sum1 - SUMMARY PHYSICS 707 Electrostatics The basic...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: SUMMARY PHYSICS 707 Electrostatics The basic differential equations of electrostatics are E ( x ) = 4 ( x ) and E ( x ) = 0 (1) where E ( x ) is the electric field and ( x ) is the electric charge density. The field is defined by the statement that a charge q at point x experiences a force F = q E ( x ) where E ( x ) is the field produced by all charge other than q itself. These equations have integral equivalents, I S d 2 x E ( x ) n = 4 Z V d 3 x ( x ) = 4 Q (2) where Q is the charge enclosed by the surface S surrounding the domain V and n is a unit outward (from V ) normal vector at a point on S ; and I C d l E ( x ) = 0 . (3) Finally, if one applies these equations on the surface of a conductor (inside of which E = 0), then one finds the surface charge density and (negative) pressure are E n ( x ) = 4 ( x ) p = 2 2 (4) There is an integral solution for E ( x ) if one knows everywhere, E ( x ) = Z d 3 x ( x )( x- x ) | x- x | 3 . (5) Introduce a scalar potential ( x ) such that E ( x ) =- ( x ) (This can be done because E ( x ) = 0 everywhere). Then 2 ( x ) =- 4 ( x ) (6) which has the integral solution ( x ) = Z d 3 x ( x ) | x- x | . (7) Note in particular that the solution for a unit point charge is 1 | x- x | and that this function is such that 2 1 | x- x | ! =- 4 ( x- x ) . (8) 1 The meaning of ( x ) is that q ( x ) is the energy of interaction of q , located at x , with the charges that produce the potential. The energy of a localized charge distribution can be written as W = 1 8 Z d 3 x E ( x ) E ( x ) = 1 2 Z d 3 x ( x )( x ) . (9) Define the electrostatic energy density as w ( x ) = 1 8 E ( x ) E ( x ) . (10) Solution of Boundary Value Problems We learned how to solve boundary value problems by a variety of methods (For the actual boundary conditions, see the section on macrostatic electrostatics)....
View Full Document

Page1 / 6

sum1 - SUMMARY PHYSICS 707 Electrostatics The basic...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online