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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 non-zero 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