Maxwell’s Equations

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# Maxwell’s Equations - PowerPoint PPT Presentation

Maxwell’s Equations. In the electric field E, and the magnetic field B , a charge q will experience a force: the Lorentz force:. Electromagnetic. F = q{E + v × B}. Static Charges produces E fields and Moving charges produces B fields. Maxwell’s Equations. Electromagnetic.

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### Maxwell’s Equations

In the electric field E, and the magnetic field B, a charge qwill experience a force: the Lorentz force:

Electromagnetic

F = q{E + v × B}.

Static Charges produces E fields and Moving charges produces B fields

Maxwell’s Equations

Electromagnetic

The effects may be summarized in the expressions for the divergence and the curl of E and B:

divE = /,

curlE = 0 ,

divB = 0 ,

curlB = µ0J

Maxwell’s Equations

Electromagnetic

Equations without divergence and curl express passive aspects, while with curl and divergence express active aspects.

A field with a curl but no divergence is

called a solenoidalfield, while one with a divergence but no curl is called an irrotationalfield.

Electrostatic Field

Equipotentials and Electric Field Vectors of

Electrostatic Field.

Electric Field Vectors

Equipotentials and Electric Field Vectors of aMicrostrip Line.

Potential Distribution

Potential Distribution associated with a Corner

Resistor.

Electric Field Magnitude

Logarithmic scaled Electric Field Magnitude

Electrodynamics

A Charged Particle

If a charged particle is set free in an electric field, it is accelerated by a force proportional to the field and charged particle

F = eE

Where F is Force

e is a charge, and

E is electric Field Intensity

Electrodynamics

Newton’s Second Law

d(mv)

dv

dm

F =

= m

+ v

dt

dt

dt

Where m = mass of particle, kg

V = velocity of particle, m-1

Electrodynamics

Newton’s Second Law

F = m

dv

= ma

dt

ma = eE

• Velocity is very small as compared to velocity of light
• Mass is essentially constant
Electrodynamics

Energy

Integrating the force over the distance traveled by charged particle is

2

2

W = m  a •dL = e E • dL

1

1

While the Integral of E between points of 1 and 2 is a potential difference V

2

W = m v •dv = eV

1

W = ½ m( v22 – v12) = eV

Electrodynamics

Particle Energy

W = eV = ½ mv2

where

W = energy acquired by particle, J

v2 = velocity of particle at point 2, or final velocity, ms-1

V1 = velocity of particle at point 1, or initial velocity, ms-1

e = charge on particle, C

m = mass of particle, kg

V = magnitude of potential difference between points 1 & 2, V

Electrodynamics

Final velocity

Considering a charged particle e starting from rest and passing through a potential of V, willattain the final velocity of :-

 =  2eV/m

Electrodynamics

Final velocity

While

e = 1.6 x 10-19C falling through

V = 1 volt

Energy = 1.6 x 10-19 Joules

m = mass of 0.91 x 10-30kg, will attain Velocity = v = 5.9 x 105 V

at 1 volt the charge attains 590 kms-1

Electrodynamics

ay =

eVd

eVdL

vy

; vy = ayt =

;  = tan-1

vx

md

mvxd

L

Vd

y

vy

v

Ed

vx

d

Electrodynamics

Problem:-

A CRT with Va = 1500V,

Deflecting space d = 10mm,

Deflecting plate length = 10mm,

Distance x = 300mm,

Find Vd to deflection of 10mm:-

Electrodynamics

Moving particle in static magnetic field

Force on a current element dL in a magnetic field is given by:

dF = (I x B)dL (N) …Motor equation

I = q/t

IL = qL/t = qv

IdL = dqv

dF = dq(v x B)

F = e(v x B) Lorentz force

Electrodynamics

Moving conductor in a magnetic field

E = F/e = v x B

V12 =  E • dL =  (v x B) • dL

2

2

1

1

1

Generating Equation

B

dL

v

2

E = v x B

Electrodynamics

Magnetic Brake

Electrodynamics

Magnetic Brake

I, B, & PUSH

Therefore F due to I is opposing to PUSH

Conductive Plate

Magnet Assembly

Electrodynamics

Magnetic Levitation

How does the LEVITRON¨ work?

When the top is spinning, the torque acts gyroscopically and the axis does not overturn but rotates about the (nearly vertical) direction of the magnetic field.

Electrodynamics

levitation

Electrodynamics

levitation

"We may perhaps learn to deprive large masses of their gravity and give them absolute levity, for the sake of easy transport."

- Benjamin Franklin

Electrodynamics

Maglev Trains

Electrodynamics

Maglev Train

A maglev train floats about 10mm above the guidway on a magnetic field. It is propelled by the guidway itself rather than an onboard engine by changing magnetic fields (see right). Once the train is pulled into the next section the magnetism switches so that the train is pulled on again. The Electro-magnets run the length of the guideway

Electrodynamics

Maglev Train Track

Maglev Train

Aerodynamics Brakes

Electrodynamics