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20092ee1_1_HW2_Solutions

# 20092ee1_1_HW2_Solutions - EE 1 Homework#2 Solutions Spring...

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EE 1 - Homework #2 Solutions Spring 2009 (1) (1pt each) 1

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(2) (10 points) The gradient of a scalar function is given by: V = ˆ ze - 2 z If V = 20 V at z = 0, find V ( z )? The gradient of V is essentially the derivative of V . Integrating the gradient will result in the general form of V ( z ): V 2 - V 1 = Z P 2 P 1 V · dl = Z P 2 P 1 ˆ ze - 2 z · xdx + ˆ ydy + ˆ zdz ) Integrating, and choosing P 1 = 0 and P 2 = z the general result is: V ( z ) - V (0) = Z z 0 e - 2 z dz = - 1 2 e - 2 z | z 0 V ( z ) = 1 - e - 2 z 2 + V (0) = 41 - e - 2 z 2 ( V ) 2
(3) (15 Points) For the vector field: ~ E = ˆ r 5 e - r - ˆ z 6 z verify the divergence theorem for the cylindrical region enclosed by r = 2, z = 0, and z = 4. To verify the divergence theorem, we merely have to show that the surface integral of the vector field is equal to the volume integral of the divergence of the same vector field: Z S ~ E · d ~ S = Z V ∇ · ~ EdV First, we set-up the surface integral. For a cylinder, there are three surfaces to worry about, the top and bottom ( S 1 and S 2, respectively) of the cylinder and the side surface of the cylinder ( S 3).

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