# Lecture 5 - 5 Curl-free fields and electrostatic potential...

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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 ) = ˆ x ˆ y ˆ z ∂ ∂x ∂ ∂y ∂ ∂z ∂V ∂x ∂V ∂y ∂V ∂z = ˆ x- ˆ y- ˆ 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) =- ˆ x 2 x + ˆ 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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Lecture 5 - 5 Curl-free fields and electrostatic potential...

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