This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 Efield 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 Efield 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 Efield 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 Efield around a closed loop is zero ( ) ∫ ∫ = ∇ − = . . s d s d...
View
Full
Document
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).
 Fall '06
 RANA
 Electromagnet

Click to edit the document details