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quiz4-solutions

quiz4-solutions - ECE 220 — Quiz 4 Please write your name...

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Unformatted text preview: ECE 220 — Quiz 4 (03/29/04) Please write your name at the top right corner before you start the exam Remember to write the units of the various quantities that you calculate Use the back side of each page if you need extra space. All formulas and units that you need are given on the last page 1. (6 points) Give a short deﬁnition of the following a) Electric Field (E) It is the Force; warheol on 0» unit PCS-\l’lVC CIAQVSC b) Voltage (V) ‘ charge 6’0”) One; Poi/1} {—1) another 2. (6 points) Can any of the following vectors represent a valid magnetic ﬂux density? (Hint: Look at Maxwell’s equations) 3 A “:40”? Check i? KLE =0. b) B = (3x)5c + (2x + z)? at) 133405? FY7751? ‘ £403 : 0. ° W is & Val/5‘1 9% M neb‘c 10W} % gap/13% b). Baﬁgxyra‘cwzxnﬁ 7 a ﬁx} _+ “algae-k2) .; 3 +0 :33’10 9 1 a . 5 l N O k V 0L [ t & 3. (6 points) The voltage in a cubic region deﬁned by -1 < x,y,z < 1 is given by V = xzyzz (V) Find the charge density in the region (assume a = so) 7. V V 7’ v 8 :_ v / _ 2 z I /g p ED ' V- DC a z 4. (12 points) The magnetic ﬁeld through a 1 turn circular loop of radius 1 cm grows with time according to the equation B = Bo(1+(t/T)) where BD=100pT and T=300s. Find the emf induced in the loop. (Direction of B — Coming out of the paper) (9 ® @ © 1 B=B° ML AHT m“ ( T 3 w ; ' 7, TotoJ mm '0 Tr (m) Cb: BA 3 KW: IO'LWF T EMF 8;: — 'vBo |O"t.TT— ACl+gF E oLt 5. (20 points) A parallel plate capacitor with plate area 10mm X 10mm and distance between the plates of 1mm, contains between its plates a dielectric with permittivity e = 490. Suppose the plate at x = 0 is at zero potential, the plate at i: = 0.001 m is at 4V, ﬁnd V(x) (potential distribution) between the plates assuming the free charge density in the dielectric , pv = x (C/m’) 764M"! 76:0 List of units 1) 2) 3) 4) 5) 6) 7) 8) J. Electric Flux Density(D) & Surface Charge Density(ps) — Coulombs/meter2(C/m2) Volume Charge Density (pv) —— Coulombs/meter3 (C/m3) . Electric Field Intensity (E) — Newtons/Coulomb (N/C) or Volts/meter (V/m) Voltage — Volts(V) Magnetic Field ~ Amperes/meter(A/m) Magnetic Flux Density — Tesla ( T ) Pennjttivity (8) — Farads/meter (F/m) Permeability (u) —- Henry/meter (H/In) List of formulas and quantities 1) 2) 3) 4) 5) 6) 7) p0 — permeability of free space (air) = 41: * 10‘7 (H/m) so - permittivity of free space (air) = 8.854 * 10‘12 (F/m) Maxwell’s equations VI) = p 313 vxE=—— at Vxﬁ=j+a—D a: Gauss’s law for magnetic fields: {113 B—Jﬁ = 0 Relationship between Electric field E and voltage V E = —VV dV If V varies only along X direction, then E = ~E Poisson’s equation V 2V = —pv ls Faraday’s law Induced emf (8) = —£12 dt Vector operations in Cartesian coordinates GradientofascalarV: VV=an+aV 9...an ax 3y Bz DivergenceofavectorD: V.D = a DX + a DY + 6 DZ ax 8y dz 2 2 V20 eratoronascalarV: VZV = 62V 8 V + a V p 3x2 ay2 322 ...
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