ECE 303: Electromagnetic Fields and Waves Fall 2007 Solutions for Prelim 1 Reminder: Date: Tuesday 9/25 Problem 1: A current loop above a perfect conductor. a. The image current is a loop of radius a , located at z =-d , and of value-I ( t ) (i.e., opposite direction). b. The z component of the magnetic ﬁeld due to the original loop is H z ( partial ) = I ( t ) 4 π ± 2 πa a 2 + ( L-d ) 2 ¶ sin α = I ( t ) 2 ˆ a 2 [ a 2 + ( L-d ) 2 ] 3 / 2 ! Then the total ﬁeld is the superposition of the ﬁelds produced by each current loop: H z = a 2 I ( t ) 2 ˆ 1 [ a 2 + ( L-d ) 2 ] 3 / 2-1 [ a 2 + ( L + d ) 2 ] 3 / 2 ! c. By symmetry, the x component of the ﬁeld is zero. Problem 2: Finding the electric ﬁeld with a volume charge and dielectric material. a. The boundary condition at x = d is ε0 E x ( x = d + ) = ε 1 E x ( x = d-) where E x ( x = d + ) = E0 . So, E x ( x = d-) = ε0 ε 1 E0 . The diﬀerential form of Gauss’ Law in one dimension is ∂ ∂x ( ε 1 E x ( x )) = ρ ( x ) for 0
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