48 Gauss’s LawP24.69EA⋅=∈=∈zzdqarrdrrin002014πErardra rEar44422200020ππ=∈=∈=∈=zconstant magnitude(The direction is radially outward from center for positive a; radially inward for negative a.)P24.70In this case the charge density is not uniform, and Gauss’s law is written as ∈ddV10ρ. Weuse a gaussian surface which is a cylinder of radius r, length A, and is coaxial with the chargedistribution.(a)When rR<, this becomes arbdVr2000Abg=∈−FHGIKJz. The element of volume is a cylindricalshell of radius r, length A, and thicknessdrso that dVr dr=2A.rarb2223200πρAA=∈FHGIKJ−FHGIKJso inside the cylinder, Erarb=∈−FHGIKJ00223.(b)When >, Gauss’s law becomesarbrR22000AAbg=∈−FHGIKJzor outside the cylinder, ERraRb=∈−FHGIKJ020223.P24.71(a)Consider a cylindrical shaped gaussian surface perpendicularto theyzplane with one end in the yzplane and the other endcontaining the point x:Use Gauss’s law:
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