EE140 Fall 2014 Mid-Term 1 Dr. Ray Kwok Name: (Last, First) _____________________________________________________________ 1) A line charge of length 10 m along the z-axis (from z = -5 to 5 m) has a linear charge density of λ= 2|z|. What is the electric potential and electric field intensity at (1, 0, 0) m ? dq = λdz x z r -5 5 θ [ ] [ ][ ]xxkxkEkkVxxxkxxxkxVxxExxkxzkxzdzzkxVrdzkrkdqdVˆ9.28ˆ22.3ˆ125114)1(1484.1611254)1(ˆ2514ˆ1254ˆ)(2544)2(2)(2222250225022==+−===−+=+−=−+−=∂∂−=−+=+=+===∫rλGV GV/mEE140 Fall 2014 Mid-Term 1 Dr. Ray Kwok Infinite ground plane 45o wire with current I length L A Image current L/2 2) (a) A wire of length L along the z-axis (from z = -L/2 to L/2) is carrying a dc current I in the +z-direction. What is the magnetic field intensity (B) at the point (x, 0, 0) ? (b) Use this result to evaluate the B field right above the ground plane at point A as shown in the figure. use Biot-Savart Law to calculate the B field at (x,0,0) The effect of the ground plane is captured by the image current, direction as shown. As a result, the magnetic field at point A is just doubled. x = L/2. dl = dz x z r -L/2 L/2 θ [ ]( )⊗+======⊗=⊗=×=∫∫−−222/2/22222/2/2222/2/2sin24seccossec4secsectancos4cos4ˆ4LxLxIxIxdxIBxrdxdzxzrdzIBrIdzrrIdBdomoLLoLLoooπµθπµθθθθπµθθθθθπµθπµπµrrlrr( )( )( ) ( )⊗=⊗+=⊗+=LIBLLLLIBLxLxIBototalototalototalπµπµπµ22/2/2/2/2/2/222222rrrEE140 Fall 2014 Mid-Term 1 Dr. Ray Kwok 3) A dielectric sphere (with dielectric constant εr) of radius ‘a’ is filled with positive charges of non-uniform volume charge density βr, where β is a constant. A conducting shell of inner radius ‘b’ and outer radius ‘c’ is shielding the dielectric as shown in the Figure. The outer shield is grounded at r = c as shown. (a) Calculate the electric field in all regions. (b) Sketch E(r) as a function of r. Label all axes. What are the magnitude of E at r = a? at r = b? and at r = c? (a) r < a (b) a < r < b (c) b < r < c E = 0 (conductor) (d) r > c E = 0 (grounded) a b c abc r Eβa2/4εo βa3/4b2εo βa2/4εrεo ( ) ( )( )rrarErdrrrdVrEQEAorrorenclosedˆ4444424022εεβπβπβρπεεε=<====∫ ∫r( ) ( )( )rrabraEadrrrdVrEQEAoaoenclosedˆ44444244022εβπβπβρπεε=<<====∫
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