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7 Apr 2008 13:56:32 IST

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MAxwell eq!!!

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1.What is the use of MAxwell's equations?

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Manasi

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7 Apr 2008 19:46:51 IST

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The first Maxwell's Equation is known as Gauss's Law, and it (roughly) states that the electric field multiplied times the a surface around it is equal to the charge enclosed by the surface, divided by a constant (called the permitivity of free space). Now, this first equation does not actually link electricity with magnetism, it's just an electric-only equation.

The second is law is known as the "no magetic monopoles" law by many, or Gauss's Law for Magnetism by others. Nor does this one link electricity and magnetism, but rather just says that all magnetic fields need to have both a source and sink. (Unlike electrical fields, that are quite happy having only a source, and no sink.) You could compare electrical fields to a lightbulb, which just shoots light off in all directions, and it could care not a whit that the light doesn't come back to it. However magnetic fields are more like fountains in a park ... they shoot out water, but the water has to fall back down the drain and get recirculated to shoot out again.

The Third Law is where the good stuff starts ... this one is known as Faraday's Law, and this one binds electricity with magnetism. Specifically, it says that if you move a magentic field through a loop of wire, a current (i.e. electromagnetic force, or voltage) will be created in that wire, as long as you keep moving the magnetic field thorugh the loop. This law is the basis of all electric generators and electric motors; I can use moving magnets to make electricity, or I can use electricity to make magnets move.

The last Maxwell's Equation is Ampere's Law, which Maxwell modified with an additional term and thus secured his position in the history of physics by making it compatible with a future that didn't even exist at the time ... Relativity.

With just a simple line integration, Ampere's Law connects the magnetic field of a current-carying wire with the electric current in the wire. It's what allows your television coaxial cable to work, along with solonoids, rail guns, particle accelerators, etc..

The amazing thing about Maxwell's Laws is that they can all be verified, played with, enjoyed, and studied with just a battery, some wire, a toy compass and a mulitmeter

edison

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7 Apr 2008 19:51:00 IST

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The answer lies in explaining the significance and interpretation of Maxwell's equations as follows

1) Charge density and the electric field

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

where

?

is the free electric charge density (in units of C/m³), not including dipole charges bound in a material, and $\mathbf{D}$ is the electric displacement field (in units of C/m²). This equation corresponds to Coulomb's law for stationary charges in vacuum.

The equivalent integral form (by the divergence theorem), also known as Gauss' law, is:

$\oint_A \mathbf{D} \cdot d\mathbf{A} = Q_\mbox{enclosed}$

where $d\mathbf{A}$ is the area of a differential square on the closed surface A with an outward facing surface normal defining its direction, and

Q enclosed

is the free charge enclosed by the surface.

Any material can be treated as linear, as long as the electric field is not extremely strong. The permittivity of free space is referred to as

? 0

, and appears in:

$\nabla \cdot \mathbf{E} = \frac{\rho_t}{\varepsilon_0}$

2) The structure of the magnetic field

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

$\mathbf{B}$ is the magnetic flux density (in units of teslas, T), also called the magnetic induction.

Equivalent integral form:

$\oint_A \mathbf{B} \cdot d\mathbf{A} = 0$

$d\mathbf{A}$ is the area of a differential square on the surface

A

with an outward facing surface normal defining its direction.

This equation only works if the integral is done over a closed surface. This equation says, that in every volume the sum of the incoming magnetic field lines equuals the sum of the outgoing magnetical field lines. This means that the magnetic field lines must be closed loops. Another way of putting it is that the field lines cannot originate from somewhere. Mathematically formulated: "There are no magnetic monopoles".

3) A changing magnetic flux and the electric field

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

Equivalent integral Form:

$\oint_{s} \mathbf{E} \cdot d\mathbf{s} - \oint_s \mathbf{B} \times \mathbf {v} \cdot d \mathbf{l} = - \frac {d\Phi_{\mathbf{B}}} {dt}$ where $\Phi_{\mathbf{B}} = \int_{A} \mathbf{B} \cdot d\mathbf{A}$

where

?_B is the magnetic flux through the area A described by the second equation

E is the electric field generated by the magnetic flux

s is a closed path in which current is induced, such as a wire

v is the instantaneous velocity of the line element (for moving circuits).

This law corresponds to the Faraday's law of electromagnetic induction.

This equation relates the electric and magnetic fields.

This further signifies that space varying electric field gives rise to time varying magnetic field and vice versa.

4) The source of the magnetic field

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

where H is the magnetic field strength (in units of A/m), related to the magnetic flux B by a constant called the permeability, ? (B = ?H), and J is the current density, defined by: J = ??_qvdV where v is a vector field called the drift velocity that describes the velocities of that charge carriers which have a density described by the scalar function ?_q.

In free space, the permeability ? is the permeability of free space, ?₀, which is defined to be exactly 4?×10^-7 W/A·m. Also, the permittivity becomes the permittivity of free space ?₀. Thus, in free space, the equation becomes:

$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0\varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}$

Which shows that space varying magnetic field gives rise to time varying electric field and vice versa.

edison

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7 Apr 2008 19:53:00 IST

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In short Maxwell's equations explain the exsistence and propagation of electromagnetic waves in space and medium

"Electromagnetic radiation is a self-propagating wave in space with electric and magnetic components. An oscillating electric field generates an oscillating magnetic field, and a magnetic field in turn generates an oscillating electric field, and so on. These oscillating fields together form an electromagnetic wave."

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