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Unformatted text preview: September 11, 2011 Physics for Scientists & Engineers 2, Chapter 22 1 General Charge Distributions We have determine the electric eld from point charges Consider the electric eld due to a charge distribution We divide the charge into differential elements of charge, dq , and nd the electric eld from each differential charge element as if it were a point charge e magnitude of the electric eld is then September 11, 2011 Physics for Scientists & Engineers 2, Chapter 22 2 dq = ! dx dq = " dA dq = # dV $ % & ' & for a charge distribution along a line over a surface throughout a volume ( ) & * & dE = k dq r 2 Finite Line of Charge To nd the electric eld along a line bisecting a nite length of wire with linear charge density , we integrate the contributions to the electric eld from all the charge We assume that the wire lies along the xaxis September 11, 2011 Physics for Scientists & Engineers 2, Chapter 22 3 Finite Line of Charge e symmetry of the situation allows us to conclude that there cannot be any electric force parallel to the wire We can calculate the electric eld due to all the charge for x 0 and multiply the result by 2 Consider a differential charge dq on the xaxis e magnitude of the the electric eld dE at a point (0, y ) due to this charge is e component of the electric eld perpendicular to the wire is September 11, 2011 Physics for Scientists & Engineers 2, Chapter 22 4 dE = k dq r 2 r = x 2 + y 2 dE y = k dq r 2 cos ! = k dq r 2 y r " # $ % & ' = k dqy r 3 Finite Line of Charge Taking dq = dx , the differential electric eld is e electric eld at a distance y from the wire is then e integral is September 11, 2011 Physics for Scientists & Engineers 2, Chapter 22 5 dE y = k ! dxy r 3 E y = 2 dE y a ! = 2 k " dxy r 3 a ! = 2 k " y dx x 2 + y 2 ( ) 3 a ! = 2 k " y dx x 2 + y 2 ( ) 3/2 a ! dx x 2 + y 2 ( ) 3/2 a ! = 1 y 2 x x 2 + y 2 " # $ $ % & ' ' a = 1 y 2 a y 2 + a 2 Finite Line of Charge e electric eld is then When the wire becomes in nitely long e electric eld a distance y from an in nitely long wire is September 11, 2011 Physics for Scientists & Engineers 2, Chapter 22 6 E y = 2 k !...
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