Lecture 3 - 3 Gauss's law and static charge densities We...

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3 Gauss’s law and static charge densities We continue with examples illustrating the use of Gauss’s law in macroscopic Feld calculations: ρ S E x ( x )= ρ s 2 ± o A z S x y E x ( x ) ρ s 2 ± o sgn( x ) x Example 1: Point charges Q are distributed over x =0 plane with an average surface charge density of ρ s C/m 2 . Determine the macroscopic electric Feld E of this charge distribution using Gauss’s law. Solution: ±irst, invoking Coulomb’s law, we convince ourselves that the Feld produced by surface charge density ρ s C/m 2 on x plane will be of the form E xE x ( x ) where E x ( x ) is an odd function of x because y - and z -components of the Feld will cancel out due to the symmetry of the charge distribution. In that case we can apply Gauss’s law over a cylindrical integration surface S having circular caps of area A parallel to x , and obtain ± S D · d S = Q V ± o E x ( x ) A - ± o E x ( - x ) A = s , which leads, with E x ( - x - E x ( x ) , to E x ( x ρ s 2 ± o for x> 0 . Hence, in vector form E x ρ s 2 ± o sgn ( x ) , where sgn ( x ) is the signum function, equal to ± 1 for x 0 . Note that the macroscopic Feld calculated above is discontinuous at x plane containing the surface charge ρ s , and points away from the same surface on both sides. 1
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ρ E x ( x )= ρx ± o A z x W 2 - W 2 E x ( x ) ρW 2 ± o - W 2 W 2 x y Example 2: Point charges Q are distributed throughout an infnite slab oF width W located over - W 2 <x< W 2 with an average charge density oF ρ C/m 3 . Determine the macroscopic electric feld E oF the charged slab inside and outside. Solution: Symmetry arguments based on Coulomb’s law once again indicates that we expect a solution oF the Form E xE x ( x ) where E x ( x ) is an odd Function oF x . In that case, applying Gauss’s law with a cylindrical surFace S having circular caps oF area A parallel to x =0 extending between - x and x< W 2 , we obtain ± S D · d S = Q V ± o E x ( x ) A - ± o E x ( - x ) A = ρ 2 xA, which leads, with E x ( - x - E x ( x ) , to E x ( x ρx ± o For 0 W 2 .
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Lecture 3 - 3 Gauss's law and static charge densities We...

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