20092ee1_1_HW1_Solutions

20092ee1_1_HW1_Solutions - EE 1 - Homework #1 Solutions (1)...

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EE 1 - Homework #1 Solutions (1) R π 0 R 2 π 0 ˆ R dφdθ =? (R, θ , and φ define the spherical coordinate system) Physically, the above integral is summing the unit vector ˆ R over the entire sphere. If we inspect the figure below, one can see that for every vector, there is exactly one other vector in the exact opposite direction with the same magnitude ( | ˆ R | = 1). Therefore, every vector will have one other that will cancel it out. The sum total would be equal to zero: Z π 0 Z 2 π 0 ˆ R dφdθ = 0 . Mathematically, the same can be obtained by revealing the dependence of ˆ R on θ and φ . ˆ R = ˆ rsinθ + ˆ zcosθ = ˆ xcosφsinθ + ˆ ysinφsinθ + ˆ zcosθ The integral would then be: Z π 0 Z 2 π 0 ˆ R dφdθ = ˆ x Z π 0 Z 2 π 0 cosφsinθ dφdθ y Z π 0 Z 2 π 0 sinφsinθ dφdθ z Z π 0 Z 2 π 0 cosθ dφdθ = - ˆ xsinφ | 2 π 0 cosθ | π 0 + ˆ ycosφ | 2 π 0 cosθ | π 0 + ˆ zsinθ | π 0 = ˆ x (0 - 0)(2) + ˆ y (1 - 1)( - 2) + ˆ z (0 - 0) = 0 . (2) Draw the vectors: 3
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This note was uploaded on 06/06/2009 for the course EE EE 1 taught by Professor Ozcan during the Spring '09 term at UCLA.

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20092ee1_1_HW1_Solutions - EE 1 - Homework #1 Solutions (1)...

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