Lecture03_Part1_V1.2

Lecture03_Part1_V1.2 - PHY501 Lecture III—Part 1 Multipoles and Dielectrics V1.2 Multipole Expansions It is often useful to expand the potential

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Unformatted text preview: PHY501 Lecture III—Part 1 Multipoles and Dielectrics V1.2 Multipole Expansions It is often useful to expand the potential due to a localized charge distribution in terms of spherical harmonics ( x ) = 1 1 2 +1 q m Y m ( , ) r + 1 ,m (1) We also have R x’ x x x’ x y ( x ) = 1 4 ( x ') x x ' V dV ' (2) which, with the addition theorem 1 x x ' = 4 2 + 1 r < r > + 1 Y m * ( ', ') Y m ( , ) , m (3) where, , becomes r > = r = x and r < = r ' = x ' ( x ) = 1 1 2 +1 Y m ( , ) r + 1 r ' ( x ') Y m * ( ', ') dV ' V ,m (4) Comparing Eqs. 1 and 4 we see that q m = r ' ( x ') Y m * ( ', ') dV ' V (5) Multipole Expansions Example I: Point charge at the origin x ' ( ) = q x ' ( ) Also Why? q 00 = r ' ( x ') Y m * ( ', ') dV ' V = q 4 (6) q m = 0 for > 0 Example II: Dipole centered on origin x ' ( ) = q r ' 2 r ' a ( ) ' ( ) cos ' 1 ( ) cos ' + 1 ( ) [ ] (7) q 00 = q 4 x ' a ˆ z ( ) dV ' x ' + a ˆ z ( ) dV ' V V = 0 q 10 = q r ' r ' 2 Y m * ( ', ') cos ' 1 ( ) cos ' + 1 ( ) [ ] V ' ( ) d ' ( r ' a ) r ' 2 dr ' d cos ' ( ) = q a 3 4 cos ' 1 1 cos ' 1 ( ) cos ' + 1 ( ) [ ] d cos ' ( ) = 3 4 2 qa = 3 4 p z (8) Exercise: Show that for the distribution in the sketch. q 11 = q 1 1 = z q +q a a +z Field Due to a Dipole We digress slightly to obtain a result for future use. For a pure dipole at the orgin given by where p = p z ˆ z ( x ) = 1 1 2 +1 q m Y m ( , ) r + 1 ,m = 1 3 q 10 Y 1 ( , ) r 2 (9) q 10 = 3 4 p z and Y 1 = 2 + 1 4 P 1 = 3 4 cos (10) Combining Eqs. 9 and 10 yields ( x ) = 1 4 p z cos r 2 = p x 4 r 3 E = = p x 4 r 3 (11) or E = = 1 4 1 r 3 p x ( ) + 3 p x ˆ n r 4 , ˆ n = x x = 1 4 p r 3 + 3 p ˆ n r 3 ˆ n = 3 p ˆ n ( ) ˆ n p 4 r 3 (12) Multipole Expansions: Quadrupoles Note that Q tot = 0 and p tot = +q q +z z a a a a +q q Recall that Y 2 2* = 1 4 15 2 sin 2 e 2 i Y 2 1* = 15 8 sin cos e i (13) Y 2 0* = 1 2 5 4 3cos 2 1 ( ) q 22 = r ' 2 Y 2 2* ( ', ') V ( x ') dV ' = 1 2 15 2 qa 2 (14) Thus also q 21 = 0 and q 20 = 3 qa 2 5 4 (15)...
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This note was uploaded on 02/04/2012 for the course PHY 501 at Princeton.

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Lecture03_Part1_V1.2 - PHY501 Lecture III—Part 1 Multipoles and Dielectrics V1.2 Multipole Expansions It is often useful to expand the potential

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