Lecture03_Part2_V1.1

Lecture03_Part2_V1.1 - PHY501 Lecture III—Part 2 Boundary...

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Unformatted text preview: PHY501 Lecture III—Part 2 Boundary Conditions at the Interface Between Dielectrics V1.1 Dielectric Boundary Conditions It is often useful to expand the potential due to a localized charge distribution in terms of spherical harmonics Since E 2 E 1 ( ) ˆ n 21 = 0 (58) Thus E || is continuous at an interface. Since there are an infinite number of contours C—each of them perpendicular to n 21 , we write 2 n 1 D 2 Gaussian Surface 1 D 1 D ˆ n da S = Q enc D 2 ˆ n 21 ( ) A D 1 ˆ n 21 ( ) A = A D 2 D 1 ( ) ˆ n 21 = (56) For = 0, D is continuous E = E = E ( ) S ˆ n¡da = E d l = C by Stokes Theorem E 2 ( ) || d E 1 ( ) || d = 0 (57) Also, if we assume that is not infinite we have E 1 = 2 (59) d 2 1 1 2 Stokes Contour E E Dielectric Boundary Conditions Example I: Charge near a planar interface. D 1 = 1 4 q cos d 2 + s 2 q 'cos d 2 + s 2 = 1 4 q q ' ( ) d d 2 + s 2 ( ) 3 2 q 1 2 d We use the method of images. In effect, we have two image problems: one in region 1 and one in region 2. Recall that in image problems, all images must exist outside of the region of interest. In region 1, we see the real charge q and image q’, while in Region 2, we see only the image q’’. Region 2, we see only the image q’’....
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Lecture03_Part2_V1.1 - PHY501 Lecture III—Part 2 Boundary...

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