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

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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 = µ 0 J

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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 solenoidal field, while one with a divergence but no curl is called an irrotational field.

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Electrostatic Field Potential Distribution.

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Electrostatic Field Equipotentials and Electric Field Vectors of Electrostatic Field.

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Electric Field Vectors Equipotentials and Electric Field Vectors of a Microstrip Line.

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Potential Distribution Potential Distribution associated with a Corner Resistor.

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Electric Field Magnitude Logarithmic scaled Electric Field Magnitude

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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

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Electrodynamics Newton’s Second Law F = d(mv) dt = m dv dt + v dt dm Where m = mass of particle, kg V = velocity of particle, m -1

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Electrodynamics Newton’s Second Law F = m dv dt = ma Velocity is very small as compared to velocity of light Mass is essentially constant ma = eE

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Electrodynamics Energy W = m a dL = e E dL Integrating the force over the distance traveled by charged particle is While the Integral of E between points of 1 and 2 is a potential difference V W = m v dv = eV 1 2 W = ½ m( v 2 2 – v 1 2 ) = eV

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Electrodynamics Particle Energy W = eV = ½ mv 2 where W = energy acquired by particle, J v 2 = velocity of particle at point 2, or final velocity, ms -1 V 1 = 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

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Electrodynamics Final velocity Considering a charged particle e starting from rest and passing through a potential of V, will attain the final velocity of :- = 2eV/m

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Electrodynamics Final velocity While e = 1.6 x C falling through V = 1 volt Energy = 1.6 x Joules m = mass of 0.91 x kg, will attain Velocity = v = 5.9 x 10 5 V at 1 volt the charge attains 590 kms -1

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Electrodynamics a y = eV d md y v vxvx vyvy L d EdEd VdVd ; v y = a y t = eV d L mv x d ; = tan -1 vyvy vxvx

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Electrodynamics Problem:- A CRT with V a = 1500V, Deflecting space d = 10mm, Deflecting plate length = 10mm, Distance x = 300mm, Find V d to deflection of 10mm:- Deflection y = V d Lx/2V a d V d = 2V a dy/Lx= 100 V

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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

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Electrodynamics Moving conductor in a magnetic field E = F/e = v x B V 12 = E dL = (v x B) dL B v E = v x B dLdL Generating Equation

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Electrodynamics Magnetic Brake

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Electrodynamics Magnetic Brake Magnet Assembly Conductive Plate I, B, & PUSH Therefore F due to I is opposing to PUSH

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Electrodynamics Magnetic Levitation

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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.

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How does the LEVITRON¨ work? levitionta

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Electrodynamics levitation

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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

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Electrodynamics Maglev Trains

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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

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Electrodynamics Maglev Train Track

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Maglev Train Aerodynamics Brakes

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Electrodynamics Advantages: no components that would wear out there is no friction. Note that there will still be air resistance. less noise The final advantage is speed

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