329fall09hw3sol

# 329fall09hw3sol - ECE 329 Fall 2009 Homework 3 Solution 1 a...

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Unformatted text preview: ECE 329 Fall 2009 Homework 3 Solution September 12, 2009 1. a) We can write E as E = Q 4 π ˆ x x r 3 + ˆ y y r 3 + ˆ z z r 3 . Therefore, ∇ × E = Q 4 π ˆ x ˆ y ˆ z ∂ ∂x ∂ ∂y ∂ ∂z x r 3 y r 3 z r 3 = Q 4 π ˆ x ∂ ∂y z r 3- ∂ ∂z y r 3 + ˆ y ∂ ∂z x r 3- ∂ ∂x z r 3 + ˆ z ∂ ∂x y r 3- ∂ ∂y x r 3 . Take the calculation of ∂ ∂y ( z r 3 ) for example: ∂ ∂y z r 3 = ∂ ∂y z p x 2 + y 2 + z 2 3 =- 3 zy p x 2 + y 2 + z 2 5 . Similarly, we can obtain the other partial di erentials. Finally, ∇ × E = ˆ x Q 4 π - 3 zy p x 2 + y 2 + z 2 5 + 3 yz p x 2 + y 2 + z 2 5 +ˆ y Q 4 π - 3 xz p x 2 + y 2 + z 2 5 + 3 zx p x 2 + y 2 + z 2 5 +ˆ z Q 4 π - 3 yx p x 2 + y 2 + z 2 5 + 3 xy p x 2 + y 2 + z 2 5 = 0 . b) In this spherically symmetric situation, ∇ V = ∂V ∂r ˆ r....
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