lecture4 - 1 ECE 303 – Fall 2006 – Farhan Rana –...

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Unformatted text preview: 1 ECE 303 – Fall 2006 – Farhan Rana – Cornell University Lecture 4 Electric Potential In this lecture you will learn: • Electric Scalar Potential • Laplace’s and Poisson’s Equation • Potential of Some Simple Charge Distributions ECE 303 – Fall 2006 – Farhan Rana – Cornell University Conservative or Irrotational Fields Irrotational or Conservative Fields: Vector fields for which are called “irrotational” or “conservative” fields F r = × ∇ F r • This implies that the line integral of around any closed loop is zero F r . = ∫ s d F r r Equations of Electrostatics: Recall the equations of electrostatics from a previous lecture: ρ ε = ∇ E o r . = × ∇ E r ⇒ In electrostatics or electroquasistatics , the E-field is conservative or irrotational (But this is not true in electrodynamics) 2 ECE 303 – Fall 2006 – Farhan Rana – Cornell University Conservative or Irrotational Fields More on Irrotational or Conservative Fields: • If the line integral of around any closed loop is zero ….. . = ∫ s d F r r F r …. 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) B r r A r r B r r A r r B r r A r r s d F s d F s d F s d F s d F s d F s d F path path path path path path 2 1 2 1 2 1 2 1 1 2 2 1 . . . . . . . ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∫ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∫ ⇒ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∫ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∫ ⇒ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∫ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∫ ⇒ = ∫ r r r r r r r r r r r r r r r r r r r r r r r r r r F r 1 r r 2 r r path A path B ECE 303 – Fall 2006 – Farhan Rana – Cornell University The Electric Scalar Potential - I The scalar potential: Any conservative field can always be written (up to a constant) as the gradient of some scalar quantity. This holds because the curl of a gradient is always zero. For the conservative E-field one writes: (The –ve sign is just a convention) φ −∇ = E r ( ) ( ) Then = ∇ × ∇ = × ∇ ϕ F r ϕ ∇ = F r If Where φ 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) ( ) r r φ ( ) c r + r φ The absolute value of potential in a problem is generally fixed by some physical reasoning that essentially fixes the value of the constant c 3 ECE 303 – Fall 2006 – Farhan Rana – Cornell University The Electric Scalar Potential - II This immediately suggests that: • The line integral of E-field between any two points is the difference of the potentials at those points ( ) ( ) ( ) 2 1 2 1 2 1 . . r r s d s d E r r r r r r r r r r r r r φ φ φ − = ∫ ∇ − = ∫ 1 r r 2 r r • The line integral of E-field around a closed loop is zero ( ) ∫ ∫ = ∇ − = . . s d s d...
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This note was uploaded on 02/02/2008 for the course ECE 3030 taught by Professor Rana during the Fall '06 term at Cornell University (Engineering School).

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lecture4 - 1 ECE 303 – Fall 2006 – Farhan Rana –...

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