Laplace's equation

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In electrostatics, a potential function can be always defined for a given electric field function, and it has to satisfy Poisson's equation

[math]\nabla^2 V = -\frac{\rho}{\epsilon_0}[/math]

and in case there is no charge distributed in the space of interest, following Laplace's equation holds

[math]\nabla^2 V = 0[/math].

1D Laplace's equation

In one-dimension, Laplace's equation becomes

[math]\frac{d^2V}{dx^2} = 0[/math]

and its general solution is [math]V(x) = mx + b[/math]. This solution of course satisfies the property [math]V(x) = \frac12 [ V(x-a) + V(x+a) ][/math].

2D Laplace's equation

In two-dimension, Laplace's equation becomes

[math]\frac{\partial^2V}{\partial x^2}+\frac{\partial^2V}{\partial y^2} = 0[/math]

which is a partial differential equation.

Following properties should hold for harmonic functions in 2-dim.

  • If you draw a circle of any radius R about the point (x,y), following equation should hold
[math]V(x,y) = \frac{1}{2\pi R} \oint_{circle} V dl[/math]
  • V has no local maxima or minima.
  • Harmonic function in 2-dim minimizes the surface area spanning the given boundary line.

3D Laplace's equation

[math]\frac{\partial^2V}{\partial x^2}+\frac{\partial^2V}{\partial y^2}+\frac{\partial^2V}{\partial z^2} = 0[/math]
  • If you imagine a sphere of any radius R about the point [math]\textbf{r}[/math], following equation should hold
[math]V(\textbf{r}) = \frac{1}{4\pi R^2} \oint_{sphere} V da[/math]
  • Therefore, V can have no local maxima or minima.

Separation of variables

Cartesian coordinates

Use [math]V(\mathbf{r})=X(x)Y(y)Z(z)[/math], then Laplace's equation becomes

[math]\nabla^2 V(\mathbf{r}) = YZ\frac{dX}{dx} + XZ\frac{dY}{dy} + XY\frac{dZ}{dz}=0[/math]

which, after dividing by V, becomes

[math]\frac1X\frac{dX}{dx} + \frac1Y\frac{dY}{dy} + \frac1Z\frac{dZ}{dz}=0[/math]

which can be only true when each term becomes constant.

Spherical coordinates

Use [math]V(\mathbf{r})=R(r)\Theta(\theta)\Phi(\phi)[/math], then Laplace's equation becomes

[math]\frac1{r^2}\frac{\partial}{\partial r}\left( r^2\frac{\partial V}{\partial r}\right)+\frac1{r^2\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial V}{\partial\theta}\right) + \frac1{r^2\sin^2\theta}\frac{\partial^2V}{\partial\phi^2}=0[/math]
[math]\Theta\Phi\frac{d}{dr}\left( r^2\frac{dR}{dr}\right) + \frac{R\Phi}{\sin\theta}\frac{d}{d\theta}\left(\sin\theta\frac{d\Theta}{d\theta}\right) + \frac{R\Theta}{\sin^2\theta}\frac{d^2\Phi}{d\phi^2}=0[/math]

which, after dividing by V, becomes

[math]\underbrace{\frac1R\frac{d}{dr}\left( r^2\frac{dR}{dr}\right)}_{\alpha} + \underbrace{\frac1{\Theta}\frac1{\sin\theta}\frac{d}{d\theta}\left(\sin\theta\frac{d\Theta}{d\theta}\right)}_{-\alpha+\frac{\beta}{\sin^2\theta}} + \underbrace{\frac1{\Phi}\frac1{\sin^2\theta}\frac{d^2\Phi}{d\phi^2}}_{-\frac{\beta}{\sin^2\theta}}=0[/math]

where [math]\alpha[/math] and [math]\beta[/math] are arbitrary constants.

Radial equation

The radial equation is usually formulated as below:

[math]\frac{d}{dr}\left( r^2\frac{dR}{dr}\right) = l(l+1)R[/math]

where we replaced [math]\alpha[/math] with l(l+1) and l is integer. Then it has general solution of

[math]R(r)=Ar^l + \frac{B}{r^{l+1}}[/math]
Angular equation

The angular part reduces to following differential equation, which is Associated Legendre Equation:

[math]\frac1{\sin\theta}\frac{d}{d\theta}\left(\sin\theta\frac{d\Theta}{d\theta}\right)+\left(l(l+1)-\frac{m^2}{\sin^2\theta}\right)\Theta=0[/math]
Azimuthal equation

The azimuthal part becomes simple ODE whose solution is an oscillation with m nodes:

[math]\frac{d^2\Phi}{d\phi^2} + m^2\Phi = 0[/math]