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# 450fall09hw2 - ECE 450 Homework 2 Due Tue Sept 8 2009 5 PM...

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ECE 450 Homework 2 Due: Tue, Sept 8, 2009, 5 PM 1. Consider the time-harmonic charge distribution ρ ( r , t ) = q [ δ ( r - h ˆ x ) - δ ( r + h ˆ x ) + δ ( r - h ˆ y ) - δ ( r + h ˆ y )] cos( ω t ) C m 3 in free space. a) Write down the phasors for the charge density ρ ( r , t ) and its time derivative ∂ρ t . b) Calculate the divergence · J of the current density associated with ρ ( r , t ) above. Hint: conti- nuity equation. c) What is the retarded scalar potential phasor ˜ V generated by ρ ( r , t ) above? 2. Give the time-harmonic expressions with the frequency ω for the fields A ( r , t ) with the following phasors: a) ˜ A ( r ) = j x - y b) ˜ A ( r ) = j ω ˆ y + 2 ω 2 ˆ z c) ˜ A ( r ) = ( j ˆ x + 2ˆ y ) e - jkz where k is a constant. 3. Consider the time-harmonic vector field A ( r , t ) = ˆ xA x ( r ) cos ( ω t + φ x ( r )) + ˆ yA y ( r ) cos ( ω t + φ y ( r )) + ˆ zA z ( r ) cos ( ω t + φ z ( r )) . a) Show that its complex valued phasor ˜ A ( r ) can be written in the form ˜ A ( r ) = A R ( r ) + j A I ( r ) , where A R ( r ) and A I ( r ) are real vectors and identify A R ( r ) and A I ( r ) .

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450fall09hw2 - ECE 450 Homework 2 Due Tue Sept 8 2009 5 PM...

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