Lecture 08

Lecture 08 - Uniformly Charged Rod At distance r from...

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At distance r from midpoint along a line perpendicular to the rod: ! E y = 0 = 1 4 !" 0 Q r r 2 + L / 2 ( ) 2 # \$ % % & ( ( ˆ r For very long rod: ! E = 1 4 !" 0 2 Q / L ( ) r # \$ % & ( ˆ r Field at the ends: Numerical calculation Uniformly Charged Rod

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Far from the ring ( z >> R ) : Close to the ring ( z << R ) : E z = 1 4 !" 0 Qz R 3 E z ~ z 2 0 4 1 z Q E z !" = E z ~1/ z 2 Uniformly Charged Thin Ring E z = 1 4 !" 0 Qz R 2 + z 2 ( ) 3/2
Uniformly Charged Disk A = ! R 2 E z = ( Q / A ) 2 ! 0 1 " z R 2 + z 2 ( ) 1/2 # \$ % % & ( ( Close to the disk (0 < z << R) E z ! ( Q / A ) 2 " 0 1 # z R \$ % & ( ) E z ! ( Q / A ) 2 " 0 If z / R is extremely small Very close to disk (0 < z << R)

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E - E + E net s z 0 E 2 ! Q / A " 0 E 1 = E 3 ! Q / A 2 " 0 s R Inside: Fringe: Capacitor + Q - Q s
Field inside: E =0 Field outside: r r Q E ˆ 4 1 2 0 !" = ! (like point charge) Spherical Shell of Charge

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What if charges are distributed throughout an object? Step 1: Cut up the charge into pieces r E R For each sphere: r r dQ E d ˆ 4 1 2 0 !" = ! outside: inside: dE = 0 Outside a solid sphere of charge: r r Q E ˆ 4 1 2 0 !" = ! for r>R A Solid Sphere of Charge
Inside a solid sphere of charge: E r r Q E ˆ 4 1 2 0 ! = "# ! ! Q = Q (volume of inner shells) (volume of sphere) ! Q = Q 4 / 3 " # r 3 4 / 3 " # R 3 = Q r 3 R 3 R r ! E = 1 4 !" 0 Qr R 3 ˆ r for r<R Why is E~r ? On surface: ! E = 1 4 !" 0 QR R 3 ˆ r = 1 4 !" 0 Q R 2 ˆ r A Solid Sphere of Charge

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