User:CaseyLeung/sandbox

Laplace's Equation

Multiple Expansion

Consider two charges q1 and q2 which are pretty close together. If you are very far away from the two charges, then your potential can be estimate by $V={\frac {1}{4\pi \varepsilon _{0}}}{\frac {q_{1}+q_{2}}{r}}$

But what if q1 and q2 are equal charges such that they will cancel each other? Then you might say, the potential will be estimate equals to zero. This is true in some senses, but can we obtained something more useful?

At an arbitrary point far away from +q and -q, we consider its potential $V={\frac {1}{4\pi \varepsilon _{0}}}{\Bigl (}{\frac {+q}{m}}-{\frac {-q}{n}}{\Bigr )}$

Figure 1: Dipole with charge +q and -q

Using cosine law, we have

$m^{2}=r^{2}+(d/2)^{2}-rd\cos \theta =r^{2}(1-d\cos \theta /r+d^{2}/4r^{2})$

$n^{2}=r^{2}+(d/2)^{2}+rd\cos \theta =r^{2}(1+d\cos \theta /r+d^{2}/4r^{2})$