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Separation of Variables in Spherical Coordinates

Developed by J. D. McDonnell---- - Published July 16, 2016

This set of exercises will guide the student through solving Laplace's equation for the electric potential in spherical coordinates via separation of variables. They will perform numerical integration and produce plots of the electric potential for situations with non-trivial boundary conditions.
Subject Areas Electricity & Magnetism and Mathematical/Numerical Methods Beyond the First Year IPython/Jupyter Notebook Students who complete this set of exercises will - be able to use separation of variables to solve Laplace's equation in spherical coordinates for a boundary sphere with given potential (**Exercise 1**), - be able to use separation of variables to solve Laplace's equation in spherical coordinates for a boundary sphere with given surface charge density (**Exercise 2**), - gain familiarity with the Legendre polynomials and their usefulness in physical problems (**Exercises 1 and 2**), - be able to supplement an analytical solution with numerical methods, such as numerical integration (**Exercises 1 and 2**), - and be able to produce and analyze visualizations for the electric potential (**Exercises 1 and 2**). 90 min

These exercises are not tied to a specific programming language. Example implementations are provided under the Code tab, but the Exercises can be implemented in whatever platform you wish to use (e.g., Excel, Python, MATLAB, etc.).

### Exercise 1: A Sphere with Given Potential Consider a sphere of radius $R$. On the surface of this sphere, a potential $V_0(\theta)$ will be specified. 1. Solve Laplace's equation in spherical coordinates. Remember that the series solution will have one form inside the sphere, and a different form outside the sphere. 2. Set up a numerical integral to find the coefficients $A_\ell$ and $B_\ell$ in terms of the surface potential $V_0(\theta)$. 3. Produce a plot of $V(r, \theta)$ for all $r$ and $\theta$, for each of the following surface potentials $V_0(\theta)$: - $V_0(\theta) = \cos\theta$. **Note**: This case can be solved exactly as a convenient check. - $V_0(\theta) = \sqrt{\cos\frac{\theta}{2}}$. - $V_0(\theta) = \theta^2$. ### Exercise 2: A Sphere with Given Surface Charge Density Consider a sphere of radius $R$. On the surface of this sphere, a *surface charge density* $\sigma_0(\theta)$ will be specified. 1. Solve Laplace's equation in spherical coordinates. Remember that the series solution will have one form inside the sphere, and a different form outside the sphere. Due to the charge on the surface of the sphere, the boundary condition at $r=R$ must be handled differently. 2. Set up a numerical integral to find the undetermined coefficients in terms of the surface charge density $\sigma_0(\theta)$. 3. Produce a plot of $V(r, \theta)$ for all $r$ and $\theta$, for each of the following surface charge densities $\sigma_0(\theta)$: - $\sigma_0(\theta) = \cos\theta$. **Note**: Again, this case can be solved exactly as a convenient check. - $\sigma_0(\theta) = \cos^3 \theta$. **Note**: This case can also be solved by hand. - $\sigma_0(\theta) = \cosh(\cos\theta)$.