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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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Presentation Transcript

### 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

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

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.

Equipotentials and Electric Field Vectors of

Electrostatic Field.

Equipotentials and Electric Field Vectors of aMicrostrip Line.

Potential Distribution associated with a Corner

Resistor.

Logarithmic scaled Electric Field Magnitude

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

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

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

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

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

Final velocity

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

 =  2eV/m

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

ay =

eVd

eVdL

vy

; vy = ayt =

;  = tan-1

vx

md

mvxd

L

Vd

y

vy

v

Ed

vx

d

Problem:-

A CRT with Va = 1500V,

Deflecting space d = 10mm,

Deflecting plate length = 10mm,

Distance x = 300mm,

Find Vd to deflection of 10mm:-

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

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

Magnetic Brake

Magnetic Brake

I, B, & PUSH

Therefore F due to I is opposing to PUSH

Conductive Plate

Magnet Assembly

Magnetic Levitation

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.

levitionta

levitation

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

Maglev Trains

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

Maglev Train Track

Aerodynamics Brakes