2024-02-08 Multipole Expansion

Covered In this class:

• Review of solving Laplace’s equation with Spherical Coordinates
• Solving Laplace’s equation With Multi-pole Expansion

In class problems:

3.27

$$V_2=\frac{1}{4\pi {ϵ}_0}\frac{1}{r^3}{\int }_{V^{\prime }}{r}^2\left(\frac{3}{2}{\cos }^2\theta -\frac{1}{2}\right)k\frac{R}{r^2}(R-2r){\sin }^2\theta r^{2}d{V}^{\prime }$$

The integral

$$\int r^2\left(\frac{3}{2}{\cos }^2\theta -\frac{1}{2}\right)(R-2r){\sin }^2\theta dV$$ $$\iint -\frac{R^4\cdot \left(3{\cos }^2\left(\theta \right)-1\right){\sin }^2\left(\theta \right)}{12}d\theta ,\varphi$$ $$\underset{0}{\overset{2\pi }{\int }}\frac{\pi R^4}{96}d\varphi$$ $$\frac{{\pi }^2{R}^4}{48}$$

Put the constants back in front

$$kR\frac{1}{4\pi {ϵ}_0}\frac{1}{r^3}\frac{{\pi }^2{R}^4}{48}$$ $$\frac{k}{4\pi {ϵ}_0}\frac{{\pi }^2{R}^5}{48}$$