This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full 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