PHY2049ch21B%281-11-10%29

# PHY2049ch21B%281-11-10%29 - Last time we derived the force...

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3 2 2 o 3 qqd d ˆ Fk 1 k r2 r ⎡⎤ ⎛⎞ =− + ⎢⎥ ⎜⎟ ⎝⎠ ⎣⎦ G Last time we derived the force on +q o due to +q and q arranged as shown, getting, d d/2 r + r y q + q +q o r F + G F G F G F G Such a pair of equal but opposite charges is called a dipole . The force on + q o decreases not only as its distance from the dipole, r , increases but also if the dipole separation, d shrinks (note that if + q and – q merge they become a neutral body and the electrostatic force ). F0 = G To see this behavior more readily we use the binomial theorem to expand the term in square brackets. The binomial theorem is (HRW page A10), 2 n2 nx n(n 1)x (1 x) 1 ..., x 1 1! 2! + =+ + + <

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3 22 4 2 33 3 2 dd ( 1 ) d 1 1 ... 2r 1! 2! ⎡⎤ −− ⎛⎞ += + + + ⎢⎥ ⎜⎟ ⎝⎠ ⎣⎦ Now as d shrinks the term falls off rapidly while the higher the order terms fall off more rapidly still . Ignoring all but the 1 gives for as d becomes much less than r, 2 d o 3 qqd ˆ Fk k r =− G Then with 2 d x = F G which clearly goes to 0 as d does. Note the fall off of the force with 1/r 3 . This is a general feature of dipoles which we’ll revisit in Chapter 22.
For the subject of this course we will need one further property of materials related to their atomic basis . When atoms assemble into solids (or liquids) their outermost electrons typically participate in the bonding between the atoms. In some materials , these electrons are rigidly held in place , while in other materials one or more of these electrons (per atom) become shared over the entire solid and are free to move . In the first case the material is an electrical insulator (like glass or plastic), which does not permit charge to flow through it. In the second case the material is an electrical conductor (like metals), which readily permit the flow of charge through them. insulator conductor

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Because electrons that are uncompensated by very near lying positive charge repel each other, excess electrons that are forced onto a conductor will arrange themselves to be as far apart as possible. They manage this by existing as a surface layer of negative charge on the conductor. For a spherical conducting body these excess electrons make up a spherical “skin” of negative charge at the surface . Less obvious is that if some quantity of electrons are forced off of a conductor leaving behind an excess positive charge the remaining electrons will arrange themselves to have the excess positive charge also exist as a thin layer at the surface .
In most cases excess positive charge remaining when there is a deficit of electrons behaves as if positive charge had been added when in fact it is electrons that were subtracted. Indeed the flow of electrons off the conductor (electron flow to the right on the wire below) can just as readily be interpreted as a flow of positive charges (+e) flowing (to the left) onto the conductor.

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PHY2049ch21B%281-11-10%29 - Last time we derived the force...

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