MAXWELL'S EQUATION

 

Introduction

Maxwell’s equations are the foundation of electromagnetism. They explain how electric and magnetic fields are produced and how they interact.

They combine:

• Electrostatics
• Magnetism
• Electromagnetic induction
• Electric currents

Derivation of Maxwell’s First Equation

Gauss’s Law for Electricity

The total electric flux through a closed surface is equal to the charge enclosed divided by permittivity.

Step 1: Start with Coulomb’s Law

The electric field due to a point charge is:

\[ \vec{E} = \frac{1}{4\pi \varepsilon_0} \frac{q}{r^2}\hat{r} \]

• \(q\) → Point charge
• \(r\) → Distance from charge
• \(\varepsilon_0\) → Permittivity of free space

Step 2: Consider a Gaussian Surface

Consider a spherical surface of radius \(r\) enclosing the charge \(q\).

Since the electric field is symmetrical, the magnitude of electric field remains constant over the surface.

\[ E = \frac{1}{4\pi \varepsilon_0} \frac{q}{r^2} \]

Step 3: Electric Flux Through the Surface

Electric flux is defined as:

\[ \Phi_E = \oint \vec{E}\cdot d\vec{A} \]

Since \(\vec{E}\) and \(d\vec{A}\) are parallel:

\[ \Phi_E = E \oint dA \]

Surface area of sphere:

\[ \oint dA = 4\pi r^2 \]

Step 4: Substitute the Values

\[ \Phi_E = \left( \frac{1}{4\pi \varepsilon_0} \frac{q}{r^2} \right) (4\pi r^2) \]

Simplifying:

\[ \Phi_E = \frac{q}{\varepsilon_0} \]

Maxwell’s First Equation

\[ \boxed{ \oint \vec{E}\cdot d\vec{A} = \frac{Q_{enc}}{\varepsilon_0} } \]

The total electric flux passing through a closed surface equals the enclosed electric charge divided by permittivity of free space.

Derivation of Maxwell’s Second Equation

Understand Magnetic Field Lines

Magnetic field lines always form closed loops.

• Field lines emerge from the north pole
• Field lines enter the south pole
• Inside the magnet, they return back to the north pole

Unlike electric charges, isolated magnetic monopoles do not exist.

Consider a Closed Surface

Imagine a closed surface surrounding a magnet.

The magnetic field lines entering the surface are equal to the magnetic field lines leaving the surface.

\[ \text{Magnetic flux entering} = \text{Magnetic flux leaving} \]

Therefore, the net magnetic flux through the closed surface becomes zero.

Magnetic Flux

Magnetic flux through a surface is given by:

\[ \Phi_B = \oint \vec{B}\cdot d\vec{A} \]

• \(\vec{B}\) → Magnetic field
• \(d\vec{A}\) → Small area vector

Net Magnetic Flux

Since magnetic field lines form continuous closed loops, the total magnetic flux through any closed surface is always zero.

\[ \Phi_B = 0 \]

Maxwell’s Second Equation

\[ \boxed{ \oint \vec{B}\cdot d\vec{A} = 0 } \]

This equation states that the total magnetic flux through a closed surface is always zero because magnetic monopoles do not exist.

Derivation of Maxwell’s Third Equation

Faraday’s Law of Electromagnetic Induction

Electromagnetic Induction

Michael Faraday discovered that whenever magnetic flux linked with a conductor changes, an emf is induced in the conductor.

• Changing magnetic field produces electric field
• This phenomenon is called electromagnetic induction
• The induced current opposes the change producing it

Magnetic Flux

Magnetic flux through a surface is:

\[ \Phi_B = \int \vec{B}\cdot d\vec{A} \]

• \(\Phi_B\) → Magnetic flux
• \(\vec{B}\) → Magnetic field
• \(d\vec{A}\) → Area vector

Faraday’s Experimental Observation

According to Faraday’s experiment:

\[ \mathcal{E} = -\frac{d\Phi_B}{dt} \]

where:

• \(\mathcal{E}\) → Induced emf
• \(\frac{d\Phi_B}{dt}\) → Rate of change of magnetic flux

The negative sign represents Lenz’s Law.

Relation Between emf and Electric Field

The induced emf around a closed loop is:

\[ \mathcal{E} = \oint \vec{E}\cdot d\vec{l} \]

• \(\vec{E}\) → Electric field
• \(d\vec{l}\) → Small length element of loop

Substitute emf Equation

Substituting Faraday’s emf equation:

\[ \oint \vec{E}\cdot d\vec{l} = -\frac{d\Phi_B}{dt} \]

Maxwell’s Third Equation

\[ \boxed{ \oint \vec{E}\cdot d\vec{l} = -\frac{d\Phi_B}{dt} } \]

This equation states that a changing magnetic field produces an electric field.

Derivation of Maxwell’s Fourth Equation

Ampere–Maxwell Law

Start with Ampere’s Circuital Law

Ampere discovered that electric current produces a magnetic field.

\[ \oint \vec{B}\cdot d\vec{l} = \mu_0 I \]

• \(\vec{B}\) → Magnetic field
• \(d\vec{l}\) → Small length element
• \(I\) → Current enclosed
• \(\mu_0\) → Permeability of free space

Problem with Charging Capacitor

Consider a charging capacitor.

• Current flows through the wire
• No physical current flows between capacitor plates
• But magnetic field still exists between the plates

Ampere’s original law could not explain this situation completely.

Maxwell’s Correction

James Clerk Maxwell introduced the concept of displacement current.

A changing electric field also produces a magnetic field.

\[ I_d = \varepsilon_0 \frac{d\Phi_E}{dt} \]

• \(I_d\) → Displacement current
• \(\Phi_E\) → Electric flux

Total Current

Total current becomes:

\[ I_{total} = I + I_d \]

Substituting displacement current:

\[ I_{total} = I + \varepsilon_0 \frac{d\Phi_E}{dt} \]

Modified Ampere’s Law

Substituting total current into Ampere’s law:

\[ \oint \vec{B}\cdot d\vec{l} = \mu_0 \left( I + \varepsilon_0 \frac{d\Phi_E}{dt} \right) \]

Maxwell’s Fourth Equation

\[ \boxed{ \oint \vec{B}\cdot d\vec{l} = \mu_0 \left( I + \varepsilon_0 \frac{d\Phi_E}{dt} \right) } \]

This equation states that magnetic fields are produced by both electric current and changing electric fields.