Module 3a - 2/9/2009 Module 3 Lecture 1 From Maxwells...

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2/9/2009 1 Module 3 Lecture 1 From Maxwell’s equations to electromagnetic waves and light Page 2 Topics for today Calculation of displacement current (example) Vectorial phasors Coordinate systems Calculations of E from H and vice versa Derivation of a partial differential equation for E(x,y,z,t) in infinite linear, isotropic space The particular solutions when the time dependence is harmonic.
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2/9/2009 2 Page 3 Displacement current calculation A parallel plate capacitor with plate area S = 5 cm 2 and separation d=3 mm has a voltage of 50 sin(1000t) volts applied to its plates. Calculate the displacement current assuming e = 2 e 0 Solution: D = e E = e V/d then J D = ∂D/∂t = ( e /d) dV/dt So the total displacement current between the plates will be given by: I D = J D S = ( e S/d) dV/dt (by the way, this is equal to CdV/dt where C is the capacitance of this capacitor. Answer: I D = ( e S/d) dV/dt = (2X10 -9 /36 p )(5X10 -4 /3X10 -3 ) 50X10 3 cos10 3 t ...= 147.4 cos10 3 t nA Trick: we used e 0 = 10 -9 /36 p (useful approximate value) Page 4 Scalar vs Vectorial phasors A scalar harmonic wave is characterized by the following time and space dependence (for propagation along direction x) A(x,y,z,t) = A 0 cos( w t- b x+ f ) (sinus function is included because sin(x) = cos(x- p /2)) For propagation an arbitrary direction r , this becomes: A(x,y,z,t) = A 0 cos( w t- (k r) + f ) Where k = (k x ,k y ,k z ) and r = (x,y,z) So (k r) = k x x + k y y +k z z
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2/9/2009 3 Page 5 Finally for vector waves A vector wave has three components (along the 3 coordinate axes), A x , A y , A z When all waves (and consequently all their components) have the same oscillating frequency f, we can use phasors for each of the components. E.g.: B = 20 sin( w t) a x + 10 cos( w t+2 p x/3) a y The corresponding phasor is also vectorial: B s = (20/j) a x +10e j2 p x/3 a y = B sx a x + B sy a y Page 6 How did we do the last part? B
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Module 3a - 2/9/2009 Module 3 Lecture 1 From Maxwells...

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