Chapter 3 Fields of Stationary Electric Charges: II Solid angles a Gauss Law Conductors a Poissons equation Laplace's equation a Uniqueness theorem -Images
Chapter 3 Fields of Stationary Electric Charges : II ◼ Solid Angles ◼ Gauss’ Law ◼ Conductors ◼ Poisson’s Equation ◼ Laplace’s Equation ◼ Uniqueness Theorem ◼ Images
3. 1 Solid angles (1) Angle subtended by a curve (see Fig 3. 1 a small segment of curve dl subtends a small angle at p sin t integrating over the curve C yields d l sin e 2 raglan (2 ) Solid angle subtended by y a surtace(see Fig3.3 a small element of area da subtends a small solid angle at p do cos Ada r1. da integrating over a finite area s yields cos oda .2, Steradian
If s is a closed surface containing P cos eda 2 4丌, steradian If P is situated outside of S, (see Fig 3.4) cos ede 92=/s
3.2 Gauss’LaW This law relates the flux of e thru a closed surface to e charge Inside By using this law one can find e of simple charge dis tributions easily Let a point charge be at the point p inside the closed surface s. the flux of e thru a small element of area da is Q ri da Q a 4丌∈0 4∈0 Integrating over S yields /E·da=4丌e0 /ds 4丌 4丌∈0
If several point charges Q i are inside the fields is E=∑E The flux of e thru a small element of area da is E·da=∑E;:da Integrating over S yields Is E. da Q ∈0 ∈0 For a general distribution of charge inside s Q=/ p(r) so we have Is E da The gauss' law in integral form
By the divergence theorem the surface integral can be written as a volume integral /sE.da=V·Edr Thus we have hV.E′1 ∈ Since the volume is arbitrary E(r)=plr) i. e. at any point of space, the divergence of e is equal to the charge density divided by the permittivity Eo The gauss law in differential form
Remarks The integral form of the gauss law is very useful to get the e, especially when the charge distribution has some symmetry Example es a point charge a infinite line oT charge a InfinIte sheet of charge a spherical charge The differential form of the gauss law is of funda mental importance, as it tells how the charge density determines the electric field e
3. 3 Conductors a conductor is a material inside which charges can flow freely. (no resistance) or electrostaticS equ librium,- fixed in space zero electric field, -all points are at the same potential.(otherwise, charges will move. If a conductor is placed in an electric field charges flow within it, so as to produce a second electric field that cancels the first one in the conductor Applying Gauss' law V. E= p/Eo inside conductors, SInce E=0,→p=0 Conclusion: the charge density is 0 inside a conductor Corollary: any net charge on a conductor must reside on its surface
At the surface of a conductor. e must be normal ie Eu=0, otherwise charges would flow along the sur- face.(see Fig 3-8) Applying Gauss 'law to the surface, we have e= o/E0, different from an infinite sheet with E=o/2E0
da Figure 3-8 Portion of a charged conductor carrying a surface charge density a. The charge enclosed by the imaginary box is o da. There is zero field inside the conductor. Then from Gausss law e=d
