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Theorems from Vector Calculus
In the following </>, if,, and A are well-behaved scalar or vector functions, Vis a
three-dimensional volume with volume element d3x, S is a closed twodimensional
surface bounding V, with area element da and unit outward normal
n at da.
L V · A d 3x = LA· n da (Divergence theorem)
L Vi(., d 3x = L if.,n da
L V x A d3x = L n x A da
L (</>V2 !(., + V<f> • Vi(.,) d 3x = L <f>n • Vi(., da (Green’s first identity)
L (<f>V2 !(., – l(.,V2</>) d 3x = L (</>Vi(., – l(.,V</>) · n da (Green’s theorem)
lrfthe following S is an open surface and C is the contour bounding it, with line
element di. The normal n to Sis defined by the right-hand-screw rule in relation
to the sense of the line integral around C.
L (V x A) · n da = f c A · di
L n x V if, da = fc if, di
A point charge q is brought to a position a distance d away from an
innite plane conductor held at zero potential. Using the method
of images, nd:
(a) the surface-charge density induced on the plane, and plot it;
(b) the force between the plane and the charge by using Coulomb’s
law for the force between the charge and its image;
(c) the total force acting on the plane by integrating ۲=۲۰ over
the whole plane;
(d) the work necessary to remove the charge q from its position to
(e) the potential energy between the charge q and its image (com-
pare the answer to part d and discuss).
(f) Find the answer to part d in electron volts for an electron
originally one angstrom from the surface.