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Unformatted text preview: 1. (25 points) A very long wire of radius R is carrying a current density a? = a (R2 — 32) it}, Where a is a constant and s is the cylindrical coordinate that measures the distance ﬁom the zaacis. (a) What is the total current carried by this Wire? @ (b) Determine ﬁﬁeld at the point (x = g, y = 523, z = 0). 2. (25 points) ' ‘ “529 g :32; (w) 4%.) The charge density on the surface of a sphere of radius R is given by cr = an 0082 6. (a) Find the electric potential inside and out side the sphere. g g 113 I. 3 anan‘acm V:..=V..,+lrgw
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Using boundary conditions: A planar sheet (2 = O) carries uniform surface charge density or and a steady surface current K = K053. (a) If Fields E" and B; on two sides (very near the surface) are given by Etc}, 2 Emir. + Elygj + E122; EDP 2
Bus + 313,3": + B132“: and Ebottom = Egmﬁ + Egyz? + E292; 3.130th = ngﬁ + Bgyé + Bgyé, respectively. Determine relations between components of ﬁelds Etop, Btop and Ebottom, Bbomm. (b) Potential inside a spherical cavity (in a conductor) of radius R is given by V[r, I9) = jag—£0 f5 (1 — {335) cos 9. y 3, aging _cr
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W k B. 4. (15 points) Show that Maxwell’s equations in free space, i.e., in absence of charge and current deneities, lead to the following two wave equations: 6. (25 points) From Coulomb’s Law, the potential due to a uniformly charged ring of radius 3% and charge g, at a point on the axis (a distance z from the center of the ring) is given by _ q 1
V(0101z) _ 41r60(3%2+z2)1/2. '(a) For large values of z, expand the above potential in powers of %; (1)) Compare the above expansion with 2:0 F355 Pdcos 9) evaluated on the zaxis, and determine ﬁrst three nonzero coefﬁcients by. Hint: 133(1) = 1 for every 3. (0) Write ﬁrst three nonvanishing terms for the potential at (33, R, 1, I Q 7 "/2
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b. 3'32 < r (region III) (4 points) a“. (b) Determine undetermined constants of part a) using continuity of potential and the diSCOntinuity of the normal component of E ﬁeld. (10 points) {c} Determine the charge density on the conducting cylingr.(8 points) Q i) VI: = (d.°+boinr)+ 2(anv”+h.f")cg1n15 "W n —“ n
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 Spring '09
 Gangopashyaya
 Magnetism

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