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Unformatted text preview: Phys 411 Set 1 Special problems
1. (a) By using Gauss’ law (Eq. ref: Gauss ) and the definition of the Dirac delta function show that if q EÝr Þ = k 1 3 r r then the charge density is _Ýr Þ = q N Ý3 Þ Ýr Þ. (b) Show that if _Ý r Þ = and EÝr Þ = k 1 X then EÝr Þ = k 1 >
i=1 N all space > q i N Ý3Þ Ýr ? r i Þ
i=1 N _Ý r v Þ Ýr ? r v Þ r ? r v  3 qi Ýr ? r i Þ. r ? r i  3 2. The charge density for a line charge is given by _Ýr Þ = Ý5 statC/cm Þ NÝx Þ NÝz Þ for y  < 3 cm _Ýr Þ = 0 statC/cm otherwise (a) evaluate, exactly, the electric field along the straight line y = +1 cm, z = 0 cm. (b) Imagine cutting the line charge into N (N = 1, 4, 8, and 12) equal length segments. Replace each segment by a point particle having the charge of the segment and located at the center of the segment. Using a spreadsheet evaluate the value of the components of the electric field at points along the straight line between Ý0.2 cm, 1 cm, 0 cm Þ and Ý15 cm, 1 cm, 0 cm Þ for the each value of N. (c) Plot the difference between these values and those obtained from the in part (a). Use enough points to obtain a smooth curve. 3. Show that the curl of the electric field due to a point charge, q EÝr Þ = k 1 3 r r vanishes. Be sure to check that it is not a delta function. 4. A spherical object is solid except for an interior spherical hole. The sphere has a radius R and the hole has a radius R/3. The object is composed of a material with a charge volume density _. Place the center of the spherical object at the origin of a xyz coordinate system and the center of the hole on the z axis at z = R/4. Using the integral form of Gauss’ law and the superposition principle obtain the equations for and plot the values of the nonzero components of the electric field at points along (a) the z axis with z  < 2R; (b) the y axis with y  < 2R. The field will be in units of _R/3P 0 and the coordinate in units of R.
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 Spring '10
 G
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