EE243 Advanced Electromagnetic Theory Lec #3: Electrostatics (Apps., Form),

Transcription

1 EE4 Advanced Electromagnetic Theory Lec #: Electrostatics Apps., Form, Electrostatic Boundary Conditions Energy, Force and Capacitance Electrostatic Boundary Conditions on Φ Image Solutions Eample Green s Functions Integral Formulation Reading: Jackson.,.-.5,.7-.

2 Div D = ρ Electrostatic Boundary Conditions D terminates on surface charge on a conductor dφ/dn = σ/ε Φ Φ E ˆ E n = = σ / ε n n How about for Φ? Jackson.6 evaluates dipole layer D Φ ˆ ε Φ n = D / Thus Φ is continuous unless there is a surface dipole layer

3 Energy Jackson. Electrostatic potential is potential energy of a charge Add a charge to m- charges = m- terms Repeat to add more charges leaving out selfinteractions to get N charges Put in symmetric form un-nest do loops to get ½ of regular double sum W = ρ Φ d

4 Energy Cont. W = ρ Φ d Use Poisson s Equation Integrate by parts Rewrite as E field W ε ε = Φ Φ = ε d Φ d = [ ] E d Physical interpretation: The electrostatic energy is stored in space as /DE and there is stored energy any time that the electric field is non-zero. 4

5 Force Calculated from change in energy for a small virtual displacement W = F. Force per unit area a due to surface charge Volume a ε σ = ε w = E σ W = ε a Outward force per unit area F = σ ε 5

6 Capacitance Capacitance is defined as the charge per unit voltage when all other conductors are grounded Mutual capacitance is charge per unit voltage difference when a pair have equal and opposite charge and all other conductors are grounded Potential is sum over charges Potential Energy found by adding new potential to m- => half double sum /C ij V i V j 6

7 Method of Images Under favorable and rare conditions inferred from a geometry a small number of eternal charges can simulate the required boundary conditions. Eamples for Dirichlet G = on boundary Charge above a conducting plane Charge -q at position -y Charge in a 6/n wedge Charge outside a conducting sphere Charge -aq/y at y = a /y Charge inside a spherical hole in a conductor Eamples of Neumann = Are there any? Jackson.-.4 7

8 Conducting Sphere in a Uniform E field Q -R +aq/r -aq/r -a /R a /R a θ Jackson.5 r -Q z R Consider two charges to create uniform field in limit R => infinity and Q/R constant -Q at y = R and +Q at y = -R Add images to make G = +aq/r at +a /R and aq/r at a /R Potential is 4 terms Assume R >> a; use /+ / appro. - Take limit R => infinity and Q/R constant 8

9 Conducting Sphere - Uniform E field Cont. Potential Φ = E r a r cosθ Physically interpret as dipole charge times separation Qa a D = = 4πεEa = ε E R R D is D times volume and is oriented directly opposite to the applied field Volume Surface charge density from D normal is D σ surface = ε Φ r= a = ε E r cosθ 9

10 Green s Theorem and Integral da n n d V V = φ ψ ψ φ φ ψ ψ φ Green s nd Identity Theorem Use φ = Φ and Poisson s Equation for F Use ψ = G any solution to Poisson s Equation for one point charge in the internal region and any boundary conditions on dv 4, = Ψ π da n G n G d G S V Φ Φ + = Φ π ρ πε,, 4, 4, ε ρ = Φ

11 Common Case: Integral Representation with the Free Space Green s Function, G = For a unit charge in free space the potential is proportional to da n n d S V Φ Φ + = Φ 4 4 π ρ πε Need to know: Charge distribution in interior The potential on the boundary The derivative of the potential normal to the boundary on the boundary surface charge

12 Eample Green s Function Application Surface Charge Patches Surface Potential ρ = Observation Point = Boundary Integration Point Φ = 4πε Φ ρ d + da Φ 4π n n V S Observation point is in solution region Surface integration points are on boundary Volume integration is over solution region