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MAXWELL’S

EQUATIONS

&

THEIR

APPLICATIONS

BY,

09BEC035

09BEC036GUIDE : PROF. MEHUL NAIK 1

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INDEX• BASIC CONCEPTS

• MAXWELL’S EQUATIONS

• ELECTROMAGNETIC WAVES

• SPEED OF ELECROMAGNETIC

WAVES

• APPLICATIONS OF MAXWELL’S

EQUATION

GUIDE : PROF. MEHUL NAIK2

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BASIC CONCEPTS

(1) There are two kinds of charges. These have been

labeled positive charge and negative charge.(2) Electrical charge is quantized.

(3) Equality of the positive and the negative charge quantum:

(4) In any closed system charge is conserved.

(5) Charges generate electric and magnetic fields.


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(6) The electromagnetic forces on a charged particle, q, can be obtained from a knowledge of the

fields E, B generated at the position of q by all other charges. The force in Newton is given by

F = q[E + ( V*B )]

(7) Superposition:

E=E1+E2 ; B=B1+B2.

(8) A Stationary Charged Particle(q):E=k q(R/{|R|^3} ); B=0.


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(9) The Fields generated by a Moving Charged Particle(q):

E=[k q{R/(|R|^3) }]; B=[{V*E(c^2)}].


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6 GUIDE : PROF. MEHUL NAIK

Maxwell’s Equations

• We have been examining a variety of

electrical and magnetic phenomena • James Clerk Maxwell summarized all of

electricity and magnetism in just four equations

• Remarkably, the equations predict the existence of electromagnetic waves

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• The first is Gauss’s Law which is an

extended form of Coulomb’s Law • The second is the equivalent for magnetic

fields, except that we know that magnetic poles always occur in pairs (north & south)

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• The third is Faraday’s Law that a changing

magnetic field produces an electric field • The fourth is that a changing electric field

produces a magnetic field • The latter is a bit of a stretch. We knew that a current produces a magnetic field

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http://slidepdf.com/reader/full/new-fadu 10/2510 GUIDE : PROF. MEHUL NAIK

• Start with Ampere’s Law

B||Δl = μ 0 I ∑ Earlier, we just went on a closed path

enclosing surface 1. But according to

Ampere’s Law, we could have considered surface 2. The current enclosed is the same as for surface 1.We can say that the current flowing into any volume must equal that coming out.

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• Suppose we have a charged capacitor and it

begins to discharge

Surface 1 works but

surface 2 has no current passing through the surface

yet there is a magnetic field inside the

surface.

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Same problem here. Surface 1 works, but no current passes through surface two which encloses a magnetic field.What is happening???


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• While the capacitor is discharging, a

current flows • The electric field between the plates of the

capacitor is decreasing as current flows • Maxwell said the changing electric field is

equivalent to a current • He called it the displacement current

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( )

( )

∑

∑

Δ

ΔΦ+=Δ

Δ

ΔΦ

=

=Φ

=

Δ

Δ=

Δ

Δ

=⎟ ⎠

⎞

⎜⎝

⎛ ==

+=Δ

t

I l B

t I

AE

I

t

E A

t

Q

AE Ed d

A

CV Q

I I l B

E C

E

D

E

D

DC

000||

0

0

00

0||

ε μ μ

ε

ε

ε ε

μ


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Electromagnetic Waves

• So, a magnetic field will be produced in space if there is a changing electric field

• But, this magnetic field is changing since the

electric field is changing • A changing magnetic field produces an

electric field that is also changing • We have a self-perpetuating system

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Close switch and current flows briefly. Sets

up electric field. Current flow sets up magnetic field as little circles around the wires. Fields not instantaneous, but form

in time. Energy is stored in fields and cannot move infinitely fast.


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Picture a shows first half cycle. When current reverses in picture b, the fields reverse. See the first disturbance moving outward. These are the electromagnetic waves.


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Notice that the electric and

magnetic fields are at right angles to one another! They are also perpendicular to the direction of motion of the wave.


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Speed of EM Waves

• Now that we have shown how the waves are formed from oscillating charges, we need to see if we can predict how fast the move

•

We move far away from the source so that the wave fronts are essentially flat • Just like dropping a rock in a pond and looking at the waves a few hundred feet away

from the impact point

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This picture defines the coordinate system we will

use in our discussion. Wave propagates along the x-axis. The electric field varies in the y-direction and the magnetic field in the z-direction .


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We are going to apply Faraday’s Law to the imaginary moving rectangle abcd. Compute the magnetic flux

change v By

t

t v By

t

A B

t emf B

00 =Δ

Δ=

Δ

Δ=

Δ

ΔΦ=


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• We can say the emf around the loop is the

sum of the individual emfs going along each straight line segment in the loop • We look at the work done in moving a test

charge around the loop • emf = W/q = Fd/q = Ed

• emf = Ey0 = By0v• E = Bv

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Now we are going to look at the change in electric flux. Set a new imaginary rectangle and

play the same game as before.

v Ez

t

t v Ez

t

l B E 000

00000||

ε μ ε μ ε μ =

Δ

Δ=

Δ

ΔΦ=Δ∑


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Bz0 = μ 0ε 0 Ez0v

B = μ 0ε 0 Ev

E = Bv B = μ 0ε 0( Bv)v

1= μ 0ε 0v2

v =1

μ 0ε 0

v = 1

8.85 ×10−12 × 4π ×10−7

v = 3×108


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THANK YOU

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