Chapter 3 Fields of Stationary Electric Charges: II Solid angles Gauss' Law Conductors a Poisson's equation Laplace's equation Uniqueness Theorem - lmages
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 dl d sin e do integrating over the curve C yields d l sin e a=JC g radlan. (2 )Solid angle subtended by a surface(see Fig3. 3) a small element of area da subtends a small solid angle at P cos 8da r1. da integrating over a finite area S yields cos eda 2 steradian
If s is a closed surface containing p cos eda 4丌, steradian. If P is situated outside of S,(see Fig 3.4) cO s eda g2=/s-2=0
3.2 gauss’Law This law relates the flux of e thru a closed surface to the charge inside By using this law one can find e of simple charge dis- tributions easily Let a point charge q be at the point P inside the closed surface s. The flux of e thru a small element of area da is E. da Q r1. da Q d o 4丌∈0 4丌∈0 Integrating over yIelds JE da 4丌 4m∈0 4丌∈0
If several point charges Qi are inside th e fields is E=∑E2 The flux of e thru a small element of area da is E·da=∑E;da Integrating over S yields Q2;1 /sE.da=∑ ∑Q ∈ For a general distribution of charge inside s Q=/p(r')di so we have E da=o/p(r) The gauss law in integral form
By the divergence theorem the surface integral can be written as a volume integral Is E da=h,V ed Thus we have A,v. edr'=-/p(ra ∈∩ Since the volume is arbitrary V·E(r)=-p(r) i.e. at any point of space, the divergence of e is equal to the charge density divided by the permittivity 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 Examples (1)a point charge (2 )a infinite line of charge (3 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) For electrostatics,- equilibrium,- 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 ace. ( see Fg38) Applying Gauss'law to the surface, we have e=o/Eo 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 Gauss' s law,E=ol∈o
