# HW84_2 - V ? Problem 4 In free space, a constant volume...

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Electromagnetics School of Electrical and Computer Engineering University of Tehran Homework 2 Fall Semester 1384 Due 84/8/2 Problem 1 Consider a known irrotational vector field ) ( r F r r . Using ) ( r F r r , we define the scalar field () z y x , , Φ as: = Φ ) , , ( ) 0 , 0 , 0 ( ) ( ) , , ( z y x l d r F z y x r r r a) Determine the gradient of Φ . b) Interpret the obtained result in (a). Problem 2 Inside a volume V , there is a known time-invariant volume charge distribution ) ( r r ρ . Assume the permittivity of free space, that is o ε , for all the points of V . The electric potential Φ on V is kept at a constant value. Is the static electric field ) ( r E r r generated by ) ( r r ρ within V unique? Is the corresponding electric potential Φ unique? Problem 3 Calculate the Laplacian of the function ) ( r r Φ defined by the following convolution integral: = Φ V r r v d r f r r r r r ) ' ( ) ( ) ( r f r is a known scalar field. Does your result depend on the volume

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Unformatted text preview: V ? Problem 4 In free space, a constant volume charge of 1 3 / m C has filled the region m r 1 &lt; of the spherical coordinate system. A uniform line charge of 2 m C / is located along the x-axis in the interval m x m 3 2 &lt; &lt; . Evaluate the force on the line charge. Problem 5 As we know, the vector magnetic potential ) ( r A r r in free space is related to ) ( r J r r according to: ∫ ′ − π ′ ′ µ = V o r r v d r J r A r r r r r r 4 ) ( ) ( Show that ) ( r A r r is solenoidal. Note that for a time-invariant current density, ) ( r J r r is divergence-less. Problem 6 Prove that ∫ ∫ ∂ × − = × ∇ V V S d F dv F r r r r . M. Shahabadi...
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## This note was uploaded on 02/06/2011 for the course ECE 423 taught by Professor Dolatabadi during the Spring '11 term at Amirkabir University of Technology.

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HW84_2 - V ? Problem 4 In free space, a constant volume...

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