2013S_721_HW4_sol

# Of the electric eld near the center of the sphere

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 0 dl+1 q lAl Pl (cos θ) = a 4πǫ0 l (l + 1)Bl l al − 1 Pl (cos θ) − dl+1 a al l+1 Al = l+1 − Bl d l 3 Solving, this gives al 2l + 1 l + ǫ0 /ǫ(l + 1) dl+1 (ǫ0 /ǫ − 1)l al Bl = l + ǫ0 /ǫ(l + 1) dl+1 al r l q 2l + 1 Pl (cos θ) Φin = 4πǫ l + ǫ0 /ǫ(l + 1) dl+1 al Al = l Φout = q 4πǫ0 l l r&lt; (ǫ0 /ǫ − 1)l al + l+1 l + ǫ0 /ǫ(l + 1) dl+1 r&gt; a r l+1 Pl (cos θ) (b) Calculate the rectangular components of the electric ﬁeld near the center of the sphere When r &lt;&lt; a, the high l terms drop out, the potential becomes r r2 3 cos2 θ − 1 q 1 3 5 + cos θ + + ··· 4πǫd ǫ0 /ǫ 1 + 2ǫ0 /ǫ d 2 + 3ǫ0 /ǫ d2 2 3 5 z 3z 2 − r2 q 1+ + + ··· = 4πǫ0 d 2 + ǫ/ǫ0 d 3 + 2ǫ/ǫ0 2d2 3 5 z ˆ 2z z − xx − y y ˆ ˆ ˆ q + + ··· =− 4πǫ0 d 2 + ǫ/ǫ0 d 3 + 2ǫ/ǫ0 2d2 Φin = Ein (c) Verify that, in the limit ǫ/ǫ0 → ∞, your result is the same as that for the conduc...
View Full Document

Ask a homework question - tutors are online