Lect04ElectPoten - ECE 303 Electromagnetic Fields and Waves...

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1 1 Swartz 06/5/20 ECE 303 – Electromagnetic Fields and Waves – Summer 2006 Lecture 2: ECE 303 Electromagnetic Fields and Waves Instructor: Dr. Wesley E. Swartz Fall 2008 Lecture 4 2008/9/5 Electric Potential Electric Scalar Potential Laplace’s and Poisson’s Equations Potentials of Some Simple Charge Distributions 3 Swartz 06/5/20 ECE 303 – Electromagnetic Fields and Waves – Summer 2006 Lecture 2: Conservative or Irrotational Fields Irrotational or Conservative Fields: Vector fields for which are called “irrotational” or “conservative” fields. This implies that the line integral of around any closed loop is zero. Equations of Electrostatics: Recall the equations of electrostatics from a previous lecture: In electrostatics or electroquasistatics , the E-field is conservative or irrotational. 0 = × F F 0 s d F = ρ ε = E o 0 = × E F 4 Swartz 06/5/20 ECE 303 – Electromagnetic Fields and Waves – Summer 2006 Lecture 2: Conservative or Irrotational Fields More on Irrotational or Conservative Fields: If the line integral of around any closed loop is zero … … then the line integral of between any two points is independent of any specific path (i.e. the line integral is the same for all possible paths between the two points): 0 s d F = B r r A r r B r r A r r B r r A r r 2 1 2 1 2 1 2 1 1 2 2 1 s d F s d F 0 s d F s d F 0 s d F s d F path path path path path path = = = + F 1 r 2 r path A path B 0 s d F = F 5 Swartz 06/5/20 ECE 303 – Electromagnetic Fields and Waves – Summer 2006 Lecture 2: The scalar potential: Any conservative field can always be written as the gradient of some scalar quantity. This is because the curl of a gradient is always zero. For the conservative E-field one writes: (The negative sign is just a convention) Where is the scalar electric potential . The scalar potential is defined only up to a constant: If the scalar potential gives a certain electric field, then the scalar potential will also give the same electric field (where c is a constant). Remember that the derivative of a constant is zero. The absolute value of potential in a problem is generally fixed by some physical reasoning that essentially fixes the value of the constant c . The Electric Scalar Potential (1) Φ −∇ = E ( ) () 0 F = × = × then = F If c ) r ( + ) r ( 6 Swartz 06/5/20 ECE 303 – Electromagnetic Fields and Waves – Summer 2006 Lecture 2: The Electric Scalar Potential (2) We now know that: This immediately suggests that:
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Lect04ElectPoten - ECE 303 Electromagnetic Fields and Waves...

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