Partnership for Integration of Computation into Undergraduate Physics

### 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. - $\sigma_0(\theta) = \cosh(\cos\theta)$.