5. Lagrangians and Hamiltonians

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Setup

\(Clear["Global`*"]; 
Off[General::spell]; 
Off[General::spell1]; 
    \)

Needs["Calculus`VectorAnalysis`"]

Needs["Graphics`PlotField`"]

Needs["Graphics`PlotField3D`"]

<< "D:\Mma\Inits\ch5init.m"

Clear["Global`*"];

5.2 Problems

5.2.1  Point Charges, Multipoles, and Image Problems

Problem 2:  Point Charges and a Grounded Plane

Consider a point charge q a distance d from an infinite conducting plane that is grounded.

Part (a)   Find the potential.  Expand it in powers of 1/r and find the leading-order term.  Find the electric field to the same order.

dipole= (Monopole[+q,{0,0, d}]
        + Monopole[-q,{0,0,-d}])

dipole/.z->0

dipoleR = dipole//PotentialExpansion

dipoleR//TrigToP

efieldR=Grad[dipoleR,Spherical[r,θ,ϕ]]

Part (b)   Find the induced charge density on the plane and verify that the total induced charge is -q.

charge=-1/(4π) D[dipole,z]//.z->0

int1= Integrate[charge,{y,-Infinity,Infinity}]//PowerExpand

int1[[2]]

int2= Integrate[int1[[2]],{x,-Infinity,Infinity}]

qnet=int2[[2]]//PowerExpand

Part (c)   Plot the induced charge density on the plane using ContourPlot.

vals={d->1,q->1}

ContourPlot[charge/.vals
,{y,-1,1},{x,-1,1}];

ContourPlot[charge/.vals
,{y,-1,1},{x,-1,1}
,ColorFunction->Hue];