

where:

- $\oint_C$ is the line integral operator, which means the integral is taken over a closed path $C$
- $\vec{E}$ is the electric field along the path $C$
- $\mathrm{d}\vec{l}$ is the infinitesimal length element of the path $C$
- $\iint_S$ is the surface integral operator, which means the integral is taken over any surface $S$ bounded by the closed path $C$
- $\vec{B}$ is the magnetic field passing through the surface $S$
- $\mathrm{d}\vec{A}$ is the infinitesimal area element of the surface $S$
- $t$ is time

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## Ampere’s law with Maxwell’s correction

Ampere’s Law with Maxwell’s Correction is a modification of the original Ampere’s Law that takes into account the displacement current term.

This law states that the line integral of the magnetic field around any closed loop is equal to the sum of the electric current passing through the loop and the displacement current enclosed by the loop. Mathematically, it can be expressed as:

\begin{equation} \oint_C \vec{B} \cdot \mathrm{d}\vec{l} = \mu_0 \left(I_{enc} + \epsilon_0 \frac{\mathrm{d}}{\mathrm{d}t} \iint_S \vec{E} \cdot \mathrm{d}\vec{A} \right) \end{equation}

where:

- $\oint_C$ is the line integral operator, which means the integral is taken over a closed loop $C$
- $\vec{B}$ is the magnetic field along the loop $C$
- $\mathrm{d}\vec{l}$ is the infinitesimal length element of the loop $C$
- $\mu_0$ is the magnetic constant (also known as the permeability of free space)
- $I_{enc}$ is the total electric current passing through any surface bounded by the loop $C$
- $\epsilon_0$ is the electric constant (also known as the permittivity of free space)
- $\vec{E}$ is the electric field passing through the surface $S$

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# Maxwell’s equations explained

## What is Gauss’s law for the electric field?

Gauss’s law for electric fields is one of four of Maxwell’s equations, which describes the relationship between electric fields and electric charges. It states that the total electric flux through any closed surface is proportional to the total charge enclosed within the surface.

Mathematically, it can be expressed as: $$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$$ where $\nabla \cdot \mathbf{E}$ is the divergence of the electric field $\mathbf{E}$, $\rho$ is the charge density, and $\epsilon_0$ is the permittivity of free space.

In simpler terms, Gauss’s law for electric fields means that electric fields emanate from electric charges, and their strength depends on the amount of charge present. The law also implies that the electric field lines always begin on positive charges and end on negative charges, forming continuous loops.

The law has several important applications, such as calculating the electric field of a point charge, determining the electric field inside a uniformly charged sphere, and finding the electric flux through various shapes and configurations of charged objects. Overall, Gauss’s law for electric fields is a fundamental principle in the study of electromagnetism and plays a crucial role in many practical applications, including the design of electrical circuits, motors, and generators.

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## What is Gauss’s law for magnetic fields?

Gauss’s law for magnetic fields states that the magnetic flux through any closed surface is always zero.

Magnetic flux is a measure of the amount of magnetic field passing through a given area. It is calculated by taking the dot product of the magnetic field vector and the surface area vector. In symbols, magnetic flux $\Phi_B$ is given by: $$\Phi_B = \int_S \mathbf{B} \cdot d\mathbf{A}$$

where $S$ is a closed surface, $\mathbf{B}$ is the magnetic field vector, and $d\mathbf{A}$ is a small element of surface area. The dot product of $\mathbf{B}$ and $d\mathbf{A}$ gives the component of the magnetic field that is perpendicular to the surface. Integrating over the entire surface $S$ gives the total magnetic flux passing through the surface.

Gauss’s law for magnetic fields says that no matter what closed surface $S$ you choose, the magnetic flux passing through it is always zero. Mathematically, this is expressed as: $$\nabla \cdot \mathbf{B} = 0$$

where $\nabla \cdot \mathbf{B}$ is the divergence of the magnetic field. Divergence is a mathematical operation that measures the tendency of a vector field to either converge or diverge at a given point. If the divergence is zero, it means that the field lines do not diverge or converge, but instead form closed loops.

In simpler terms, Gauss’s law for magnetic fields means that magnetic fields always form closed loops, and there are no isolated magnetic charges, also known as magnetic monopoles. It implies that any magnetic field that begins in one place must end up somewhere else, forming a closed loop.

It is important to note that Gauss’s law for magnetic fields is not an independent law, but rather a consequence of the other three of Maxwell’s equations. Unlike electric fields, magnetic fields do not interact directly with electric charges, but they are produced by moving charges or current-carrying wires. Therefore, the other three Maxwell’s equations are required to fully describe the behavior of magnetic fields in the presence of electric fields and charges.

## What is faraday’s law of electromagnetic induction?

Faraday’s law of electromagnetic induction is one of the four Maxwell’s equations that describe the behavior of electric and magnetic fields. It states that when the magnetic flux through a closed loop changes over time, an electric field is induced in the loop, which in turn causes an electric current to flow.

Mathematically, it can be expressed as: $$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$$ where $\nabla \times \mathbf{E}$ is the curl of the electric field $\mathbf{E}$, and $\frac{\partial \mathbf{B}}{\partial t}$ is the time derivative of the magnetic field $\mathbf{B}$.

In simpler terms, Faraday’s law of electromagnetic induction means that a changing magnetic field creates an electric field, which can induce an electric current in a closed loop. The strength of the induced electric field depends on the rate of change of the magnetic field and the area of the loop. This phenomenon is the basis for many important technologies, such as electric generators, transformers, and motors.

Faraday’s law has several important consequences. First, it implies that a changing magnetic field induces an electric field, even in the absence of any charges or currents. This is known as electromagnetic induction. Second, it explains why electric generators work: by rotating a coil of wire in a magnetic field, a changing magnetic flux is created which induces an electric current in the coil. Finally, it shows that electric and magnetic fields are closely related and that changes in one field can cause changes in the other.

It is important to note that Faraday’s law is not a stand-alone law, but rather a consequence of the other three of Maxwell’s equations. Together with Gauss’s law for electric fields, Gauss’s law for magnetic fields, and Ampere’s law with Maxwell’s modification, they form a complete set of equations that describe the behavior of electric and magnetic fields in the presence of charges and currents.

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## What is Ampere’s law with Maxwell’s correction?

Ampere’s Law is one of the four Maxwell’s equations that describe the behavior of electric and magnetic fields. It relates the magnetic field to the current density and the rate of change of the electric field.

The original version of Ampere’s Law states that the line integral of the magnetic field $\mathbf{B}$ around any closed loop is equal to the current passing through the loop:

$$\begin{equation} \oint_{\partial S} \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enc} \end{equation}$$

where $\partial S$ is a closed loop that encloses a current $I_{enc}$, $\mu_0$ is the magnetic constant, and $d\mathbf{l}$ is an infinitesimal vector element of the path.

However, this original version of the law does not take into account the effects of a changing electric field, which can also produce a magnetic field. To account for this, Maxwell introduced a correction term that includes the time derivative of the electric field:

$$\begin{equation} \oint_{\partial S} \mathbf{B} \cdot d\mathbf{l} = \mu_0 \left(I_{enc} + \varepsilon_0 \frac{d}{dt} \int_S \mathbf{E} \cdot d\mathbf{A}\right) \end{equation}$$

where $\varepsilon_0$ is the electric constant, $\mathbf{E}$ is the electric field, and $d\mathbf{A}$ is an infinitesimal vector element of surface area. The integral of the electric field over a surface $S$ gives the total electric flux through the surface.

This equation states that the line integral of the magnetic field around any closed loop is equal to the sum of the electric current passing through the loop and the rate of change of electric flux passing through any surface bounded by the loop. This correction reflects the fact that a time-varying electric field can induce a magnetic field, as described by Faraday’s Law of Electromagnetic Induction.

This corrected version of Ampere’s Law is known as Ampere’s Law with Maxwell’s Correction. It takes into account the fact that a changing electric field can produce a magnetic field and provides a more complete description of the relationship between electric and magnetic fields. It is an important equation in the study of electromagnetism and is used in a wide range of applications, from electrical engineering to particle physics.

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## Maxwell’s equations differential form

### Gauss’s Law for Electric Fields in differential form:

$$\begin{equation} \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} \end{equation}$$

This equation relates the divergence of the electric field to the charge density in a given region of space. In other words, it tells us how much electric flux is emanating from a point charge or a distribution of charges.

The symbol $\nabla$ (pronounced “del”) is the gradient operator, which is a vector operator that tells us how a scalar or vector field is changing in space. The dot product with the electric field $\mathbf{E}$ and the divergence operator $\nabla \cdot$ results in a scalar value that represents the net flow of electric flux out of a region of space. The right-hand side of the equation represents the charge density $\rho$ in that same region, divided by the electric constant $\varepsilon_0$, which relates the strength of the electric field to the amount of charge present.

### Gauss’s Law for Magnetic Fields in differential form:

$$\begin{equation} \nabla \cdot \mathbf{B} = 0 \end{equation}$$

This equation states that the divergence of the magnetic field is always zero, which means that there are no magnetic monopoles (analogous to electric charges) and that magnetic field lines always form closed loops. In other words, the magnetic field is always “sourced” by a current or a changing electric field, and it always “sinks” somewhere else.

The symbol $\nabla$ and the divergence operator $\nabla \cdot$ have the same meaning as in the previous equation. The magnetic field $\mathbf{B}$ is a vector field that describes the direction and magnitude of the magnetic force experienced by a moving charge or a current. The fact that the divergence is always zero means that magnetic field lines never originate from or terminate at a single point, but always form closed loops.

### Faraday’s Law of Electromagnetic Induction in differential form:

$$\begin{equation} \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \end{equation}$$

This equation relates the curl of the electric field to the time rate of change of the magnetic field. In other words, it tells us how a changing magnetic field induces an electric field and vice versa.

The symbol $\nabla$ and the curl operator $\nabla \times$ represent the rotational behavior of a vector field in space. The curl of the electric field $\mathbf{E}$ is a measure of how the electric field is changing in space, while the time derivative of the magnetic field $\frac{\partial \mathbf{B}}{\partial t}$ is a measure of how the magnetic field is changing over time. The minus sign on the right-hand side of the equation indicates that a changing magnetic field induces an electric field that opposes the change, in accordance with Lenz’s Law.

### Ampere’s Law with Maxwell’s Correction in differential form:

$$\begin{equation} \nabla \times \mathbf{B} = \mu_0 \left(\mathbf{J} + \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}\right) \end{equation}$$

This equation relates the curl of the magnetic field to the current density and the time rate of change of the electric field. In other words, it tells us how a current or a changing electric field produces a magnetic field.

The symbol $\nabla$ and the curl operator $\nabla \times$ have the same meaning as in the previous equation. The magnetic field $\mathbf{B}$ is related to the current density $\mathbf{J}$ and the time rate of change of the electric field $\frac{\partial \mathbf{E}}{\partial t}$ through the constants $\mu_0$ and $\varepsilon_0$. The constant $\mu_0$ is the magnetic constant, which relates the strength of the magnetic field to the amount of current present, while the constant $\varepsilon_0$ is the electric constant, which relates the strength of the electric field to the amount of charge present.

The term $\mathbf{J}$ represents the density of electric current flowing in a given region of space, while the term $\varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}$ represents the displacement current, which is a kind of “fictitious” current that arises from the time-varying electric field. This correction to Ampere’s Law was added by Maxwell to account for the fact that a changing electric field can produce a magnetic field, just as a current does.

So, these four equations summarize the fundamental laws of electromagnetism and govern the behavior of electric and magnetic fields in space. They are essential tools for understanding a wide range of phenomena, from the behavior of electric circuits and electromagnetic waves to the structure of atoms and the nature of light.

## Maxwell’s equations in integral form

### Gauss’s Law for Electric Fields in integral form:

$$\begin{equation} \oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{enc}}{\varepsilon_0} \end{equation}$$

This equation relates the electric field to the net charge enclosed within a closed surface. In other words, it tells us how much electric flux is emanating from a charge or a distribution of charges.

The symbol $\oint_S$ represents the closed surface integral, which tells us how much of the electric field is penetrating a closed surface $S$. The dot product with the electric field $\mathbf{E}$ and the differential surface area $d\mathbf{A}$ results in a scalar value that represents the amount of electric flux emanating from the surface. The right-hand side of the equation represents the total charge $Q_{enc}$ enclosed within that same surface, divided by the electric constant $\varepsilon_0$, which relates the strength of the electric field to the amount of charge present.

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- What is the difference between a permanent magnet and an electromagnet?
- Solenoid, definition, working principle, types, and applications

### Gauss’s Law for Magnetic Fields in integral form:

$$\begin{equation} \oint_S \mathbf{B} \cdot d\mathbf{A} = 0 \end{equation}$$

This equation states that the magnetic field lines always form closed loops and that there are no magnetic monopoles (analogous to electric charges).

The symbol $\oint_S$ and the surface integral have the same meaning as in the previous equation. The magnetic field $\mathbf{B}$ is a vector field that describes the direction and magnitude of the magnetic force experienced by a moving charge or a current. The fact that the surface integral is always zero means that magnetic field lines never originate from or terminate at a single point, but always form closed loops.

### Faraday’s Law of Electromagnetic Induction in integral form:

$$\begin{equation} \oint_C \mathbf{E} \cdot d\mathbf{l} = -\frac{d}{dt} \int_S \mathbf{B} \cdot d\mathbf{A} \end{equation}$$

This equation relates the electric field to the time rate of change of the magnetic field through a closed loop. In other words, it tells us how a changing magnetic field induces an electric field and vice versa.

The symbol $\oint_C$ represents the closed loop integral, which tells us how much of the electric field is circulating around a closed loop $C$. The dot product with the differential path length $d\mathbf{l}$ results in a scalar value that represents the line integral of the electric field along the path. The right-hand side of the equation represents the time derivative of the surface integral of the magnetic field $\frac{d}{dt} \int_S \mathbf{B} \cdot d\mathbf{A}$, which is a measure of how the magnetic field is changing over time.

### Ampere’s Law with Maxwell’s Correction in integral form:

$$\begin{equation} \oint_C \mathbf{B} \cdot d\mathbf{l} = \mu_0 \left(I_{enc} + \varepsilon_0 \frac{d}{dt} \int_S \mathbf{E} \cdot d\mathbf{A}\right) \end{equation}$$

This equation relates the magnetic field to the current enclosed within a closed loop and the time rate of change of the electric field through a surface. In other words, it tells us how a current or a changing electric field produces a magnetic field.

The symbol $\oint_C$ and the line integral have the same meaning as in the previous equation. The magnetic field $\mathbf{B}$ is related to the current density $\mathbf{J}$ and the time rate of change of the electric field $\frac{\partial \mathbf{E}}{\partial t}$ through the constants $\mu_0$ and $\varepsilon_0$. The constant $\mu_0$ is the magnetic constant, which relates the strength of the magnetic field to the amount of current present, while the constant $\varepsilon_0$ is the electric constant, which relates the strength of the electric field to the amount of charge present.

The symbol $\oint_C$ represents the closed loop integral, which tells us how much of the magnetic field is circulating around a closed loop $C$. The dot product with the differential path length $d\mathbf{l}$ results in a scalar value that represents the line integral of the magnetic field along the path. The right-hand side of the equation represents the sum of two terms. The first term, $I_{enc}$, represents the current enclosed within the loop.

The second term, $\varepsilon_0 \frac{d}{dt} \int_S \mathbf{E} \cdot d\mathbf{A}$, represents the displacement current, which is a kind of “fictitious” current that arises from the time-varying electric field. This correction to Ampere’s Law was added by Maxwell to account for the fact that a changing electric field can produce a magnetic field, just as a current does.

These four equations provide a concise mathematical description of the behavior of electric and magnetic fields. They are essential tools for understanding a wide range of phenomena, from the behavior of electric circuits and electromagnetic waves to the structure of atoms and the nature of light. By expressing these laws in both differential and integral forms, we can study the behavior of electric and magnetic fields at both the microscopic and macroscopic levels, and we can apply these principles to a wide variety of physical systems.

# Derive maxwell’s equations

## Derive maxwell’s first equation (Gauss’s Law for Electric Fields)

Gauss’s law for electric field is one of the four Maxwell’s equations that relates the electric flux through a closed surface to the charge enclosed within that surface. It states that the electric flux through a closed surface is proportional to the charge enclosed within that surface.

The mathematical expression of Gauss’s law for electric field is:

$$\begin{equation} \oint_{\partial V} \mathbf{E}\cdot d\mathbf{A} = \frac{Q_{enc}}{\epsilon_0} \end{equation}$$

where $\mathbf{E}$ is the electric field, $Q_{enc}$ is the total charge enclosed within the closed surface, $\epsilon_0$ is the electric constant (also known as vacuum permittivity), $\oint_{\partial V}$ denotes the closed surface integral over the boundary of the surface $V$, and $d\mathbf{A}$ is the differential area vector pointing outward from the surface.

### Derivation of first maxwell’s equation

Gauss’s law for electric field can be derived using the divergence theorem, also known as Gauss’s theorem. The divergence theorem relates a volume integral of a vector field over a region in space to the surface integral of the same field over the boundary of that region.

Let us consider a closed surface $S$ enclosing a volume $V$. The electric field $\mathbf{E}$ can be represented as a vector function of position in space. The divergence of the electric field at any point in space is given by:

$$\begin{equation} \nabla \cdot \mathbf{E} = \frac{\partial E_x}{\partial x} + \frac{\partial E_y}{\partial y} + \frac{\partial E_z}{\partial z} \end{equation}$$

where $E_x$, $E_y$, and $E_z$ are the components of the electric field along the $x$, $y$, and $z$ axes, respectively.

Now, we can apply the divergence theorem to the electric field $\mathbf{E}$ over the volume $V$ enclosed by the surface $S$:

$$\begin{equation} \oint_S \mathbf{E} \cdot d\mathbf{A} = \iiint_V (\nabla \cdot \mathbf{E}) ,dV \end{equation}$$

where $\oint_S$ denotes the closed surface integral over the boundary of the surface $S$, $d\mathbf{A}$ is the differential area vector pointing outward from the surface, and $\iiint_V$ denotes the triple integral over the volume $V$.

Since the surface $S$ is closed, the outward pointing differential area vector $d\mathbf{A}$ at every point on the surface has the same magnitude and direction. Therefore, we can take the magnitude of the dot product $\mathbf{E} \cdot d\mathbf{A}$ outside the integral, and we can simplify the surface integral on the left-hand side of the equation to obtain:

$$\begin{equation} \oint_S \mathbf{E} \cdot d\mathbf{A} = E \oint_S dA = E A \end{equation}$$

where $E$ is the magnitude of the electric field and $A$ is the total area of the surface $S$.

Similarly, we can simplify the triple integral on the right-hand side of the equation using the definition of the divergence:

$$\begin{equation} \iiint_V (\nabla \cdot \mathbf{E}) ,dV = \iiint_V \frac{\rho}{\epsilon_0} ,dV = \frac{Q_{enc}}{\epsilon_0} \end{equation}$$

where $\rho$ is the charge density, $Q_{enc}$ is the total charge enclosed within the volume $V$, and $\epsilon_0$ is the electric constant.

Substituting these simplifications into the divergence theorem equation, we obtain:

$$\begin{equation} EA = \frac{Q_{enc}}{\epsilon_0} \end{equation}$$

Dividing both sides by the area $A$ of the surface $S$, we arrive at the final form of Gauss’s law for electric field:

$$\begin{equation} \oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{enc}}{\epsilon_0} \end{equation}$$

This equation states that the electric flux through a closed surface is proportional to the total charge enclosed within that surface, and is known as Gauss’s law for electric field.

## Derive second maxwell’s equation (Gauss’s law for magnetic field)

The second of Maxwell’s equations is Gauss’s law for magnetism, which relates the magnetic field to the sources of magnetism, i.e., the currents and the changing electric fields. It states that the magnetic flux through any closed surface is always zero. Mathematically, this is expressed as: $$\nabla \cdot \mathbf{B} = 0$$

### Derivation of second maxwell’s equation

To derive Gauss’s law for magnetism, let us consider a closed surface $S$ and a small loop $L$ inside the surface. We can apply Faraday’s law of electromagnetic induction to the loop $L$ to obtain:

$$\begin{equation} \oint_L \mathbf{E} \cdot d\mathbf{l} = -\frac{d\Phi_B}{dt} \end{equation}$$

where $\mathbf{E}$ is the electric field, $d\mathbf{l}$ is the differential length vector along the loop $L$, and $\Phi_B$ is the magnetic flux through the surface $S$ enclosed by the loop $L$.

Now, using Stokes’ theorem, we can convert the line integral on the left-hand side of the equation to a surface integral over the boundary of the surface $S$ enclosed by the loop $L$:

$$\begin{equation} \oint_L \mathbf{E} \cdot d\mathbf{l} = \int_S (\nabla \times \mathbf{E}) \cdot d\mathbf{A} \end{equation}$$

where $\nabla \times \mathbf{E}$ is the curl of the electric field and $d\mathbf{A}$ is the differential area vector pointing outward from the surface $S$.

Substituting this expression into Faraday’s law, we obtain:

$$\begin{equation} \int_S (\nabla \times \mathbf{E}) \cdot d\mathbf{A} = -\frac{d\Phi_B}{dt} \end{equation}$$

Since the magnetic flux through any closed surface is always zero, the time derivative of the magnetic flux through the surface enclosed by the loop $L$ is also zero. Therefore, we can simplify the right-hand side of the equation to obtain:

$$\begin{equation} \int_S (\nabla \times \mathbf{E}) \cdot d\mathbf{A} = 0 \end{equation}$$

This equation must hold for any closed surface $S$, and since the surface integral on the left-hand side of the equation can only be zero if the integrand is zero at every point on the surface, we arrive at the final form of Gauss’s law for magnetism:

$$\begin{equation} \nabla \cdot \mathbf{B} = 0 \end{equation}$$

where $\mathbf{B}$ is the magnetic field. This equation states that the magnetic flux through any closed surface is always zero, and is known as Gauss’s law for magnetism.

**Suggested reading:**

- Magnetic moment of a revolving electron, and Bohr magneton, class 12
- Magnetic moment of atoms, electrons, nucleus, molecules, and elementary particles

## Derive third maxwell’s equation (Faraday’s law of electromagnetic induction)

The third of Maxwell’s equations is Faraday’s law of electromagnetic induction, which relates a changing magnetic field to the generation of an electric field. It states that the electric field induced in a closed loop is proportional to the rate of change of magnetic flux through the loop.

Mathematically, it can be expressed as: $$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$$ where $\nabla \times \mathbf{E}$ is the curl of the electric field $\mathbf{E}$, and $\frac{\partial \mathbf{B}}{\partial t}$ is the time derivative of the magnetic field $\mathbf{B}$

### Derivation of third maxwell’s equation

To derive Faraday’s law of electromagnetic induction, let us consider a closed loop $L$ in a magnetic field $\mathbf{B}$ that is changing with time. The magnetic flux through the loop is given by:

$$\begin{equation} \Phi_B = \int_S \mathbf{B} \cdot d\mathbf{A} \end{equation}$$

where $S$ is the surface enclosed by the loop $L$, $d\mathbf{A}$ is the differential area vector pointing outward from the surface $S$, and $\mathbf{B}$ is the magnetic field.

If the magnetic field changes with time, then the flux through the loop changes as well, and we can express the time derivative of the flux as:

$$\begin{equation} \frac{d\Phi_B}{dt} = \frac{d}{dt} \int_S \mathbf{B} \cdot d\mathbf{A} = \int_S \frac{\partial \mathbf{B}}{\partial t} \cdot d\mathbf{A} \end{equation}$$

where we have used the fact that the surface $S$ is fixed in space, and only the magnetic field changes with time.

Now, applying Faraday’s law of electromagnetic induction, we can relate the electric field induced in the loop to the time derivative of the magnetic flux through the loop as follows:

$$\begin{equation} \oint_L \mathbf{E} \cdot d\mathbf{l} = -\frac{d\Phi_B}{dt} = -\int_S \frac{\partial \mathbf{B}}{\partial t} \cdot d\mathbf{A} \end{equation}$$

where $\mathbf{E}$ is the electric field, $d\mathbf{l}$ is the differential length vector along the loop $L$, and we have used the negative sign to indicate that the induced electric field opposes the change in magnetic flux.

Using Stokes’ theorem, we can convert the line integral on the left-hand side of the equation to a surface integral over the boundary of the surface $S$ enclosed by the loop $L$:

$$\begin{equation} \oint_L \mathbf{E} \cdot d\mathbf{l} = \int_S (\nabla \times \mathbf{E}) \cdot d\mathbf{A} \end{equation}$$

Substituting this expression into Faraday’s law, we obtain:

$$\begin{equation} \int_S (\nabla \times \mathbf{E}) \cdot d\mathbf{A} = -\frac{\partial}{\partial t} \int_S \mathbf{B} \cdot d\mathbf{A} \end{equation}$$

Since the surface $S$ is arbitrary, we can equate the integrands on both sides of the equation to obtain the final form of Faraday’s law of electromagnetic induction:

$$\begin{equation} \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \end{equation}$$

where $\mathbf{E}$ is the electric field and $\mathbf{B}$ is the magnetic field. This equation states that a changing magnetic field induces an electric field, and is known as Faraday’s law of electromagnetic induction.

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## Derive fourth maxwell’s equation (Ampere’s law with Maxwell’s correction)

The fourth of Maxwell’s equations is Ampere’s law with Maxwell’s correction, which relates the curl of the magnetic field to the current density and the rate of change of the electric field. It states that the circulation of the magnetic field around a closed loop is proportional to the current passing through the loop, as well as the rate of change of the electric field.

Mathematically, it can be expressed as: $$\begin{equation} \nabla \times \mathbf{B} = \mu_0 \left(\mathbf{J} + \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}\right) \end{equation}$$

### Derivation of fourth maxwell’s equation

To derive Ampere’s law with Maxwell’s correction, let us consider a closed loop $L$ carrying a steady current $I$, and let $\mathbf{H}$ be the magnetic field due to this current. Applying Ampere’s law, we have:

$$\begin{equation} \oint_L \mathbf{H} \cdot d\mathbf{l} = I \end{equation}$$

where $d\mathbf{l}$ is the differential length vector along the loop $L$.

Now, let us consider a time-varying electric field $\mathbf{E}$, and the corresponding displacement current density $\epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$, where $\epsilon_0$ is the permittivity of free space. Applying Ampere’s law to a loop $L’$ that encloses the changing electric field, we have:

$$\begin{equation} \oint_{L’} \mathbf{H} \cdot d\mathbf{l} = \int_S \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \cdot d\mathbf{A} \end{equation}$$

where $S$ is the surface enclosed by the loop $L’$, $d\mathbf{A}$ is the differential area vector pointing outward from the surface $S$, and we have used the fact that the time-varying electric field induces a displacement current.

Now, applying Stokes’ theorem to the left-hand side of the equation, we can convert the line integral to a surface integral over the boundary of the surface $S$ enclosed by the loop $L’$:

$$\begin{equation} \int_S (\nabla \times \mathbf{H}) \cdot d\mathbf{A} = \int_S \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \cdot d\mathbf{A} \end{equation}$$

Substituting $\mathbf{B} = \mu_0 \mathbf{H}$, where $\mu_0$ is the permeability of free space, and using the fact that $\nabla \cdot \mathbf{B} = 0$, we can rewrite the left-hand side of the equation as:

$$\begin{equation} \int_S (\nabla \times \mathbf{H}) \cdot d\mathbf{A} = \int_S (\nabla \times \mathbf{B}) \cdot d\mathbf{A} = \mu_0 \int_S \mathbf{J} \cdot d\mathbf{A} \end{equation}$$ |

where $\mathbf{J}$ is the current density.

Therefore, we can rewrite Ampere’s law with the addition of Maxwell’s correction as:

$$\begin{equation} \nabla \times \mathbf{B} = \mu_0 \left(\mathbf{J} + \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}\right) \end{equation}$$

where $\mathbf{B}$ is the magnetic field, $\mathbf{J}$ is the current density, and $\mathbf{E}$ is the electric field. This equation relates the curl of the magnetic.

# Applications of maxwell’s equations

Maxwell’s equations have many applications in different fields of science and technology. Here are some applications of maxwell’s equations:

**Electromagnetic Waves:**Maxwell’s equations predict the existence of electromagnetic waves, which include radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays, and gamma rays. These waves have a wide range of applications in communications, medicine, and many other fields.**Electrical Engineering:**Maxwell’s equations are used in the design and analysis of electrical circuits, antennas, and electromagnetic devices such as transformers, motors, and generators.**Material Science:**The behavior of materials can be studied using Maxwell’s equations. For example, they can be used to understand how materials respond to electromagnetic waves and to design materials with specific electrical and magnetic properties.**Plasma Physics:**Maxwell’s equations are used to describe the behavior of plasmas, which are partially ionized gases that are found in many natural and man-made environments, such as stars, lightning, and plasma TVs.**Astrophysics:**Maxwell’s equations are used to study the behavior of electromagnetic fields in space, including the magnetic fields of planets and stars.**Medical Imaging:**Maxwell’s equations are used in medical imaging techniques such as magnetic resonance imaging (MRI) and computed tomography (CT) scans.**Quantum Field Theory:**Maxwell’s equations are used in the development of quantum field theory, which is a framework for understanding the behavior of subatomic particles.

# Physical significance of maxwell’s equations

Maxwell’s equations describe the behavior of electromagnetic fields and their interactions with charged particles. There are four Maxwell equations, and each one has a specific physical significance:

**Gauss’s law for electric fields:**This equation relates the electric field to the charge density in a given region of space. It states that the flux of the electric field through any closed surface is proportional to the total charge enclosed within the surface. This equation has important implications for the behavior of electric fields around charged objects, such as conductors.**Gauss’s law for magnetic fields:**This equation states that there are no magnetic monopoles and that the total magnetic flux through any closed surface is zero. This equation has important implications for the behavior of magnetic fields around magnets and in electromagnetic waves.**Faraday’s law of electromagnetic induction:**This equation relates a changing magnetic field to the creation of an electric field. It states that an electromotive force (EMF) is induced in a closed loop whenever the magnetic flux through the loop changes over time. This equation is the basis for many electrical devices, such as transformers and generators.**Ampere’s law:**This equation relates a changing electric field to the creation of a magnetic field. It states that the circulation of the magnetic field around a closed loop is proportional to the electric current passing through the loop. This equation is the basis for the behavior of magnetic fields in circuits and in electromagnetic waves.

If taken together, these four equations provide a complete description of the behavior of electromagnetic fields and their interactions with charged particles. They have important practical applications in many areas of science and engineering, including telecommunications, electronics, and materials science.

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# Frequently Asked Questions – FAQs

##### What are electromagnetic waves?

Electromagnetic waves are waves that are created by the oscillation or acceleration of charged particles, such as electrons. These waves consist of oscillating electric and magnetic fields that are perpendicular to each other and to the direction of wave propagation. They can travel through a vacuum, as well as through different materials, such as air, water, and solids.

##### What is the electromagnetic theory?

The electromagnetic theory is a fundamental physical theory that describes the behavior of electric and magnetic fields and their interactions with charged particles. It is based on the four Maxwell’s equations that were developed by James Clerk Maxwell in the 19th century. These equations describe the relationship between electric and magnetic fields, as well as their sources, and provide a complete description of electromagnetic phenomena.

Electromagnetic theory is a cornerstone of modern physics and is used to explain a wide range of phenomena, including the behavior of electric and magnetic fields, the propagation of electromagnetic waves, and the interaction between electromagnetic fields and matter. It has practical applications in many areas of science and technology, including telecommunications, electronics, and materials science.

##### Why is Maxwell’s equation important?

Maxwell’s equations are important for several reasons:

1. They provide a complete and accurate description of the behavior of electromagnetic fields and their interactions with charged particles. This description has been shown to be consistent with experimental results and has stood the test of time, making Maxwell’s equations a cornerstone of modern physics.

2. They unify the previously separate fields of electricity and magnetism, demonstrating that they are fundamentally the same phenomenon. This unification led to the development of the electromagnetic theory and helped to lay the groundwork for the development of special relativity.

3. They predict the existence of electromagnetic waves, which led to the discovery of radio waves and the development of radio communication.

4. They have practical applications in many areas of science and technology, including telecommunications, electronics, and materials science. For example, Maxwell’s equations are used to design antennas, transmission lines, and other electronic devices.

##### Why Ampere’s law is modified by Maxwell?

Ampere’s law, as originally formulated, stated that the circulation of the magnetic field around a closed loop is proportional to the electric current passing through the loop. However, this law was found to be incomplete when Maxwell incorporated the concept of a changing electric field into his electromagnetic theory.

Maxwell’s modification to Ampere’s law involves adding an additional term, known as the displacement current, to the equation. The displacement current represents the rate of change of the electric field through a surface bounded by the loop, and is given by the equation:

$$\frac{\partial D}{\partial t}$$

where $D$ is the electric displacement field.

Maxwell’s modification to Ampere’s law is necessary because it allows for the propagation of electromagnetic waves, which consist of both electric and magnetic fields. Without the displacement current term, Ampere’s law would not be able to account for the creation of magnetic fields due to the changing electric fields associated with electromagnetic waves.

Maxwell’s modification to Ampere’s law is thus an important step in the development of the electromagnetic theory, as it allows for a complete and accurate description of the behavior of electromagnetic fields and their interactions with charged particles.

##### What are 4 Maxwell’s equations?

Maxwell’s equations are a set of four fundamental equations that describe the behavior of electric and magnetic fields. The four equations are:

1.** Gauss’s law for electric fields:** This equation relates the electric flux through a closed surface to the electric charge enclosed within the surface.

2. **Gauss’s law for magnetic fields: **This equation states that there are no magnetic monopoles and that the total magnetic flux through any closed surface is zero.

3. **Faraday’s law of electromagnetic induction:** This equation relates a changing magnetic field to the creation of an electric field.

4. **Ampere’s law with Maxwell’s correction:** This equation relates a changing electric field to the creation of a magnetic field.

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