Spherical coordinate for Poisson and Laplace equation

Question

ScanSolve AI Tutor · Accepted Answer

Here are Poisson's and Laplace's equations in spherical coordinates:

The Laplacian operator $\nabla^2$ in spherical coordinates $(r, \theta, \phi)$ is given by:

\nabla^2 V = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial V}{\partial r} \right) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial V}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 V}{\partial \phi^2}

Poisson's Equation in Spherical Coordinates
Poisson's equation is $\nabla^2 V = -\frac{\rho}{\epsilon_0}$ , where $V$ is the electric potential, $\rho$ is the volume charge density, and $\epsilon_0$ is the permittivity of free space.
$\boxed{\frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial V}{\partial r} \right) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial V}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 V}{\partial \phi^2} = -\frac{\rho}{\epsilon_0}}$
Laplace's Equation in Spherical Coordinates
Laplace's equation is $\nabla^2 V = 0$ , which is a special case of Poisson's equation where the charge density $\rho = 0$ .
$\boxed{\frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial V}{\partial r} \right) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial V}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 V}{\partial \phi^2} = 0}$

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