I need to solve Laplace's equation: $$\nabla^2\Phi = 0$$ with the boundary conditions: $$\Phi(\theta=0)=0$$ $$\Phi(\theta=\pi)=a_1$$
In cylindrical coordinates ($r,\theta,z$), for $\Phi =\Phi(\theta)$ only. My first attempt at this yielded the following: $$\require{cancel}\nabla^2\Phi = \cancel{\frac{1}{r}\frac{\partial}{\partial r} r\frac{\partial\Phi}{\partial r}} + \frac{1}{r^2}\frac{\partial^2\Phi}{\partial\theta^2}+\cancel{\frac{\partial^2\Phi}{\partial z^2}}=0$$ $$\nabla^2\Phi = \frac{1}{r^2}\frac{\partial^2\Phi}{\partial \theta^2} = 0$$ $$\frac{\partial^2\Phi}{\partial \theta^2} = 0$$ $$\frac{\partial\Phi}{\partial\theta} = c$$ $$\Phi = c\theta + b$$ $$b = 0$$ $$a_1 = c\pi$$ $$c = \frac{a_1}{\pi}$$ $$\Phi = \frac{a_1}{\pi}\theta$$ Which looks nice, but I'm applying it to a physics problem (half-cylinder of uniform resistance $\rho$ with above boundary conditions), and I'm not coming out with the right units. I know that in evaluating a triple integral over the volume, I would need
$$\int_0^\pi\int_0^h\int_{r_1}^{r_2}\Phi\ r\ dr\ dz\ d\theta$$ but can I even do what I'm trying to?