329lect05 - 5 Curl-free elds and electrostatic potential...

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5 Curl-free Felds and electrostatic potential Mathematically, we can generate a curl-free vector Feld 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 deFned to be the vector V ( ∂V ∂x , ∂V ∂y , ∂V ∂z ) , pointing in the direction of increasing V ; in abbreviated notation, curl- free Felds E can be indicated as E = -∇ V. – Verifcation: Curl of vector V is ∇× ( V )= ± ± ± ± ± ± ± ˆ x ˆ y ˆ z ∂x ∂y ∂z ∂V ∂x ∂V ∂y ∂V ∂z ± ± ± ± ± ± ± x 0 - ˆ y 0 - ˆ z 0=0 . If E = -∇ V represents an electrostatic feld , then V is called the electrostatic potential . Simple dimensional analysis indicates that units of electro- static potential must be volts (V). 1
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The prescription E = -∇ V , including the minus sign (optional, but taken by convention in electrostatics), ensures that electro- static feld E points From regions oF “high potential” to “low po- tential” as illustrated in the next example. Electrostatic felds 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 Feld E = -∇ V in the same region. Solution: The electrostatic Feld is E = -∇ ( x 2 - 6 y )= - ( ∂x , ∂y , ∂z )( x 2 - 6 y )=( - 2 x, 6 , 0) = - ˆ x 2 x y 6 V/m . Note that this Feld is directed from regions of high potential to low potential. Also note that electric Feld vectors are perpendicular everywhere to “equipotential” contours. ± 4 ± 2 0 2 4 ± 4 ± 2 0 2 4 Light colors indicate “high V dark colors “low V Given an electrostatic potential V ( x, y, z ) , fnding the corresponding elec- trostatic feld E ( x, y, z ) is a straightForward procedure (taking the negative gradient) as already illustrated in Example 1. The reverse operation oF fnding V ( x, y, z ) From a given E ( x, y, z ) can be accomplished by perForming a vector line integral ± p o E · d l 2
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in 3D space, since, as shown below, such integrals are “path independent” for curl-free Felds E = -∇ V .
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329lect05 - 5 Curl-free elds and electrostatic potential...

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