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EE1 HW 1 Sol

# EE1 HW 1 Sol - EE 1 Homework#1 Solutions(1 2 0 0 ^ R d d...

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EE 1 - Homework #1 Solutions (1) π 0 2 π 0 ˆ R dφ dθ =? (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: π 0 2 π 0 ˆ R dφ dθ = 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: π 0 2 π 0 ˆ R dφ dθ = ˆ x π 0 2 π 0 cosφsinθ dφ dθ y π 0 2 π 0 sinφsinθ dφ dθ z π 0 2 π 0 cosθ dφ dθ = - 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 ˆ R + 3 ˆ θ + 3 ˆ φ and 3ˆ r + 3 ˆ φ + 3ˆ z and 3ˆ x + 3ˆ y + 3ˆ z .

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EE1 HW 1 Sol - EE 1 Homework#1 Solutions(1 2 0 0 ^ R d d...

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