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Unformatted text preview: 5 Curl-free fields and electrostatic potential Mathematically, we can generate a curl-free vector field E ( x, y, z ) as E =- ( V x , V y , V z ) , by taking the gradient of any scalar function V ( r ) = V ( x, y, z ) . The gradient of V ( x, y, z ) is defined to be the vector V ( V x , V y , V z ) , pointing in the direction of increasing V ; in abbreviated notation, curl- free fields E can be indicated as E =- V. Verification: Curl of vector V is ( V ) = x y z x y z V x V y V z = x- y- z 0 = 0 . If E =- V represents an electrostatic field , then V is called the electrostatic potential . Simple dimensional analysis indicates that units of electro- static potential must be volts (V). 1 The prescription E =- V , including the minus sign (optional, but taken by convention in electrostatics), ensures that electro- static field E points from regions of high potential to low po- tential as illustrated in the next example. Electrostatic fields E point from regions of high V to low V Example 1: Given an electrostatic potential V ( x, y, z ) = x 2- 6 y V in a certain region of space, determine the corresponding electrostatic field E =- V in the same region. Solution: The electrostatic field is E =- ( x 2- 6 y ) =- ( x , y , z )( x 2- 6 y ) = (- 2 x, 6 , 0) =- x 2 x + y 6 V/m . Note that this field is directed from regions of high potential to low potential. Also note that electric field vectors are perpendicular everywhere to equipotential contours. 4 2 2 4 4 2 2 4 Light colors indicate high V dark colors low V Given an electrostatic potential V ( x, y, z ) , finding the corresponding elec- trostatic field E ( x, y, z ) is a straightforward procedure (taking the negative gradient) as already illustrated in Example 1....
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This note was uploaded on 05/05/2010 for the course ECE ece 442 taught by Professor Kim during the Spring '10 term at University of Illinois at Urbana–Champaign.
- Spring '10