XVI Contents
6.8 CurrentFlowInsidethe Earth............................139
6.9 Refractionof CurrentLines ..............................143
6.10 DipoleField............................................144
6.11 BasicEquationsin DirectCurrentFlowField ..............149
6.12 Units..................................................150
7 Solution of Laplace Equation ..............................151
7.1 EquationsofPoissonandLaplace.........................151
7.2 LaplaceEquationinDirect CurrentFlowDomain...........152
7.3 Laplace Equation in Generalised Curvilinear Coordina tes . . . . 153
7.4 LaplaceEquationinCartesianCoordinates ................156
7.4.1 When Potential is a Function of Vertical Axis z, i.e.,
φ =f(z) .........................................156
7.4.2 When Potential is a Function of Both x and y, i.e.,
φ =f(x, y).......................................157
7.4.3 Solution of Boundary Value Problems in Cartisian
Coordinates by the Method of Separation of Variables 158
7.5 Laplace Equation in Cylindrical Polar Coordinates . . . . . . . . . . 162
7.5.1 When Potential is a Function of z ,i.e., φ =f(z).......164
7.5.2 When Potential is a Function of Azimuthal Angle
Only i.e., φ =f(ψ)................................164
7.5.3 When the Potential is a Function of Radial Distance,
i.e., Φ = f(ρ).....................................164
7.5.4 When Potential is a Function of Both ρ and ψ,
i.e., φ =f(ρ, ψ)...................................165
7.5.5 When Potential is a Function of all the Three
Coordinates, i.e., φ =f(ρ, ψ, z) .....................171
7.5.6 BesselEquationandBessel’s Functions..............172
7.5.7 Modified Bessel’sFunctions........................177
7.5.8 Some RelationofBessel’s Function .................181
7.6 Solution of Laplace E quation in Spherical Polar Co-ordinates . 183
7.6.1 When Potential is a Function of Radial Distance r
i.e., φ =f(r) .....................................183
7.6.2 When Potential is a Function of Polar Angle, i.e.,
φ =f(θ).........................................184
7.6.3 When Potential is a Function of Azimuthal Angle
i.e., ϕ =f(ψ).....................................185
7.6.4 When Potential is a Function of Both the Radial
Distance and Polar Angle i.e., φ =f(r, θ) ............185
7.6.5 Legender’s Equation and Legender’s polynomial . . . . . . 187
7.6.6 When Potential is a Function of all the Three
Coordina tes Viz, Radial Distance, Polar Angle and
Azimuthal Angle,i.e., φ =f(r, θ, ψ)..................198
7.6.7 AssociatedLegendrePolynomial ...................200
7.7 Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
7.7.1 Zonal, Sectoral and Tesseral Harmonics . . . . . . . . . . . . . 202