Maxwell's Equation's

Maxwell's equations are a set of four fundamental laws that describe how electric and magnetic fields interact and propagate. These equations form the foundation of classical electrodynamics, optics, and electric circuits. They are named after the Scottish physicist James Clerk Maxwell.

Maxwell First Equation

Maxwell’s first equation is based on the Gauss law of electrostatic, which states that “when a closed surface integral of electric flux density is always equal to charge enclosed over that surface”

Mathematically Gauss law can be expressed as,

Over a closed surface, the product of the electric flux density vector and surface integral is equal to the charge enclosed.

$$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$$

where:

• $\mathbf{E}$ is the electric field.
• $\rho$ is the electric charge density.
• $\epsilon_0$ is the permittivity of free space.

or,

$\nabla \cdot \mathbf{D} = \rho$

Here $D$ is electric displacement field and $D = E\epsilon_0$.

Derivation

Let us consider a surface $S$ bounding a volume $V$ in a dielectric medium, which is kept in the $\mathbf{E}$ field. The application of external field $\mathbf{E}$ polarises the dielectric medium and charges are induced, called bound charges or charges due to polarisation. The total charge density at a point in a small volume element $dV$ would then be $(\rho + \rho_p)$, where $\rho_p$ is the polarisation charge density (the same as $\rho_p$), given by $\rho_p = -\nabla \cdot \mathbf{P}$ and $\rho$ is the free charge density at that point in the small volume element $dV$.

Thus, the total charge density at that point will be $\rho - (\nabla \cdot \mathbf{P})$. Then Gauss’s theorem can be expressed as

$$\oint_S \mathbf{E} \cdot d\mathbf{S} = \int_V (\nabla \cdot \mathbf{E}) dV$$

and

$$\int_V (\nabla \cdot \mathbf{E}) dV = \frac{1}{\epsilon_0} \int_V (\rho - \nabla \cdot \mathbf{P}) dV$$

or

$$\epsilon_0 \int_V (\nabla \cdot \mathbf{E}) dV = \int_V (\rho - \nabla \cdot \mathbf{P}) dV$$

or

$$\int_V (\epsilon_0 \nabla \cdot \mathbf{E} + \nabla \cdot \mathbf{P}) dV = \int_V \rho dV$$

The quantity $(\epsilon_0 \mathbf{E} + \mathbf{P})$ is denoted by a quantity $\mathbf{D}$, called the electric displacement. Therefore,

$$\int_V (\nabla \cdot \mathbf{D}) dV = \int_V \rho dV$$

Since this equation is true for all the arbitrary volumes, the integrands in this equation must be equal, i.e.,

$$\nabla \cdot \mathbf{D} = \rho$$

This is the Maxwell’s first equation.

When the medium is isotropic, the three vectors $\mathbf{D}$, $\mathbf{E}$ and $\mathbf{P}$ are in the same direction and for small field $\mathbf{E}$, $\mathbf{D}$ is proportional to $\mathbf{E}$, i.e.,

$$\mathbf{D} = \epsilon \mathbf{E}$$

where $\epsilon$ is called the permittivity of the dielectric medium. The ratio $\epsilon / \epsilon_0$ is called the dielectric constant of the medium.

Integral Form of Maxwell’s First Equation

Differential form of the Maxwell's first equation is

$\mathbf{\nabla} \cdot \mathbf{D} = \rho$

On integrating above eq over a volume $V$, we have

$\int_V (\mathbf{\nabla} \cdot \mathbf{D} \,) dv = \int_V \rho \, dV$

Using Gauss's divergence theorem, the above equation reads

$\oint_S \mathbf{D} \cdot d\mathbf{S} = \int_V \rho \, dV$

or

$\oint_S \mathbf{D} \cdot d\mathbf{S} = q$

Here $q$ is the total charge contained in the volume $V$ and $S$ is the surface bounding the volume $V$. This integral form of the Maxwell's first equation says that the total electric displacement through the surface $S$ enclosing the volume $V$ is equal to the total charge contained within this volume $V$.

This observation can be used to give the following form: The total outward flux corresponding to the displacement vector $\mathbf{D}$ through a closed surface $S$ is equal to the total charge $q$ within the volume $V$ enclosed by the surface $S$.

Maxwell Second Equation

Maxwell second equation is based on Gauss law on magnetostatics.

Gauss law on magnetostatics states that “closed surface integral of magnetic flux density is always equal to total scalar magnetic flux enclosed within that surface of any shape or size lying in any medium.”

Gauss's law for magnetism states that the net magnetic flux through a closed surface is zero, implying that there are no magnetic monopoles. It is given by:

$$\nabla \cdot \mathbf{B} = 0$$

where:

• $\mathbf{B}$ is the magnetic field.

Derivation

Since the magnetic lines of force are either closed loops or extend infinitely, the flux of magnetic induction $\mathbf{B}$ across any closed surface is always zero. This can be expressed as:

$\oint_S \mathbf{B} \cdot d\mathbf{S} = 0$

Applying Gauss’s divergence theorem, which relates the surface integral of a vector field to the volume integral of its divergence, we have:

$\oint_S \mathbf{B} \cdot d\mathbf{S} = \int_V (\nabla \cdot \mathbf{B}) \, dV = 0$

Since the surface integral vanishes for any closed surface (where the magnetic field lines neither start nor end), it follows that:

$\nabla \cdot \mathbf{B} = 0$

This equation represents Maxwell’s second equation. It indicates that the divergence of the magnetic field $\mathbf{B}$ is zero everywhere in space, implying that magnetic field lines neither have sources nor sinks (magnetic monopoles). Instead, they form continuous loops or extend to infinity, ensuring the conservation of magnetic flux.

Integral Form of Maxwell’s Second Equation

Differential form of the Maxwell's second equation is

$\mathbf{\nabla} \cdot \mathbf{B} = 0$

Exactly in a manner adopted for maxwell's first equation integral form, we can show that

$\oint_S \mathbf{B} \cdot d\mathbf{S} = 0$

which signifies that the total outward flux of magnetic induction $\mathbf{B}$ through any closed surface $S$ is equal to zero.

Maxwell Third Equation

Maxwell’s 3rd equation is derived from Faraday’s law of Electromagnetic Induction. It states that “Whenever there are n-turns of a conducting coil in a closed path placed in a time-varying magnetic field, an alternating electromotive force (emf) gets induced in each coil.” This is described by Lenz’s law, which states, “An induced electromotive force always opposes the time-varying magnetic flux.”

Consider two coils with N turns each, one being the primary coil connected to an alternating current (AC) source, and the other being the secondary coil connected in a closed loop placed near the primary coil. When an AC current passes through the primary coil, an alternating electromotive force gets induced in the secondary coil.

Faraday's law of induction states that the induced electromotive force (emf) around a closed loop is equal to the negative rate of change of the magnetic flux through the loop. It is given by:

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

where:

• $\mathbf{E}$ is the electric field.
• $\mathbf{B}$ is the magnetic field.
• $t$ is time.

Derivation

To derive Faraday's law of electromagnetic induction, we start with the concept of magnetic flux and apply the integral form of Maxwell's equations.

According to Faraday's law, the emf induced in a closed loop is given by

$$\mathcal{E}_{\text{emf}} = -\frac{\partial \Phi}{\partial t} = -\frac{\partial}{\partial t} \int_S \mathbf{B} \cdot \mathbf{dS}$$

Here the flux $\Phi = \int_S \mathbf{B} \cdot \mathbf{dS}$ where $S$ is any surface having the loop as boundary. The emf $(\mathcal{E}_{\text{emf}})$ can also be found by calculating the work done in carrying a unit charge completely around the loop. Thus,

$$\mathcal{E}_{\text{emf}} = \oint \mathbf{E} \cdot \mathbf{dl}$$

Here $\mathbf{E}$ is the intensity of the field associated with the induced emf. On equating the above two equations, we get

$$\oint \mathbf{E} \cdot \mathbf{dl} = - \int_S \frac{\partial B}{\partial t} \cdot \mathbf{dS}$$

According to Stokes' theorem, the line integral can be transformed into surface integral with the help of

$$\oint \mathbf{E} \cdot \mathbf{dl} = \int_S (\nabla \times \mathbf{E}) \cdot \mathbf{dS}$$

Therefore

$$\int_S (\nabla \times \mathbf{E}) \cdot \mathbf{dS} = -\int_S \frac{\partial \mathbf{B}}{\partial t} \cdot \mathbf{dS}$$

This equation must be true for any surface whether small or large in the field. So the two vectors in the integrands must be equal at every point, i.e.,

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

This is the Maxwell's third equation.

Integral Form of Maxwell's Third Equation

Differential form of the Maxwell's third equation is

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

On integrating above eq over an open surface $S$ bounded by a closed path, we have

$\int_S (\mathbf{\nabla} \times \mathbf{E}) \cdot d\mathbf{S} = -\int_S \frac{\partial \mathbf{B}}{\partial t} \cdot d\mathbf{S}$

Converting surface integral into a line integral by Stokes' theorem, we get

$\oint_l \mathbf{E} \cdot d\mathbf{l} = - \frac{\partial}{\partial t}\int_S B \cdot dS$

which signifies that the electromotive force around a closed path $l$ is equal to the time derivative of the magnetic flux through any surface bounded by that path.

Maxwell's Fourth Equation

It is based on Ampere’s circuit law. To understand Maxwell’s fourth equation, it is crucial to understand Ampere’s circuit law,

Consider a wire of a current-carrying conductor with the current I. Since there is an electric field, there has to be a magnetic field vector around it. Ampere’s circuit law states that “The closed line integral of magnetic field vector is always equal to the total amount of scalar electric field enclosed within the path of any shape”, which means the current flowing along the wire(which is a scalar quantity) is equal to the magnetic field vector (which is a vector quantity)

Ampère's law, with Maxwell's correction, states that the magnetic field around a closed loop is related to the electric current and the rate of change of the electric field through the loop. It is given by:

$$\nabla \times \mathbf{H} = \left( \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t} \right)$$

where:

• The $\nabla \times \mathbf{H}$ term represents the curl of the magnetic field $H$, which describes the rotational behavior of the magnetic field.
• $\mathbf{H}$ is the magnetic field.
• $\mathbf{J}$ is the conduction current density, representing the current due to the movement of charges (such as electrons) in a conductor.

Derivation

According to the Ampere's law, the work done in carrying a unit magnetic pole once around a closed arbitrary path linked with the current is expressed by

$\oint_l \mathbf{H} \cdot d\mathbf{l} = I$

or

$\oint_l \mathbf{H} \cdot d\mathbf{l} = \int_s \mathbf{J} \cdot d\mathbf{S}$

where $J$ is current density.

By Stokes' theorem, the line integral of $H$ around a closed loop $l$ can be expressed as:

$\oint_l \mathbf{H} \cdot d\mathbf{l} = \int_s (\nabla \times \mathbf{H}) \cdot d\mathbf{S}$

Therefore,

$\int_s (\nabla \times \mathbf{H}) \cdot d\mathbf{S} = \int_s \mathbf{J} \cdot d\mathbf{S}$

This gives

$\nabla \times \mathbf{H} = \mathbf{J}$

The above relation is derived on the basis of Ampere's law, which holds good only for the steady current. However, for the changing electric fields, the current density should be modified. The difficulty with the above equation is that, if we take divergence of this equation, then

$\nabla \cdot (\nabla \times \mathbf{H}) = \nabla \cdot \mathbf{J}$

$0 = \nabla \cdot \mathbf{J} \quad \text{[Since divergence of a curl = 0]}$

$\nabla \cdot \mathbf{J} = 0$

which conflicts with the continuity equation, as

$\nabla \cdot \mathbf{J} = -\frac{\partial \rho}{\partial t}$

Therefore, Maxwell realised that the definition of the current density is incomplete and suggested to add another density $\mathbf{J}^*$. Therefore

$\mathbf{J}^* = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t}$

Now, taking divergence of the above equation, we get

$\nabla \cdot (\nabla \times \mathbf{H}) = \nabla \cdot \mathbf{J} + \nabla \cdot \frac{\partial \mathbf{D}}{\partial t}$

or

$0 = \nabla \cdot \mathbf{J} + \frac{\partial}{\partial t} (\nabla \cdot \mathbf{D}) - \; \text{eq 1}$

Since,

$\rho = \nabla \cdot \mathbf{D}$

$\nabla \cdot \mathbf{J} = -\frac{\partial \rho}{\partial t} = -\frac{\partial}{\partial t} (\nabla \cdot \mathbf{D})$

Substitute $\nabla \cdot J$ into eq 1:

$0 = -\frac{\partial}{\partial t} (\nabla \cdot \mathbf{D}) + \frac{\partial}{\partial t} (\nabla \cdot \mathbf{D})$

This confirms that:

$\nabla \cdot (\mathbf{J} + \frac{\partial \mathbf{D}}{\partial t}) = 0$

Therefore, the total current density $J^*$ which includes both the real current density and the displacement current density, satisfies the continuity equation.

Finally, with the inclusion of the displacement current density, the modified Ampere's law is:

$\mathbf{\nabla} \times \mathbf{H} = \mathbf{J^*}$

$\nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t}$

The last term of R.H.S. of this equation is called Maxwell's correction and is known as displacement current density. The above equation is called modified Ampere's law for unsteady or changing current which is responsible for the electromagnetic fields.

This equation now accounts for both the conduction current (real current) and the displacement current (the changing electric field), providing a complete description of the magnetic fields generated by both static and dynamic electric fields.

Integral Form of Maxwell's Fouth Equation

Differential form of the Maxwell's fourth equation is

$\mathbf{\nabla} \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t}$

Exactly in a similar manner as in methods of other maxwell's equations integral form, we have this equation in the following form

$\oint_l \mathbf{H} \cdot d\mathbf{l} = \int_S(\mathbf{J} + \frac{\partial \mathbf{D}}{\partial t}) \cdot dS$

This signifies that the circulation of the magnetic induction vector around a closed path $l$ is a measure of the conduction current and displacement current through the open surface bounded by that path.